3.461 \(\int \frac{\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{12}} \, dx\)

Optimal. Leaf size=796 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right ) b^3}{24 a^{3/2}}+\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (b d-4 a g) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt{\frac{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{7/3}}{896 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{b x^3+a}}+\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (b d-4 a g)\right ) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt{\frac{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{7/3}}{49280 a^2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{b x^3+a}}-\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^{7/3}}{448 a^2 \left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )}+\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^2}{448 a^2 x}+\frac{27 (7 b c-22 a f) \sqrt{b x^3+a} b^2}{7040 a^2 x^2}-\frac{e \sqrt{b x^3+a} b^2}{24 a x^3}-\frac{27 d \sqrt{b x^3+a} b^2}{1120 a x^4}-\frac{27 c \sqrt{b x^3+a} b^2}{1760 a x^5}-\frac{\left (\frac{945 c}{x^8}+\frac{2970 g}{x^4}+\frac{2079 f}{x^5}+\frac{1540 e}{x^6}+\frac{1188 d}{x^7}\right ) \sqrt{b x^3+a} b}{18480}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{3960 g}{x^7}+\frac{3465 f}{x^8}+\frac{3080 e}{x^9}+\frac{2772 d}{x^{10}}\right ) \left (b x^3+a\right )^{3/2}}{27720} \]

[Out]

-(b*((945*c)/x^8 + (1188*d)/x^7 + (1540*e)/x^6 + (2079*f)/x^5 + (2970*g)/x^4)*Sq
rt[a + b*x^3])/18480 - (27*b^2*c*Sqrt[a + b*x^3])/(1760*a*x^5) - (27*b^2*d*Sqrt[
a + b*x^3])/(1120*a*x^4) - (b^2*e*Sqrt[a + b*x^3])/(24*a*x^3) + (27*b^2*(7*b*c -
22*a*f)*Sqrt[a + b*x^3])/(7040*a^2*x^2) + (27*b^2*(b*d - 4*a*g)*Sqrt[a + b*x^3]
)/(448*a^2*x) - (27*b^(7/3)*(b*d - 4*a*g)*Sqrt[a + b*x^3])/(448*a^2*((1 + Sqrt[3
])*a^(1/3) + b^(1/3)*x)) - (((2520*c)/x^11 + (2772*d)/x^10 + (3080*e)/x^9 + (346
5*f)/x^8 + (3960*g)/x^7)*(a + b*x^3)^(3/2))/27720 + (b^3*e*ArcTanh[Sqrt[a + b*x^
3]/Sqrt[a]])/(24*a^(3/2)) + (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(7/3)*(b*d - 4*a*g)*
(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqr
t[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*
x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(896*a^(5/3)*Sqrt[(a^(
1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^
3]) + (9*3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(7/3)*(7*b^(1/3)*(7*b*c - 22*a*f) + 110*(1
- Sqrt[3])*a^(1/3)*(b*d - 4*a*g))*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin
[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 -
4*Sqrt[3]])/(49280*a^2*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/
3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi [A]  time = 2.83545, antiderivative size = 796, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right ) b^3}{24 a^{3/2}}+\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (b d-4 a g) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt{\frac{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{7/3}}{896 a^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{b x^3+a}}+\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} \left (7 \sqrt [3]{b} (7 b c-22 a f)+110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (b d-4 a g)\right ) \left (\sqrt [3]{b} x+\sqrt [3]{a}\right ) \sqrt{\frac{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{7/3}}{49280 a^2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{b x^3+a}}-\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^{7/3}}{448 a^2 \left (\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )}+\frac{27 (b d-4 a g) \sqrt{b x^3+a} b^2}{448 a^2 x}+\frac{27 (7 b c-22 a f) \sqrt{b x^3+a} b^2}{7040 a^2 x^2}-\frac{e \sqrt{b x^3+a} b^2}{24 a x^3}-\frac{27 d \sqrt{b x^3+a} b^2}{1120 a x^4}-\frac{27 c \sqrt{b x^3+a} b^2}{1760 a x^5}-\frac{\left (\frac{945 c}{x^8}+\frac{2970 g}{x^4}+\frac{2079 f}{x^5}+\frac{1540 e}{x^6}+\frac{1188 d}{x^7}\right ) \sqrt{b x^3+a} b}{18480}-\frac{\left (\frac{2520 c}{x^{11}}+\frac{3960 g}{x^7}+\frac{3465 f}{x^8}+\frac{3080 e}{x^9}+\frac{2772 d}{x^{10}}\right ) \left (b x^3+a\right )^{3/2}}{27720} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^12,x]

[Out]

