\(\int \frac {(g+h x)^3 (d+e x+f x^2)}{(a+c x^2)^{3/2}} \, dx\) [108]

Optimal result

Integrand size = 29, antiderivative size = 229 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt {a+c x^2}}{6 a c^3}-\frac {\left (3 a h^2 (3 f g+e h)-2 c g \left (f g^2+3 h (e g+d h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{5/2}} \]

[Out]

-1/2*(3*a*h^2*(e*h+3*f*g)-2*c*g*(f*g^2+3*h*(d*h+e*g)))*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(5/2)-(a*e-(-a*f+c
*d)*x)*(h*x+g)^3/a/c/(c*x^2+a)^(1/2)-1/3*(-4*a*f+3*c*d)*h*(h*x+g)^2*(c*x^2+a)^(1/2)/a/c^2-1/6*h*(12*c^2*d*g^2+
16*a^2*f*h^2-4*a*c*(7*f*g^2+3*h*(d*h+3*e*g))+c*h*(-9*a*e*h-11*a*f*g+6*c*d*g)*x)*(c*x^2+a)^(1/2)/a/c^3

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1659, 847, 794, 223, 212} \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {h \sqrt {a+c x^2} \left (4 \left (4 a^2 f h^2-a c \left (3 h (d h+3 e g)+7 f g^2\right )+3 c^2 d g^2\right )+c h x (-9 a e h-11 a f g+6 c d g)\right )}{6 a c^3}+\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (-3 a h^2 (e h+3 f g)+6 c g h (d h+e g)+2 c f g^3\right )}{2 c^{5/2}}-\frac {h \sqrt {a+c x^2} (g+h x)^2 (3 c d-4 a f)}{3 a c^2}-\frac {(g+h x)^3 (a e-x (c d-a f))}{a c \sqrt {a+c x^2}} \]

[In]

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

[Out]

-(((a*e - (c*d - a*f)*x)*(g + h*x)^3)/(a*c*Sqrt[a + c*x^2])) - ((3*c*d - 4*a*f)*h*(g + h*x)^2*Sqrt[a + c*x^2])
/(3*a*c^2) - (h*(4*(3*c^2*d*g^2 + 4*a^2*f*h^2 - a*c*(7*f*g^2 + 3*h*(3*e*g + d*h))) + c*h*(6*c*d*g - 11*a*f*g -
9*a*e*h)*x)*Sqrt[a + c*x^2])/(6*a*c^3) + ((2*c*f*g^3 + 6*c*g*h*(e*g + d*h) - 3*a*h^2*(3*f*g + e*h))*ArcTanh[(
Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*c^(5/2))

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 1659

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c
*x^2, x], x, 1]}, Simp[(d + e*x)^m*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c*(p + 1))), x] + Dist[1/(2*a*c*(p
+ 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*c*(p + 1)*(d + e*x)*Q - a*e*g*m + c*d*f*(2*p
+ 3) + c*e*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && GtQ[m, 0] &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {\int \frac {(g+h x)^2 (-a (f g+3 e h)+(3 c d-4 a f) h x)}{\sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {\int \frac {(g+h x) \left (-a \left (2 (3 c d-4 a f) h^2+3 c g (f g+3 e h)\right )+c h (6 c d g-11 a f g-9 a e h) x\right )}{\sqrt {a+c x^2}} \, dx}{3 a c^2} \\ & = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt {a+c x^2}}{6 a c^3}+\frac {\left (2 c f g^3+6 c g h (e g+d h)-3 a h^2 (3 f g+e h)\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c^2} \\ & = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt {a+c x^2}}{6 a c^3}+\frac {\left (2 c f g^3+6 c g h (e g+d h)-3 a h^2 (3 f g+e h)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c^2} \\ & = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt {a+c x^2}}{6 a c^3}+\frac {\left (2 c f g^3+6 c g h (e g+d h)-3 a h^2 (3 f g+e h)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.12 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {-16 a^3 f h^3+6 c^3 d g^3 x+a c^2 \left (6 d h \left (-3 g^2-3 g h x+h^2 x^2\right )-3 e \left (2 g^3+6 g^2 h x-6 g h^2 x^2-h^3 x^3\right )+f x \left (-6 g^3+18 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right )\right )+a^2 c h \left (f \left (36 g^2+27 g h x-8 h^2 x^2\right )+3 h (4 d h+3 e (4 g+h x))\right )+3 a \sqrt {c} \left (3 a h^2 (3 f g+e h)-2 c \left (f g^3+3 g h (e g+d h)\right )\right ) \sqrt {a+c x^2} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 a c^3 \sqrt {a+c x^2}} \]

[In]

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

[Out]

