3.95 \(\int \frac {(a+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^4} \, dx\)

Optimal. Leaf size=475 \[ \frac {c \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 g^3 \left (10 f g^2-h (4 e g-d h)\right )\right )}{2 h^6 \left (a h^2+c g^2\right )^{3/2}}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{3 h (g+h x)^3 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (-x \left (3 a f h^2+c \left (5 f g^2-2 h (e g-d h)\right )\right )-3 a h (3 f g-e h)+c g \left (-d h+4 e g-\frac {10 f g^2}{h}\right )\right )}{6 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a f h^2+2 c \left (10 f g^2-h (4 e g-d h)\right )\right )}{2 h^6}-\frac {\sqrt {a+c x^2} \left (c h x \left (3 a h^2 (3 f g-e h)+c g \left (10 f g^2-h (4 e g-d h)\right )\right )+\left (a h^2+c g^2\right ) \left (3 a f h^2+2 c \left (10 f g^2-h (4 e g-d h)\right )\right )\right )}{2 h^5 (g+h x) \left (a h^2+c g^2\right )} \]

[Out]

-1/6*(c*g*(4*e*g-10*f*g^2/h-d*h)-3*a*h*(-e*h+3*f*g)-(3*a*f*h^2+c*(5*f*g^2-2*h*(-d*h+e*g)))*x)*(c*x^2+a)^(3/2)/
h^2/(a*h^2+c*g^2)/(h*x+g)^2-1/3*(d*h^2-e*g*h+f*g^2)*(c*x^2+a)^(5/2)/h/(a*h^2+c*g^2)/(h*x+g)^3+1/2*c*(3*a^2*h^4
*(-e*h+4*f*g)+2*c^2*g^3*(10*f*g^2-h*(-d*h+4*e*g))+3*a*c*g*h^2*(11*f*g^2-h*(-d*h+4*e*g)))*arctanh((-c*g*x+a*h)/
(a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1/2))/h^6/(a*h^2+c*g^2)^(3/2)+1/2*(3*a*f*h^2+2*c*(10*f*g^2-h*(-d*h+4*e*g)))*arc
tanh(x*c^(1/2)/(c*x^2+a)^(1/2))*c^(1/2)/h^6-1/2*((a*h^2+c*g^2)*(3*a*f*h^2+2*c*(10*f*g^2-h*(-d*h+4*e*g)))+c*h*(
3*a*h^2*(-e*h+3*f*g)+c*g*(10*f*g^2-h*(-d*h+4*e*g)))*x)*(c*x^2+a)^(1/2)/h^5/(a*h^2+c*g^2)/(h*x+g)

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Rubi [A]  time = 0.84, antiderivative size = 469, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1651, 813, 844, 217, 206, 725} \[ \frac {c \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 \left (10 f g^5-g^3 h (4 e g-d h)\right )\right )}{2 h^6 \left (a h^2+c g^2\right )^{3/2}}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{3 h (g+h x)^3 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (-x \left (3 a f h^2-2 c h (e g-d h)+5 c f g^2\right )-3 a h (3 f g-e h)+c g \left (-d h+4 e g-\frac {10 f g^2}{h}\right )\right )}{6 h^2 (g+h x)^2 \left (a h^2+c g^2\right )}-\frac {\sqrt {a+c x^2} \left (c h x \left (3 a h^2 (3 f g-e h)-c g h (4 e g-d h)+10 c f g^3\right )+\left (a h^2+c g^2\right ) \left (3 a f h^2-2 c h (4 e g-d h)+20 c f g^2\right )\right )}{2 h^5 (g+h x) \left (a h^2+c g^2\right )}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a f h^2-2 c h (4 e g-d h)+20 c f g^2\right )}{2 h^6} \]

Antiderivative was successfully verified.

