3.1.89 \(\int (g+h x)^2 (a+c x^2)^{3/2} (d+e x+f x^2) \, dx\) [89]

Optimal. Leaf size=346 \[ \frac {a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (8 a h^2 (2 f g+e h)+c g \left (5 f g^2-8 h (e g+7 d h)\right )\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}} \]

[Out]

1/192*(48*c^2*d*g^2+3*a^2*f*h^2-8*a*c*(f*g^2+h*(d*h+2*e*g)))*x*(c*x^2+a)^(3/2)/c^2-1/56*(-8*e*h+5*f*g)*(h*x+g)
^2*(c*x^2+a)^(5/2)/c/h+1/8*f*(h*x+g)^3*(c*x^2+a)^(5/2)/c/h-1/1680*(96*a*h^2*(e*h+2*f*g)+12*c*g*(5*f*g^2-8*h*(7
*d*h+e*g))-5*h*(7*(-3*a*f+8*c*d)*h^2-2*c*g*(-8*e*h+5*f*g))*x)*(c*x^2+a)^(5/2)/c^2/h+1/128*a^2*(48*c^2*d*g^2+3*
a^2*f*h^2-8*a*c*(f*g^2+h*(d*h+2*e*g)))*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(5/2)+1/128*a*(48*c^2*d*g^2+3*a^2*
f*h^2-8*a*c*(f*g^2+h*(d*h+2*e*g)))*x*(c*x^2+a)^(1/2)/c^2

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Rubi [A]
time = 0.32, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1668, 847, 794, 201, 223, 212} \begin {gather*} \frac {x \left (a+c x^2\right )^{3/2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{192 c^2}+\frac {a x \sqrt {a+c x^2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^2}+\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^{5/2}}-\frac {\left (a+c x^2\right )^{5/2} \left (12 \left (8 a h^2 (e h+2 f g)-8 c g h (7 d h+e g)+5 c f g^3\right )-5 h x \left (7 h^2 (8 c d-3 a f)-2 c g (5 f g-8 e h)\right )\right )}{1680 c^2 h}-\frac {\left (a+c x^2\right )^{5/2} (g+h x)^2 (5 f g-8 e h)}{56 c h}+\frac {f \left (a+c x^2\right )^{5/2} (g+h x)^3}{8 c h} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(g + h*x)^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]

[Out]

(a*(48*c^2*d*g^2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*x*Sqrt[a + c*x^2])/(128*c^2) + ((48*c^2*d*g^
2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*x*(a + c*x^2)^(3/2))/(192*c^2) - ((5*f*g - 8*e*h)*(g + h*x)
^2*(a + c*x^2)^(5/2))/(56*c*h) + (f*(g + h*x)^3*(a + c*x^2)^(5/2))/(8*c*h) - ((12*(5*c*f*g^3 - 8*c*g*h*(e*g +
7*d*h) + 8*a*h^2*(2*f*g + e*h)) - 5*h*(7*(8*c*d - 3*a*f)*h^2 - 2*c*g*(5*f*g - 8*e*h))*x)*(a + c*x^2)^(5/2))/(1
680*c^2*h) + (a^2*(48*c^2*d*g^2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(128*c^(5/2))

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int (g+h x)^2 \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}+\frac {\int (g+h x)^2 \left ((8 c d-3 a f) h^2-c h (5 f g-8 e h) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{8 c h^2}\\ &=-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}+\frac {\int (g+h x) \left (c h^2 (56 c d g-11 a f g-16 a e h)+c h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{56 c^2 h^2}\\ &=-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c^2}\\ &=\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {\left (a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \sqrt {a+c x^2} \, dx}{64 c^2}\\ &=\frac {a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {\left (a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{128 c^2}\\ &=\frac {a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {\left (a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{128 c^2}\\ &=\frac {a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 1.09, size = 332, normalized size = 0.96 \begin {gather*} \frac {\sqrt {c} \sqrt {a+c x^2} \left (-3 a^3 h (512 f g+256 e h+105 f h x)+6 a^2 c \left (28 d h (32 g+5 h x)+8 e \left (56 g^2+35 g h x+8 h^2 x^2\right )+f x \left (140 g^2+128 g h x+35 h^2 x^2\right )\right )+16 c^3 x^3 \left (14 d \left (15 g^2+24 g h x+10 h^2 x^2\right )+x \left (8 e \left (21 g^2+35 g h x+15 h^2 x^2\right )+5 f x \left (28 g^2+48 g h x+21 h^2 x^2\right )\right )\right )+8 a c^2 x \left (14 d \left (75 g^2+96 g h x+35 h^2 x^2\right )+x \left (4 e \left (168 g^2+245 g h x+96 h^2 x^2\right )+f x \left (490 g^2+768 g h x+315 h^2 x^2\right )\right )\right )\right )-105 a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{13440 c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]