(-16*a^3*f*h^3 + 6*c^3*d*g^3*x + a*c^2*(6*d*h*(-3*g^2 - 3*g*h*x + h^2*x^2) - 3*e*(2*g^3 + 6*g^2*h*x - 6*g*h^2*
x^2 - h^3*x^3) + f*x*(-6*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2*h^3*x^3)) + a^2*c*h*(f*(36*g^2 + 27*g*h*x - 8*h^2*
x^2) + 3*h*(4*d*h + 3*e*(4*g + h*x))) + 3*a*Sqrt[c]*(3*a*h^2*(3*f*g + e*h) - 2*c*(f*g^3 + 3*g*h*(e*g + d*h)))*
Sqrt[a + c*x^2]*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(6*a*c^3*Sqrt[a + c*x^2])

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.24

method	result	size
risch	\(-\frac {h \left (-2 f \,h^{2} c \,x^{2}-3 c e \,h^{2} x -9 c f g h x +10 a f \,h^{2}-6 c d \,h^{2}-18 c e g h -18 c f \,g^{2}\right ) \sqrt {c \,x^{2}+a}}{6 c^{3}}-\frac {\frac {a e \,h^{3} x}{\sqrt {c \,x^{2}+a}}-\frac {2 c^{2} d \,g^{3} x}{a \sqrt {c \,x^{2}+a}}+\frac {3 a f g \,h^{2} x}{\sqrt {c \,x^{2}+a}}+\left (3 a c e \,h^{3}+9 a c f g \,h^{2}-6 c^{2} d g \,h^{2}-6 c^{2} e \,g^{2} h -2 c^{2} f \,g^{3}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )-\frac {-2 a^{2} f \,h^{3}+2 a c d \,h^{3}+6 a c e g \,h^{2}+6 a c f \,g^{2} h -6 c^{2} d \,g^{2} h -2 c^{2} e \,g^{3}}{c \sqrt {c \,x^{2}+a}}}{2 c^{2}}\)	\(285\)
default	\(\frac {d \,g^{3} x}{a \sqrt {c \,x^{2}+a}}+f \,h^{3} \left (\frac {x^{4}}{3 c \sqrt {c \,x^{2}+a}}-\frac {4 a \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+a}}+\frac {2 a}{c^{2} \sqrt {c \,x^{2}+a}}\right )}{3 c}\right )+\left (e \,h^{3}+3 f g \,h^{2}\right ) \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )-\frac {3 d \,g^{2} h +e \,g^{3}}{c \sqrt {c \,x^{2}+a}}+\left (d \,h^{3}+3 e g \,h^{2}+3 f \,g^{2} h \right ) \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+a}}+\frac {2 a}{c^{2} \sqrt {c \,x^{2}+a}}\right )+\left (3 d g \,h^{2}+3 e \,g^{2} h +f \,g^{3}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )\)	\(292\)

[In]

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

[Out]