[In]

Int[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^4,x]

[Out]

-(((c*g^2 + a*h^2)*(20*c*f*g^2 + 3*a*f*h^2 - 2*c*h*(4*e*g - d*h)) + c*h*(10*c*f*g^3 - c*g*h*(4*e*g - d*h) + 3*
a*h^2*(3*f*g - e*h))*x)*Sqrt[a + c*x^2])/(2*h^5*(c*g^2 + a*h^2)*(g + h*x)) - ((c*g*(4*e*g - (10*f*g^2)/h - d*h
) - 3*a*h*(3*f*g - e*h) - (5*c*f*g^2 + 3*a*f*h^2 - 2*c*h*(e*g - d*h))*x)*(a + c*x^2)^(3/2))/(6*h^2*(c*g^2 + a*
h^2)*(g + h*x)^2) - ((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(3*h*(c*g^2 + a*h^2)*(g + h*x)^3) + (Sqrt[c]*(
20*c*f*g^2 + 3*a*f*h^2 - 2*c*h*(4*e*g - d*h))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*h^6) + (c*(3*a^2*h^4*(4
*f*g - e*h) + 3*a*c*g*h^2*(11*f*g^2 - h*(4*e*g - d*h)) + 2*c^2*(10*f*g^5 - g^3*h*(4*e*g - d*h)))*ArcTanh[(a*h
- c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(2*h^6*(c*g^2 + a*h^2)^(3/2))

Rule 206

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

Rule 217

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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

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

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^4} \, dx &=-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}-\frac {\int \frac {\left (-3 (c d g-a f g+a e h)-\left (3 a f h-c \left (2 e g-\frac {5 f g^2}{h}-2 d h\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{(g+h x)^3} \, dx}{3 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (c g \left (4 e g-\frac {10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac {\int \frac {\left (4 a \left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right )-\frac {4 c \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x}{h}\right ) \sqrt {a+c x^2}}{(g+h x)^2} \, dx}{8 h^2 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (\left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )+c h \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^5 \left (c g^2+a h^2\right ) (g+h x)}-\frac {\left (c g \left (4 e g-\frac {10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}-\frac {\int \frac {8 a c \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right )-\frac {8 c \left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right ) x}{h}}{(g+h x) \sqrt {a+c x^2}} \, dx}{16 h^4 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (\left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )+c h \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^5 \left (c g^2+a h^2\right ) (g+h x)}-\frac {\left (c g \left (4 e g-\frac {10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac {\left (c \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 h^6}-\frac {\left (c \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 \left (10 f g^5-g^3 h (4 e g-d h)\right )\right )\right ) \int \frac {1}{(g+h x) \sqrt {a+c x^2}} \, dx}{2 h^6 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (\left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )+c h \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^5 \left (c g^2+a h^2\right ) (g+h x)}-\frac {\left (c g \left (4 e g-\frac {10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac {\left (c \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 h^6}+\frac {\left (c \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 \left (10 f g^5-g^3 h (4 e g-d h)\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c g^2+a h^2-x^2} \, dx,x,\frac {a h-c g x}{\sqrt {a+c x^2}}\right )}{2 h^6 \left (c g^2+a h^2\right )}\\ &=-\frac {\left (\left (c g^2+a h^2\right ) \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right )+c h \left (10 c f g^3-c g h (4 e g-d h)+3 a h^2 (3 f g-e h)\right ) x\right ) \sqrt {a+c x^2}}{2 h^5 \left (c g^2+a h^2\right ) (g+h x)}-\frac {\left (c g \left (4 e g-\frac {10 f g^2}{h}-d h\right )-3 a h (3 f g-e h)-\left (5 c f g^2+3 a f h^2-2 c h (e g-d h)\right ) x\right ) \left (a+c x^2\right )^{3/2}}{6 h^2 \left (c g^2+a h^2\right ) (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{3 h \left (c g^2+a h^2\right ) (g+h x)^3}+\frac {\sqrt {c} \left (20 c f g^2+3 a f h^2-2 c h (4 e g-d h)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 h^6}+\frac {c \left (3 a^2 h^4 (4 f g-e h)+3 a c g h^2 \left (11 f g^2-h (4 e g-d h)\right )+2 c^2 \left (10 f g^5-g^3 h (4 e g-d h)\right )\right ) \tanh ^{-1}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{2 h^6 \left (c g^2+a h^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 1.23, size = 517, normalized size = 1.09 \[ \frac {\frac {3 c \log \left (\sqrt {a+c x^2} \sqrt {a h^2+c g^2}+a h-c g x\right ) \left (-3 a^2 h^4 (e h-4 f g)+3 a c g h^2 \left (h (d h-4 e g)+11 f g^2\right )+2 c^2 \left (g^3 h (d h-4 e g)+10 f g^5\right )\right )}{\left (a h^2+c g^2\right )^{3/2}}-\frac {3 c \log (g+h x) \left (-3 a^2 h^4 (e h-4 f g)+3 a c g h^2 \left (h (d h-4 e g)+11 f g^2\right )+2 c^2 \left (g^3 h (d h-4 e g)+10 f g^5\right )\right )}{\left (a h^2+c g^2\right )^{3/2}}-\frac {h \sqrt {a+c x^2} \left ((g+h x)^2 \left (6 a^2 f h^4+a c h^2 \left (h (8 d h-23 e g)+50 f g^2\right )+c^2 \left (g^2 h (11 d h-26 e g)+47 f g^4\right )\right )+2 \left (a h^2+c g^2\right )^2 \left (h (d h-e g)+f g^2\right )-(g+h x) \left (a h^2+c g^2\right ) \left (-3 a h^2 (e h-2 f g)+c g h (7 d h-10 e g)+13 c f g^3\right )+6 c (g+h x)^3 \left (a h^2+c g^2\right ) (4 f g-e h)-3 c f h x (g+h x)^3 \left (a h^2+c g^2\right )\right )}{(g+h x)^3 \left (a h^2+c g^2\right )}+3 \sqrt {c} \log \left (\sqrt {c} \sqrt {a+c x^2}+c x\right ) \left (3 a f h^2+2 c h (d h-4 e g)+20 c f g^2\right )}{6 h^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^4,x]