[Out]

(Sqrt[c]*Sqrt[a + c*x^2]*(-3*a^3*h*(512*f*g + 256*e*h + 105*f*h*x) + 6*a^2*c*(28*d*h*(32*g + 5*h*x) + 8*e*(56*
g^2 + 35*g*h*x + 8*h^2*x^2) + f*x*(140*g^2 + 128*g*h*x + 35*h^2*x^2)) + 16*c^3*x^3*(14*d*(15*g^2 + 24*g*h*x +
10*h^2*x^2) + x*(8*e*(21*g^2 + 35*g*h*x + 15*h^2*x^2) + 5*f*x*(28*g^2 + 48*g*h*x + 21*h^2*x^2))) + 8*a*c^2*x*(
14*d*(75*g^2 + 96*g*h*x + 35*h^2*x^2) + x*(4*e*(168*g^2 + 245*g*h*x + 96*h^2*x^2) + f*x*(490*g^2 + 768*g*h*x +
315*h^2*x^2)))) - 105*a^2*(48*c^2*d*g^2 + 3*a^2*f*h^2 - 8*a*c*(f*g^2 + h*(2*e*g + d*h)))*Log[-(Sqrt[c]*x) + S
qrt[a + c*x^2]])/(13440*c^(5/2))

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Maple [A]
time = 0.11, size = 320, normalized size = 0.92

method	result	size
default	\(f \,h^{2} \left (\frac {x^{3} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{8 c}-\frac {3 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )}{8 c}\right )+\left (e \,h^{2}+2 f g h \right ) \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{7 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{35 c^{2}}\right )+\left (d \,h^{2}+2 e g h +f \,g^{2}\right ) \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )+\frac {\left (2 d g h +g^{2} e \right ) \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5 c}+d \,g^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )\)	\(320\)
risch	\(-\frac {\left (-1680 c^{3} f \,h^{2} x^{7}-1920 c^{3} e \,h^{2} x^{6}-3840 c^{3} f g h \,x^{6}-2520 a \,c^{2} f \,h^{2} x^{5}-2240 c^{3} d \,h^{2} x^{5}-4480 c^{3} e g h \,x^{5}-2240 c^{3} f \,g^{2} x^{5}-3072 a \,c^{2} e \,h^{2} x^{4}-6144 a \,c^{2} f g h \,x^{4}-5376 c^{3} d g h \,x^{4}-2688 c^{3} e \,g^{2} x^{4}-210 a^{2} f \,h^{2} c \,x^{3}-3920 a \,c^{2} d \,h^{2} x^{3}-7840 a \,c^{2} e g h \,x^{3}-3920 a \,c^{2} f \,g^{2} x^{3}-3360 c^{3} d \,g^{2} x^{3}-384 a^{2} c e \,h^{2} x^{2}-768 a^{2} c f g h \,x^{2}-10752 a \,c^{2} d g h \,x^{2}-5376 a \,c^{2} e \,g^{2} x^{2}+315 a^{3} f \,h^{2} x -840 a^{2} c d \,h^{2} x -1680 a^{2} c e g h x -840 a^{2} c f \,g^{2} x -8400 a \,c^{2} d \,g^{2} x +768 a^{3} e \,h^{2}+1536 a^{3} f g h -5376 a^{2} c d g h -2688 a^{2} c e \,g^{2}\right ) \sqrt {c \,x^{2}+a}}{13440 c^{2}}+\frac {3 a^{4} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) f \,h^{2}}{128 c^{\frac {5}{2}}}-\frac {a^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) d \,h^{2}}{16 c^{\frac {3}{2}}}-\frac {a^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) e g h}{8 c^{\frac {3}{2}}}-\frac {a^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) f \,g^{2}}{16 c^{\frac {3}{2}}}+\frac {3 a^{2} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) d \,g^{2}}{8 \sqrt {c}}\)	\(489\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)^2*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x,method=_RETURNVERBOSE)