-1/6*h*(-2*c*f*h^2*x^2-3*c*e*h^2*x-9*c*f*g*h*x+10*a*f*h^2-6*c*d*h^2-18*c*e*g*h-18*c*f*g^2)/c^3*(c*x^2+a)^(1/2)
-1/2/c^2*(a*e*h^3*x/(c*x^2+a)^(1/2)-2*c^2*d*g^3*x/a/(c*x^2+a)^(1/2)+3*a*f*g*h^2*x/(c*x^2+a)^(1/2)+(3*a*c*e*h^3
+9*a*c*f*g*h^2-6*c^2*d*g*h^2-6*c^2*e*g^2*h-2*c^2*f*g^3)*(-x/c/(c*x^2+a)^(1/2)+1/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)
^(1/2)))-(-2*a^2*f*h^3+2*a*c*d*h^3+6*a*c*e*g*h^2+6*a*c*f*g^2*h-6*c^2*d*g^2*h-2*c^2*e*g^3)/c/(c*x^2+a)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 758, normalized size of antiderivative = 3.31 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (2 \, a^{2} c f g^{3} + 6 \, a^{2} c e g^{2} h - 3 \, a^{3} e h^{3} + 3 \, {\left (2 \, a^{2} c d - 3 \, a^{3} f\right )} g h^{2} + {\left (2 \, a c^{2} f g^{3} + 6 \, a c^{2} e g^{2} h - 3 \, a^{2} c e h^{3} + 3 \, {\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} g h^{2}\right )} x^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, a c^{2} f h^{3} x^{4} - 6 \, a c^{2} e g^{3} + 36 \, a^{2} c e g h^{2} - 18 \, {\left (a c^{2} d - 2 \, a^{2} c f\right )} g^{2} h + 4 \, {\left (3 \, a^{2} c d - 4 \, a^{3} f\right )} h^{3} + 3 \, {\left (3 \, a c^{2} f g h^{2} + a c^{2} e h^{3}\right )} x^{3} + 2 \, {\left (9 \, a c^{2} f g^{2} h + 9 \, a c^{2} e g h^{2} + {\left (3 \, a c^{2} d - 4 \, a^{2} c f\right )} h^{3}\right )} x^{2} - 3 \, {\left (6 \, a c^{2} e g^{2} h - 3 \, a^{2} c e h^{3} - 2 \, {\left (c^{3} d - a c^{2} f\right )} g^{3} + 3 \, {\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (a c^{4} x^{2} + a^{2} c^{3}\right )}}, -\frac {3 \, {\left (2 \, a^{2} c f g^{3} + 6 \, a^{2} c e g^{2} h - 3 \, a^{3} e h^{3} + 3 \, {\left (2 \, a^{2} c d - 3 \, a^{3} f\right )} g h^{2} + {\left (2 \, a c^{2} f g^{3} + 6 \, a c^{2} e g^{2} h - 3 \, a^{2} c e h^{3} + 3 \, {\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} g h^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, a c^{2} f h^{3} x^{4} - 6 \, a c^{2} e g^{3} + 36 \, a^{2} c e g h^{2} - 18 \, {\left (a c^{2} d - 2 \, a^{2} c f\right )} g^{2} h + 4 \, {\left (3 \, a^{2} c d - 4 \, a^{3} f\right )} h^{3} + 3 \, {\left (3 \, a c^{2} f g h^{2} + a c^{2} e h^{3}\right )} x^{3} + 2 \, {\left (9 \, a c^{2} f g^{2} h + 9 \, a c^{2} e g h^{2} + {\left (3 \, a c^{2} d - 4 \, a^{2} c f\right )} h^{3}\right )} x^{2} - 3 \, {\left (6 \, a c^{2} e g^{2} h - 3 \, a^{2} c e h^{3} - 2 \, {\left (c^{3} d - a c^{2} f\right )} g^{3} + 3 \, {\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a c^{4} x^{2} + a^{2} c^{3}\right )}}\right ] \]

[In]

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

[Out]

[-1/12*(3*(2*a^2*c*f*g^3 + 6*a^2*c*e*g^2*h - 3*a^3*e*h^3 + 3*(2*a^2*c*d - 3*a^3*f)*g*h^2 + (2*a*c^2*f*g^3 + 6*
a*c^2*e*g^2*h - 3*a^2*c*e*h^3 + 3*(2*a*c^2*d - 3*a^2*c*f)*g*h^2)*x^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)
*sqrt(c)*x - a) - 2*(2*a*c^2*f*h^3*x^4 - 6*a*c^2*e*g^3 + 36*a^2*c*e*g*h^2 - 18*(a*c^2*d - 2*a^2*c*f)*g^2*h + 4
*(3*a^2*c*d - 4*a^3*f)*h^3 + 3*(3*a*c^2*f*g*h^2 + a*c^2*e*h^3)*x^3 + 2*(9*a*c^2*f*g^2*h + 9*a*c^2*e*g*h^2 + (3
*a*c^2*d - 4*a^2*c*f)*h^3)*x^2 - 3*(6*a*c^2*e*g^2*h - 3*a^2*c*e*h^3 - 2*(c^3*d - a*c^2*f)*g^3 + 3*(2*a*c^2*d -
3*a^2*c*f)*g*h^2)*x)*sqrt(c*x^2 + a))/(a*c^4*x^2 + a^2*c^3), -1/6*(3*(2*a^2*c*f*g^3 + 6*a^2*c*e*g^2*h - 3*a^3
*e*h^3 + 3*(2*a^2*c*d - 3*a^3*f)*g*h^2 + (2*a*c^2*f*g^3 + 6*a*c^2*e*g^2*h - 3*a^2*c*e*h^3 + 3*(2*a*c^2*d - 3*a
^2*c*f)*g*h^2)*x^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (2*a*c^2*f*h^3*x^4 - 6*a*c^2*e*g^3 + 36*a^2*
c*e*g*h^2 - 18*(a*c^2*d - 2*a^2*c*f)*g^2*h + 4*(3*a^2*c*d - 4*a^3*f)*h^3 + 3*(3*a*c^2*f*g*h^2 + a*c^2*e*h^3)*x
^3 + 2*(9*a*c^2*f*g^2*h + 9*a*c^2*e*g*h^2 + (3*a*c^2*d - 4*a^2*c*f)*h^3)*x^2 - 3*(6*a*c^2*e*g^2*h - 3*a^2*c*e*
h^3 - 2*(c^3*d - a*c^2*f)*g^3 + 3*(2*a*c^2*d - 3*a^2*c*f)*g*h^2)*x)*sqrt(c*x^2 + a))/(a*c^4*x^2 + a^2*c^3)]