[Out]

(-((h*Sqrt[a + c*x^2]*(2*(c*g^2 + a*h^2)^2*(f*g^2 + h*(-(e*g) + d*h)) - (c*g^2 + a*h^2)*(13*c*f*g^3 + c*g*h*(-
10*e*g + 7*d*h) - 3*a*h^2*(-2*f*g + e*h))*(g + h*x) + (6*a^2*f*h^4 + a*c*h^2*(50*f*g^2 + h*(-23*e*g + 8*d*h))
+ c^2*(47*f*g^4 + g^2*h*(-26*e*g + 11*d*h)))*(g + h*x)^2 + 6*c*(4*f*g - e*h)*(c*g^2 + a*h^2)*(g + h*x)^3 - 3*c
*f*h*(c*g^2 + a*h^2)*x*(g + h*x)^3))/((c*g^2 + a*h^2)*(g + h*x)^3)) - (3*c*(-3*a^2*h^4*(-4*f*g + e*h) + 3*a*c*
g*h^2*(11*f*g^2 + h*(-4*e*g + d*h)) + 2*c^2*(10*f*g^5 + g^3*h*(-4*e*g + d*h)))*Log[g + h*x])/(c*g^2 + a*h^2)^(
3/2) + 3*Sqrt[c]*(20*c*f*g^2 + 3*a*f*h^2 + 2*c*h*(-4*e*g + d*h))*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (3*c*(-3
*a^2*h^4*(-4*f*g + e*h) + 3*a*c*g*h^2*(11*f*g^2 + h*(-4*e*g + d*h)) + 2*c^2*(10*f*g^5 + g^3*h*(-4*e*g + d*h)))
*Log[a*h - c*g*x + Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2]])/(c*g^2 + a*h^2)^(3/2))/(6*h^6)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^4,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.61, size = 1900, normalized size = 4.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^4,x, algorithm="giac")

[Out]