[Out]

f*h^2*(1/8*x^3*(c*x^2+a)^(5/2)/c-3/8*a/c*(1/6*x*(c*x^2+a)^(5/2)/c-1/6*a/c*(1/4*x*(c*x^2+a)^(3/2)+3/4*a*(1/2*x*
(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))))))+(e*h^2+2*f*g*h)*(1/7*x^2*(c*x^2+a)^(5/2)/c-2/3
5*a/c^2*(c*x^2+a)^(5/2))+(d*h^2+2*e*g*h+f*g^2)*(1/6*x*(c*x^2+a)^(5/2)/c-1/6*a/c*(1/4*x*(c*x^2+a)^(3/2)+3/4*a*(
1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2)))))+1/5*(2*d*g*h+e*g^2)*(c*x^2+a)^(5/2)/c+d*g
^2*(1/4*x*(c*x^2+a)^(3/2)+3/4*a*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))))

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Maxima [A]
time = 0.30, size = 387, normalized size = 1.12 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} f h^{2} x^{3}}{8 \, c} + \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d g^{2} x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d g^{2} x - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a f h^{2} x}{16 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} f h^{2} x}{64 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} a^{3} f h^{2} x}{128 \, c^{2}} + \frac {3 \, a^{2} d g^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} + \frac {3 \, a^{4} f h^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {5}{2}}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d g h}{5 \, c} + \frac {{\left (2 \, f g h + h^{2} e\right )} {\left (c x^{2} + a\right )}^{\frac {5}{2}} x^{2}}{7 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} g^{2} e}{5 \, c} + \frac {{\left (f g^{2} + d h^{2} + 2 \, g h e\right )} {\left (c x^{2} + a\right )}^{\frac {5}{2}} x}{6 \, c} - \frac {{\left (f g^{2} + d h^{2} + 2 \, g h e\right )} {\left (c x^{2} + a\right )}^{\frac {3}{2}} a x}{24 \, c} - \frac {{\left (f g^{2} + d h^{2} + 2 \, g h e\right )} \sqrt {c x^{2} + a} a^{2} x}{16 \, c} - \frac {{\left (f g^{2} + d h^{2} + 2 \, g h e\right )} a^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {3}{2}}} - \frac {2 \, {\left (2 \, f g h + h^{2} e\right )} {\left (c x^{2} + a\right )}^{\frac {5}{2}} a}{35 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="maxima")

[Out]

1/8*(c*x^2 + a)^(5/2)*f*h^2*x^3/c + 1/4*(c*x^2 + a)^(3/2)*d*g^2*x + 3/8*sqrt(c*x^2 + a)*a*d*g^2*x - 1/16*(c*x^
2 + a)^(5/2)*a*f*h^2*x/c^2 + 1/64*(c*x^2 + a)^(3/2)*a^2*f*h^2*x/c^2 + 3/128*sqrt(c*x^2 + a)*a^3*f*h^2*x/c^2 +
3/8*a^2*d*g^2*arcsinh(c*x/sqrt(a*c))/sqrt(c) + 3/128*a^4*f*h^2*arcsinh(c*x/sqrt(a*c))/c^(5/2) + 2/5*(c*x^2 + a
)^(5/2)*d*g*h/c + 1/7*(2*f*g*h + h^2*e)*(c*x^2 + a)^(5/2)*x^2/c + 1/5*(c*x^2 + a)^(5/2)*g^2*e/c + 1/6*(f*g^2 +
d*h^2 + 2*g*h*e)*(c*x^2 + a)^(5/2)*x/c - 1/24*(f*g^2 + d*h^2 + 2*g*h*e)*(c*x^2 + a)^(3/2)*a*x/c - 1/16*(f*g^2
+ d*h^2 + 2*g*h*e)*sqrt(c*x^2 + a)*a^2*x/c - 1/16*(f*g^2 + d*h^2 + 2*g*h*e)*a^3*arcsinh(c*x/sqrt(a*c))/c^(3/2
) - 2/35*(2*f*g*h + h^2*e)*(c*x^2 + a)^(5/2)*a/c^2