Sympy [F]

\[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\left (g + h x\right )^{3} \left (d + e x + f x^{2}\right )}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.51 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {f h^{3} x^{4}}{3 \, \sqrt {c x^{2} + a} c} - \frac {4 \, a f h^{3} x^{2}}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {d g^{3} x}{\sqrt {c x^{2} + a} a} - \frac {e g^{3}}{\sqrt {c x^{2} + a} c} - \frac {3 \, d g^{2} h}{\sqrt {c x^{2} + a} c} - \frac {8 \, a^{2} f h^{3}}{3 \, \sqrt {c x^{2} + a} c^{3}} + \frac {{\left (3 \, f g h^{2} + e h^{3}\right )} x^{3}}{2 \, \sqrt {c x^{2} + a} c} + \frac {{\left (3 \, f g^{2} h + 3 \, e g h^{2} + d h^{3}\right )} x^{2}}{\sqrt {c x^{2} + a} c} + \frac {3 \, {\left (3 \, f g h^{2} + e h^{3}\right )} a x}{2 \, \sqrt {c x^{2} + a} c^{2}} - \frac {{\left (f g^{3} + 3 \, e g^{2} h + 3 \, d g h^{2}\right )} x}{\sqrt {c x^{2} + a} c} - \frac {3 \, {\left (3 \, f g h^{2} + e h^{3}\right )} a \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, c^{\frac {5}{2}}} + \frac {{\left (f g^{3} + 3 \, e g^{2} h + 3 \, d g h^{2}\right )} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, f g^{2} h + 3 \, e g h^{2} + d h^{3}\right )} a}{\sqrt {c x^{2} + a} c^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.45 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left ({\left (\frac {2 \, f h^{3} x}{c} + \frac {3 \, {\left (3 \, a c^{4} f g h^{2} + a c^{4} e h^{3}\right )}}{a c^{5}}\right )} x + \frac {2 \, {\left (9 \, a c^{4} f g^{2} h + 9 \, a c^{4} e g h^{2} + 3 \, a c^{4} d h^{3} - 4 \, a^{2} c^{3} f h^{3}\right )}}{a c^{5}}\right )} x + \frac {3 \, {\left (2 \, c^{5} d g^{3} - 2 \, a c^{4} f g^{3} - 6 \, a c^{4} e g^{2} h - 6 \, a c^{4} d g h^{2} + 9 \, a^{2} c^{3} f g h^{2} + 3 \, a^{2} c^{3} e h^{3}\right )}}{a c^{5}}\right )} x - \frac {2 \, {\left (3 \, a c^{4} e g^{3} + 9 \, a c^{4} d g^{2} h - 18 \, a^{2} c^{3} f g^{2} h - 18 \, a^{2} c^{3} e g h^{2} - 6 \, a^{2} c^{3} d h^{3} + 8 \, a^{3} c^{2} f h^{3}\right )}}{a c^{5}}}{6 \, \sqrt {c x^{2} + a}} - \frac {{\left (2 \, c f g^{3} + 6 \, c e g^{2} h + 6 \, c d g h^{2} - 9 \, a f g h^{2} - 3 \, a e h^{3}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{\frac {5}{2}}} \]

[In]

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

[Out]

1/6*((((2*f*h^3*x/c + 3*(3*a*c^4*f*g*h^2 + a*c^4*e*h^3)/(a*c^5))*x + 2*(9*a*c^4*f*g^2*h + 9*a*c^4*e*g*h^2 + 3*
a*c^4*d*h^3 - 4*a^2*c^3*f*h^3)/(a*c^5))*x + 3*(2*c^5*d*g^3 - 2*a*c^4*f*g^3 - 6*a*c^4*e*g^2*h - 6*a*c^4*d*g*h^2
+ 9*a^2*c^3*f*g*h^2 + 3*a^2*c^3*e*h^3)/(a*c^5))*x - 2*(3*a*c^4*e*g^3 + 9*a*c^4*d*g^2*h - 18*a^2*c^3*f*g^2*h -
18*a^2*c^3*e*g*h^2 - 6*a^2*c^3*d*h^3 + 8*a^3*c^2*f*h^3)/(a*c^5))/sqrt(c*x^2 + a) - 1/2*(2*c*f*g^3 + 6*c*e*g^2
*h + 6*c*d*g*h^2 - 9*a*f*g*h^2 - 3*a*e*h^3)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (g+h\,x\right )}^3\,\left (f\,x^2+e\,x+d\right )}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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