1/2*sqrt(c*x^2 + a)*(c*f*x/h^4 - 2*(4*c*f*g*h^10 - c*h^11*e)/h^15) - (20*c^3*f*g^5 + 2*c^3*d*g^3*h^2 + 33*a*c^
2*f*g^3*h^2 + 3*a*c^2*d*g*h^4 + 12*a^2*c*f*g*h^4 - 8*c^3*g^4*h*e - 12*a*c^2*g^2*h^3*e - 3*a^2*c*h^5*e)*arctan(
-((sqrt(c)*x - sqrt(c*x^2 + a))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c*g^2*h^6 + a*h^8)*sqrt(-c*g^2 - a*h^2)
) - 1/3*(60*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*f*g^5*h^2 + 18*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*d*g^3*h^4 +
69*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*f*g^3*h^4 + 15*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d*g*h^6 + 12*(s
qrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c*f*g*h^6 - 36*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*g^4*h^3*e - 36*(sqrt(c)*x
- sqrt(c*x^2 + a))^5*a*c^2*g^2*h^5*e - 3*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c*h^7*e + 210*(sqrt(c)*x - sqrt(
c*x^2 + a))^4*c^(7/2)*f*g^6*h + 54*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d*g^4*h^3 + 183*(sqrt(c)*x - sqrt(c
*x^2 + a))^4*a*c^(5/2)*f*g^4*h^3 + 27*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(5/2)*d*g^2*h^5 - 18*(sqrt(c)*x - sq
rt(c*x^2 + a))^4*a^2*c^(3/2)*f*g^2*h^5 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*d*h^7 - 6*(sqrt(c)*x -
sqrt(c*x^2 + a))^4*a^3*sqrt(c)*f*h^7 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*g^5*h^2*e - 84*(sqrt(c)*x
- sqrt(c*x^2 + a))^4*a*c^(5/2)*g^3*h^4*e + 21*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*g*h^6*e + 188*(sqrt(
c)*x - sqrt(c*x^2 + a))^3*c^4*f*g^7 + 44*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*d*g^5*h^2 - 82*(sqrt(c)*x - sqrt(
c*x^2 + a))^3*a*c^3*f*g^5*h^2 - 34*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^3*d*g^3*h^4 - 276*(sqrt(c)*x - sqrt(c*x
^2 + a))^3*a^2*c^2*f*g^3*h^4 - 48*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*d*g*h^6 - 36*(sqrt(c)*x - sqrt(c*x^2
+ a))^3*a^3*c*f*g*h^6 - 104*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*g^6*h*e + 64*(sqrt(c)*x - sqrt(c*x^2 + a))^3*
a*c^3*g^4*h^3*e + 138*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*c^2*g^2*h^5*e - 354*(sqrt(c)*x - sqrt(c*x^2 + a))^2*
a*c^(7/2)*f*g^6*h - 78*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d*g^4*h^3 - 276*(sqrt(c)*x - sqrt(c*x^2 + a))
^2*a^2*c^(5/2)*f*g^4*h^3 - 36*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c^(5/2)*d*g^2*h^5 + 60*(sqrt(c)*x - sqrt(c*x
^2 + a))^2*a^3*c^(3/2)*f*g^2*h^5 + 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(3/2)*d*h^7 + 12*(sqrt(c)*x - sqrt
(c*x^2 + a))^2*a^4*sqrt(c)*f*h^7 + 192*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*g^5*h^2*e + 114*(sqrt(c)*x -
sqrt(c*x^2 + a))^2*a^2*c^(5/2)*g^3*h^4*e - 48*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(3/2)*g*h^6*e + 222*(sqrt(
c)*x - sqrt(c*x^2 + a))*a^2*c^3*f*g^5*h^2 + 48*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^3*d*g^3*h^4 + 231*(sqrt(c)*
x - sqrt(c*x^2 + a))*a^3*c^2*f*g^3*h^4 + 33*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*g*h^6 + 24*(sqrt(c)*x - sq
rt(c*x^2 + a))*a^4*c*f*g*h^6 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^3*g^4*h^3*e - 102*(sqrt(c)*x - sqrt(c*x
^2 + a))*a^3*c^2*g^2*h^5*e + 3*(sqrt(c)*x - sqrt(c*x^2 + a))*a^4*c*h^7*e - 47*a^3*c^(5/2)*f*g^4*h^3 - 11*a^3*c
^(5/2)*d*g^2*h^5 - 50*a^4*c^(3/2)*f*g^2*h^5 - 8*a^4*c^(3/2)*d*h^7 - 6*a^5*sqrt(c)*f*h^7 + 26*a^3*c^(5/2)*g^3*h
^4*e + 23*a^4*c^(3/2)*g*h^6*e)/((c*g^2*h^6 + a*h^8)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*h + 2*(sqrt(c)*x - sqrt(c
*x^2 + a))*sqrt(c)*g - a*h)^3) - 1/2*(20*c^(3/2)*f*g^2 + 2*c^(3/2)*d*h^2 + 3*a*sqrt(c)*f*h^2 - 8*c^(3/2)*g*h*e
)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/h^6

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maple [B]  time = 0.02, size = 9835, normalized size = 20.71 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^4,x)

[Out]

result too large to display

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maxima [B]  time = 1.10, size = 2415, normalized size = 5.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^4,x, algorithm="maxima")

[Out]