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Fricas [A]
time = 0.43, size = 851, normalized size = 2.46 \begin {gather*} \left [\frac {105 \, {\left (16 \, a^{3} c g h e - 8 \, {\left (6 \, a^{2} c^{2} d - a^{3} c f\right )} g^{2} + {\left (8 \, a^{3} c d - 3 \, a^{4} f\right )} h^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (1680 \, c^{4} f h^{2} x^{7} + 3840 \, c^{4} f g h x^{6} + 768 \, {\left (7 \, c^{4} d + 8 \, a c^{3} f\right )} g h x^{4} + 280 \, {\left (8 \, c^{4} f g^{2} + {\left (8 \, c^{4} d + 9 \, a c^{3} f\right )} h^{2}\right )} x^{5} + 768 \, {\left (14 \, a c^{3} d + a^{2} c^{2} f\right )} g h x^{2} + 70 \, {\left (8 \, {\left (6 \, c^{4} d + 7 \, a c^{3} f\right )} g^{2} + {\left (56 \, a c^{3} d + 3 \, a^{2} c^{2} f\right )} h^{2}\right )} x^{3} + 768 \, {\left (7 \, a^{2} c^{2} d - 2 \, a^{3} c f\right )} g h + 105 \, {\left (8 \, {\left (10 \, a c^{3} d + a^{2} c^{2} f\right )} g^{2} + {\left (8 \, a^{2} c^{2} d - 3 \, a^{3} c f\right )} h^{2}\right )} x + 16 \, {\left (120 \, c^{4} h^{2} x^{6} + 280 \, c^{4} g h x^{5} + 490 \, a c^{3} g h x^{3} + 105 \, a^{2} c^{2} g h x + 168 \, a^{2} c^{2} g^{2} - 48 \, a^{3} c h^{2} + 24 \, {\left (7 \, c^{4} g^{2} + 8 \, a c^{3} h^{2}\right )} x^{4} + 24 \, {\left (14 \, a c^{3} g^{2} + a^{2} c^{2} h^{2}\right )} x^{2}\right )} e\right )} \sqrt {c x^{2} + a}}{26880 \, c^{3}}, \frac {105 \, {\left (16 \, a^{3} c g h e - 8 \, {\left (6 \, a^{2} c^{2} d - a^{3} c f\right )} g^{2} + {\left (8 \, a^{3} c d - 3 \, a^{4} f\right )} h^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (1680 \, c^{4} f h^{2} x^{7} + 3840 \, c^{4} f g h x^{6} + 768 \, {\left (7 \, c^{4} d + 8 \, a c^{3} f\right )} g h x^{4} + 280 \, {\left (8 \, c^{4} f g^{2} + {\left (8 \, c^{4} d + 9 \, a c^{3} f\right )} h^{2}\right )} x^{5} + 768 \, {\left (14 \, a c^{3} d + a^{2} c^{2} f\right )} g h x^{2} + 70 \, {\left (8 \, {\left (6 \, c^{4} d + 7 \, a c^{3} f\right )} g^{2} + {\left (56 \, a c^{3} d + 3 \, a^{2} c^{2} f\right )} h^{2}\right )} x^{3} + 768 \, {\left (7 \, a^{2} c^{2} d - 2 \, a^{3} c f\right )} g h + 105 \, {\left (8 \, {\left (10 \, a c^{3} d + a^{2} c^{2} f\right )} g^{2} + {\left (8 \, a^{2} c^{2} d - 3 \, a^{3} c f\right )} h^{2}\right )} x + 16 \, {\left (120 \, c^{4} h^{2} x^{6} + 280 \, c^{4} g h x^{5} + 490 \, a c^{3} g h x^{3} + 105 \, a^{2} c^{2} g h x + 168 \, a^{2} c^{2} g^{2} - 48 \, a^{3} c h^{2} + 24 \, {\left (7 \, c^{4} g^{2} + 8 \, a c^{3} h^{2}\right )} x^{4} + 24 \, {\left (14 \, a c^{3} g^{2} + a^{2} c^{2} h^{2}\right )} x^{2}\right )} e\right )} \sqrt {c x^{2} + a}}{13440 \, c^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="fricas")