1/2*sqrt(c*x^2 + a)*c^3*f*g^5/(c^2*g^4*h^5 + 2*a*c*g^2*h^7 + a^2*h^9) - 1/2*sqrt(c*x^2 + a)*c^3*f*g^4*x/(c^2*g
^4*h^4 + 2*a*c*g^2*h^6 + a^2*h^8) - 1/2*sqrt(c*x^2 + a)*c^3*e*g^4/(c^2*g^4*h^4 + 2*a*c*g^2*h^6 + a^2*h^8) + 1/
6*(c*x^2 + a)^(3/2)*c^2*f*g^4/(c^2*g^4*h^4*x + 2*a*c*g^2*h^6*x + a^2*h^8*x + c^2*g^5*h^3 + 2*a*c*g^3*h^5 + a^2
*g*h^7) + 1/2*sqrt(c*x^2 + a)*c^3*e*g^3*x/(c^2*g^4*h^3 + 2*a*c*g^2*h^5 + a^2*h^7) + 1/2*sqrt(c*x^2 + a)*c^3*d*
g^3/(c^2*g^4*h^3 + 2*a*c*g^2*h^5 + a^2*h^7) - 1/6*(c*x^2 + a)^(3/2)*c^2*e*g^3/(c^2*g^4*h^3*x + 2*a*c*g^2*h^5*x
+ a^2*h^7*x + c^2*g^5*h^2 + 2*a*c*g^3*h^4 + a^2*g*h^6) - 1/6*(c*x^2 + a)^(5/2)*c*f*g^3/(c^2*g^4*h^3*x^2 + 2*a
*c*g^2*h^5*x^2 + a^2*h^7*x^2 + 2*c^2*g^5*h^2*x + 4*a*c*g^3*h^4*x + 2*a^2*g*h^6*x + c^2*g^6*h + 2*a*c*g^4*h^3 +
a^2*g^2*h^5) + 1/6*(c*x^2 + a)^(3/2)*c^2*f*g^3/(c^2*g^4*h^3 + 2*a*c*g^2*h^5 + a^2*h^7) - 1/2*sqrt(c*x^2 + a)*
c^3*d*g^2*x/(c^2*g^4*h^2 + 2*a*c*g^2*h^4 + a^2*h^6) + 1/6*(c*x^2 + a)^(3/2)*c^2*d*g^2/(c^2*g^4*h^2*x + 2*a*c*g
^2*h^4*x + a^2*h^6*x + c^2*g^5*h + 2*a*c*g^3*h^3 + a^2*g*h^5) + 1/6*(c*x^2 + a)^(5/2)*c*e*g^2/(c^2*g^4*h^2*x^2
+ 2*a*c*g^2*h^4*x^2 + a^2*h^6*x^2 + 2*c^2*g^5*h*x + 4*a*c*g^3*h^3*x + 2*a^2*g*h^5*x + c^2*g^6 + 2*a*c*g^4*h^2
+ a^2*g^2*h^4) - 1/6*(c*x^2 + a)^(3/2)*c^2*e*g^2/(c^2*g^4*h^2 + 2*a*c*g^2*h^4 + a^2*h^6) - 9/2*sqrt(c*x^2 + a
)*c^2*f*g^3/(c*g^2*h^5 + a*h^7) + 4*sqrt(c*x^2 + a)*c^2*f*g^2*x/(c*g^2*h^4 + a*h^6) - 1/6*(c*x^2 + a)^(5/2)*c*
d*g/(c^2*g^4*h*x^2 + 2*a*c*g^2*h^3*x^2 + a^2*h^5*x^2 + 2*c^2*g^5*x + 4*a*c*g^3*h^2*x + 2*a^2*g*h^4*x + c^2*g^6
/h + 2*a*c*g^4*h + a^2*g^2*h^3) + 1/6*(c*x^2 + a)^(3/2)*c^2*d*g/(c^2*g^4*h + 2*a*c*g^2*h^3 + a^2*h^5) + 3*sqrt
(c*x^2 + a)*c^2*e*g^2/(c*g^2*h^4 + a*h^6) - 1/3*(c*x^2 + a)^(5/2)*f*g^2/(c*g^2*h^4*x^3 + a*h^6*x^3 + 3*c*g^3*h
^3*x^2 + 3*a*g*h^5*x^2 + 3*c*g^4*h^2*x + 3*a*g^2*h^4*x + c*g^5*h + a*g^3*h^3) - 5/3*(c*x^2 + a)^(3/2)*c*f*g^2/
(c*g^2*h^4*x + a*h^6*x + c*g^3*h^3 + a*g*h^5) - 5/2*sqrt(c*x^2 + a)*c^2*e*g*x/(c*g^2*h^3 + a*h^5) - 3/2*sqrt(c
*x^2 + a)*c^2*d*g/(c*g^2*h^3 + a*h^5) + 1/3*(c*x^2 + a)^(5/2)*e*g/(c*g^2*h^3*x^3 + a*h^5*x^3 + 3*c*g^3*h^2*x^2
+ 3*a*g*h^4*x^2 + 3*c*g^4*h*x + 3*a*g^2*h^3*x + c*g^5 + a*g^3*h^2) + 7/6*(c*x^2 + a)^(3/2)*c*e*g/(c*g^2*h^3*x
+ a*h^5*x + c*g^3*h^2 + a*g*h^4) + (c*x^2 + a)^(5/2)*f*g/(c*g^2*h^3*x^2 + a*h^5*x^2 + 2*c*g^3*h^2*x + 2*a*g*h
^4*x + c*g^4*h + a*g^2*h^3) - (c*x^2 + a)^(3/2)*c*f*g/(c*g^2*h^3 + a*h^5) + sqrt(c*x^2 + a)*c^2*d*x/(c*g^2*h^2
+ a*h^4) - 1/3*(c*x^2 + a)^(5/2)*d/(c*g^2*h^2*x^3 + a*h^4*x^3 + 3*c*g^3*h*x^2 + 3*a*g*h^3*x^2 + 3*c*g^4*x + 3
*a*g^2*h^2*x + c*g^5/h + a*g^3*h) - 2/3*(c*x^2 + a)^(3/2)*c*d/(c*g^2*h^2*x + a*h^4*x + c*g^3*h + a*g*h^3) - 1/
2*(c*x^2 + a)^(5/2)*e/(c*g^2*h^2*x^2 + a*h^4*x^2 + 2*c*g^3*h*x + 2*a*g*h^3*x + c*g^4 + a*g^2*h^2) + 1/2*(c*x^2
+ a)^(3/2)*c*e/(c*g^2*h^2 + a*h^4) - (c*x^2 + a)^(3/2)*f/(h^4*x + g*h^3) + 3/2*sqrt(c*x^2 + a)*c*f*x/h^4 + 10
*c^(3/2)*f*g^2*arcsinh(c*x/sqrt(a*c))/h^6 - 4*c^(3/2)*e*g*arcsinh(c*x/sqrt(a*c))/h^5 + c^(3/2)*d*arcsinh(c*x/s
qrt(a*c))/h^4 + 3/2*a*sqrt(c)*f*arcsinh(c*x/sqrt(a*c))/h^4 + 1/2*c^3*f*g^5*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x +
g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^9) - 1/2*c^3*e*g^4*arcsinh(c*g*x/(sqrt(a*c)*abs(h
*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^8) + 1/2*c^3*d*g^3*arcsinh(c*g*x/(sqrt(a*c)*
abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/((a + c*g^2/h^2)^(3/2)*h^7) - 9/2*c^2*f*g^3*arcsinh(c*g*x/(sqrt(
a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/(sqrt(a + c*g^2/h^2)*h^7) + 3*c^2*e*g^2*arcsinh(c*g*x/(sqrt
(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/(sqrt(a + c*g^2/h^2)*h^6) - 3/2*c^2*d*g*arcsinh(c*g*x/(sqr
t(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/(sqrt(a + c*g^2/h^2)*h^5) - 6*sqrt(a + c*g^2/h^2)*c*f*g*a
rcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^5 + 3/2*sqrt(a + c*g^2/h^2)*c*e*arcsin
h(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^4 - 6*sqrt(c*x^2 + a)*c*f*g/h^5 + 3/2*sqrt(
c*x^2 + a)*c*e/h^4

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^4,x)

[Out]

int(((a + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(3/2)*(f*x**2+e*x+d)/(h*x+g)**4,x)

[Out]

Integral((a + c*x**2)**(3/2)*(d + e*x + f*x**2)/(g + h*x)**4, x)

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