[Out]

[1/26880*(105*(16*a^3*c*g*h*e - 8*(6*a^2*c^2*d - a^3*c*f)*g^2 + (8*a^3*c*d - 3*a^4*f)*h^2)*sqrt(c)*log(-2*c*x^
2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(1680*c^4*f*h^2*x^7 + 3840*c^4*f*g*h*x^6 + 768*(7*c^4*d + 8*a*c^3*f)*
g*h*x^4 + 280*(8*c^4*f*g^2 + (8*c^4*d + 9*a*c^3*f)*h^2)*x^5 + 768*(14*a*c^3*d + a^2*c^2*f)*g*h*x^2 + 70*(8*(6*
c^4*d + 7*a*c^3*f)*g^2 + (56*a*c^3*d + 3*a^2*c^2*f)*h^2)*x^3 + 768*(7*a^2*c^2*d - 2*a^3*c*f)*g*h + 105*(8*(10*
a*c^3*d + a^2*c^2*f)*g^2 + (8*a^2*c^2*d - 3*a^3*c*f)*h^2)*x + 16*(120*c^4*h^2*x^6 + 280*c^4*g*h*x^5 + 490*a*c^
3*g*h*x^3 + 105*a^2*c^2*g*h*x + 168*a^2*c^2*g^2 - 48*a^3*c*h^2 + 24*(7*c^4*g^2 + 8*a*c^3*h^2)*x^4 + 24*(14*a*c
^3*g^2 + a^2*c^2*h^2)*x^2)*e)*sqrt(c*x^2 + a))/c^3, 1/13440*(105*(16*a^3*c*g*h*e - 8*(6*a^2*c^2*d - a^3*c*f)*g
^2 + (8*a^3*c*d - 3*a^4*f)*h^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (1680*c^4*f*h^2*x^7 + 3840*c^4*f
*g*h*x^6 + 768*(7*c^4*d + 8*a*c^3*f)*g*h*x^4 + 280*(8*c^4*f*g^2 + (8*c^4*d + 9*a*c^3*f)*h^2)*x^5 + 768*(14*a*c
^3*d + a^2*c^2*f)*g*h*x^2 + 70*(8*(6*c^4*d + 7*a*c^3*f)*g^2 + (56*a*c^3*d + 3*a^2*c^2*f)*h^2)*x^3 + 768*(7*a^2
*c^2*d - 2*a^3*c*f)*g*h + 105*(8*(10*a*c^3*d + a^2*c^2*f)*g^2 + (8*a^2*c^2*d - 3*a^3*c*f)*h^2)*x + 16*(120*c^4
*h^2*x^6 + 280*c^4*g*h*x^5 + 490*a*c^3*g*h*x^3 + 105*a^2*c^2*g*h*x + 168*a^2*c^2*g^2 - 48*a^3*c*h^2 + 24*(7*c^
4*g^2 + 8*a*c^3*h^2)*x^4 + 24*(14*a*c^3*g^2 + a^2*c^2*h^2)*x^2)*e)*sqrt(c*x^2 + a))/c^3]

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Sympy [A]
time = 49.78, size = 1304, normalized size = 3.77 \begin {gather*} - \frac {3 a^{\frac {7}{2}} f h^{2} x}{128 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} d h^{2} x}{16 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} e g h x}{8 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} f g^{2} x}{16 c \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {a^{\frac {5}{2}} f h^{2} x^{3}}{128 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {3}{2}} d g^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {a^{\frac {3}{2}} d g^{2} x}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} d h^{2} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} e g h x^{3}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} f g^{2} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {13 a^{\frac {3}{2}} f h^{2} x^{5}}{64 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 \sqrt {a} c d g^{2} x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c d h^{2} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c e g h x^{5}}{12 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c f g^{2} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {5 \sqrt {a} c f h^{2} x^{7}}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 a^{4} f h^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{128 c^{\frac {5}{2}}} - \frac {a^{3} d h^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} - \frac {a^{3} e g h \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 c^{\frac {3}{2}}} - \frac {a^{3} f g^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} + \frac {3 a^{2} d g^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 \sqrt {c}} + 2 a d g h \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + a e g^{2} \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + a e h^{2} \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a f g h \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 c d g h \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c e g^{2} \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c e h^{2} \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 2 c f g h \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + \frac {c^{2} d g^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} d h^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} e g h x^{7}}{3 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} f g^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} f h^{2} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)**2*(c*x**2+a)**(3/2)*(f*x**2+e*x+d),x)

[Out]

-3*a**(7/2)*f*h**2*x/(128*c**2*sqrt(1 + c*x**2/a)) + a**(5/2)*d*h**2*x/(16*c*sqrt(1 + c*x**2/a)) + a**(5/2)*e*
g*h*x/(8*c*sqrt(1 + c*x**2/a)) + a**(5/2)*f*g**2*x/(16*c*sqrt(1 + c*x**2/a)) - a**(5/2)*f*h**2*x**3/(128*c*sqr
t(1 + c*x**2/a)) + a**(3/2)*d*g**2*x*sqrt(1 + c*x**2/a)/2 + a**(3/2)*d*g**2*x/(8*sqrt(1 + c*x**2/a)) + 17*a**(
3/2)*d*h**2*x**3/(48*sqrt(1 + c*x**2/a)) + 17*a**(3/2)*e*g*h*x**3/(24*sqrt(1 + c*x**2/a)) + 17*a**(3/2)*f*g**2
*x**3/(48*sqrt(1 + c*x**2/a)) + 13*a**(3/2)*f*h**2*x**5/(64*sqrt(1 + c*x**2/a)) + 3*sqrt(a)*c*d*g**2*x**3/(8*s
qrt(1 + c*x**2/a)) + 11*sqrt(a)*c*d*h**2*x**5/(24*sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*e*g*h*x**5/(12*sqrt(1 + c
*x**2/a)) + 11*sqrt(a)*c*f*g**2*x**5/(24*sqrt(1 + c*x**2/a)) + 5*sqrt(a)*c*f*h**2*x**7/(16*sqrt(1 + c*x**2/a))
+ 3*a**4*f*h**2*asinh(sqrt(c)*x/sqrt(a))/(128*c**(5/2)) - a**3*d*h**2*asinh(sqrt(c)*x/sqrt(a))/(16*c**(3/2))
- a**3*e*g*h*asinh(sqrt(c)*x/sqrt(a))/(8*c**(3/2)) - a**3*f*g**2*asinh(sqrt(c)*x/sqrt(a))/(16*c**(3/2)) + 3*a*
*2*d*g**2*asinh(sqrt(c)*x/sqrt(a))/(8*sqrt(c)) + 2*a*d*g*h*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)
**(3/2)/(3*c), True)) + a*e*g**2*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + a*
e*h**2*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/
5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + 2*a*f*g*h*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(
a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + 2*c*d*g*h*Piecewise((-2*a**
2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x
**4/4, True)) + c*e*g**2*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4
*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + c*e*h**2*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**
3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c,
0)), (sqrt(a)*x**6/6, True)) + 2*c*f*g*h*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a +
c*x**2)/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, Tr
ue)) + c**2*d*g**2*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*d*h**2*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + c**
2*e*g*h*x**7/(3*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*f*g**2*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*f*h**2*x*
*9/(8*sqrt(a)*sqrt(1 + c*x**2/a))

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Giac [A]
time = 5.04, size = 452, normalized size = 1.31 \begin {gather*} \frac {1}{13440} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, c f h^{2} x + \frac {8 \, {\left (2 \, c^{7} f g h + c^{7} h^{2} e\right )}}{c^{6}}\right )} x + \frac {7 \, {\left (8 \, c^{7} f g^{2} + 8 \, c^{7} d h^{2} + 9 \, a c^{6} f h^{2} + 16 \, c^{7} g h e\right )}}{c^{6}}\right )} x + \frac {48 \, {\left (14 \, c^{7} d g h + 16 \, a c^{6} f g h + 7 \, c^{7} g^{2} e + 8 \, a c^{6} h^{2} e\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (48 \, c^{7} d g^{2} + 56 \, a c^{6} f g^{2} + 56 \, a c^{6} d h^{2} + 3 \, a^{2} c^{5} f h^{2} + 112 \, a c^{6} g h e\right )}}{c^{6}}\right )} x + \frac {192 \, {\left (28 \, a c^{6} d g h + 2 \, a^{2} c^{5} f g h + 14 \, a c^{6} g^{2} e + a^{2} c^{5} h^{2} e\right )}}{c^{6}}\right )} x + \frac {105 \, {\left (80 \, a c^{6} d g^{2} + 8 \, a^{2} c^{5} f g^{2} + 8 \, a^{2} c^{5} d h^{2} - 3 \, a^{3} c^{4} f h^{2} + 16 \, a^{2} c^{5} g h e\right )}}{c^{6}}\right )} x + \frac {384 \, {\left (14 \, a^{2} c^{5} d g h - 4 \, a^{3} c^{4} f g h + 7 \, a^{2} c^{5} g^{2} e - 2 \, a^{3} c^{4} h^{2} e\right )}}{c^{6}}\right )} - \frac {{\left (48 \, a^{2} c^{2} d g^{2} - 8 \, a^{3} c f g^{2} - 8 \, a^{3} c d h^{2} + 3 \, a^{4} f h^{2} - 16 \, a^{3} c g h e\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)^2*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="giac")

[Out]

1/13440*sqrt(c*x^2 + a)*((2*((4*(5*(6*(7*c*f*h^2*x + 8*(2*c^7*f*g*h + c^7*h^2*e)/c^6)*x + 7*(8*c^7*f*g^2 + 8*c
^7*d*h^2 + 9*a*c^6*f*h^2 + 16*c^7*g*h*e)/c^6)*x + 48*(14*c^7*d*g*h + 16*a*c^6*f*g*h + 7*c^7*g^2*e + 8*a*c^6*h^
2*e)/c^6)*x + 35*(48*c^7*d*g^2 + 56*a*c^6*f*g^2 + 56*a*c^6*d*h^2 + 3*a^2*c^5*f*h^2 + 112*a*c^6*g*h*e)/c^6)*x +
192*(28*a*c^6*d*g*h + 2*a^2*c^5*f*g*h + 14*a*c^6*g^2*e + a^2*c^5*h^2*e)/c^6)*x + 105*(80*a*c^6*d*g^2 + 8*a^2*
c^5*f*g^2 + 8*a^2*c^5*d*h^2 - 3*a^3*c^4*f*h^2 + 16*a^2*c^5*g*h*e)/c^6)*x + 384*(14*a^2*c^5*d*g*h - 4*a^3*c^4*f
*g*h + 7*a^2*c^5*g^2*e - 2*a^3*c^4*h^2*e)/c^6) - 1/128*(48*a^2*c^2*d*g^2 - 8*a^3*c*f*g^2 - 8*a^3*c*d*h^2 + 3*a
^4*f*h^2 - 16*a^3*c*g*h*e)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (g+h\,x\right )}^2\,{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g + h*x)^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x)

[Out]

int((g + h*x)^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2), x)

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