-(b*((945*c)/x^8 + (1188*d)/x^7 + (1540*e)/x^6 + (2079*f)/x^5 + (2970*g)/x^4)*Sq
rt[a + b*x^3])/18480 - (27*b^2*c*Sqrt[a + b*x^3])/(1760*a*x^5) - (27*b^2*d*Sqrt[
a + b*x^3])/(1120*a*x^4) - (b^2*e*Sqrt[a + b*x^3])/(24*a*x^3) + (27*b^2*(7*b*c -
22*a*f)*Sqrt[a + b*x^3])/(7040*a^2*x^2) + (27*b^2*(b*d - 4*a*g)*Sqrt[a + b*x^3]
)/(448*a^2*x) - (27*b^(7/3)*(b*d - 4*a*g)*Sqrt[a + b*x^3])/(448*a^2*((1 + Sqrt[3
])*a^(1/3) + b^(1/3)*x)) - (((2520*c)/x^11 + (2772*d)/x^10 + (3080*e)/x^9 + (346
5*f)/x^8 + (3960*g)/x^7)*(a + b*x^3)^(3/2))/27720 + (b^3*e*ArcTanh[Sqrt[a + b*x^
3]/Sqrt[a]])/(24*a^(3/2)) + (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(7/3)*(b*d - 4*a*g)*
(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqr
t[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*
x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(896*a^(5/3)*Sqrt[(a^(
1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^
3]) + (9*3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(7/3)*(7*b^(1/3)*(7*b*c - 22*a*f) + 110*(1
- Sqrt[3])*a^(1/3)*(b*d - 4*a*g))*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*
b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin
[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 -
4*Sqrt[3]])/(49280*a^2*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/
3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**12,x)

[Out]

Timed out

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Mathematica [C]  time = 4.69395, size = 1017, normalized size = 1.28 \[ \frac{b^{7/3} \left (35640 \sqrt{2} g \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{3 i+\sqrt{3}}} \left (-\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )-F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right ) a^{4/3}+12474 \sqrt [3]{b} f \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right ) a+6160 b^{2/3} e \sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3+a} \tanh ^{-1}\left (\frac{\sqrt{b x^3+a}}{\sqrt{a}}\right ) \sqrt{a}-8910 \sqrt{2} b d \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{3 i+\sqrt{3}}} \left (-\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )-F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right ) \sqrt [3]{a}-3969 b^{4/3} c \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt{\frac{\sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{\frac{\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )\right )}{147840 a^2 \sqrt{\frac{(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt{b x^3+a}}-\frac{\sqrt{b x^3+a} \left (-243 b^3 (49 c+110 d x) x^9+6 a b^2 (1134 c+11 x (162 d+x (280 e+81 x (7 f+20 g x)))) x^6+8 a^2 b (7875 c+11 x (828 d+x (980 e+9 x (133 f+170 g x)))) x^3+16 a^3 (2520 c+11 x (252 d+5 x (56 e+9 x (7 f+8 g x))))\right )}{443520 a^2 x^{11}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^12,x]

[Out]

-(Sqrt[a + b*x^3]*(-243*b^3*x^9*(49*c + 110*d*x) + 16*a^3*(2520*c + 11*x*(252*d
+ 5*x*(56*e + 9*x*(7*f + 8*g*x)))) + 6*a*b^2*x^6*(1134*c + 11*x*(162*d + x*(280*
e + 81*x*(7*f + 20*g*x)))) + 8*a^2*b*x^3*(7875*c + 11*x*(828*d + x*(980*e + 9*x*
(133*f + 170*g*x))))))/(443520*a^2*x^11) + (b^(7/3)*(6160*Sqrt[a]*b^(2/3)*e*Sqrt
[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3]*Ar
cTanh[Sqrt[a + b*x^3]/Sqrt[a]] - 3969*b^(4/3)*c*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)
*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[((-1)^(1/3)*(a^(1/3
) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^
(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] + 12474*
a*b^(1/3)*f*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-
1)^(1/3))*a^(1/3))]*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1
)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 +
(-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] - 8910*Sqrt[2]*a^(1/3)*b*d*((-1)^(1/3)*a^(1/
3) - b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1
/3))*a^(1/3))]*Sqrt[(I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3])]*(-((-1 + (-1)
^(2/3))*EllipticE[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)
^(1/3)/(-1 + (-1)^(1/3))]) - EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^
(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]) + 35640*Sqrt[2]*a^(4/3)*g*((-1)^
(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((
1 + (-1)^(1/3))*a^(1/3))]*Sqrt[(I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3])]*(-
((-1 + (-1)^(2/3))*EllipticE[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(
1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]) - EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*b^
(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))])))/(147840*a^2*Sqrt[(a
^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3])

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Maple [B]  time = 0.046, size = 1773, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^12,x)

[Out]

c*(-1/11*a*(b*x^3+a)^(1/2)/x^11-25/176*b*(b*x^3+a)^(1/2)/x^8-27/1760/a*b^2*(b*x^
3+a)^(1/2)/x^5+189/7040/a^2*b^3*(b*x^3+a)^(1/2)/x^2-63/7040*I*b^3/a^2*3^(1/2)*(-
a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*
b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(
1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^
2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*
(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1
/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-
a*b^2)^(1/3)))^(1/2)))+d*(-1/10*a*(b*x^3+a)^(1/2)/x^10-23/140*b*(b*x^3+a)^(1/2)/
x^7-27/1120/a*b^2*(b*x^3+a)^(1/2)/x^4+27/448/a^2*b^3*(b*x^3+a)^(1/2)/x+9/448*I*b
^3/a^2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2
)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2
)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I
*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/
2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1
/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1
/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^
(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/
3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*
(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))))+
e*(-1/9*a*(b*x^3+a)^(1/2)/x^9-7/36*b*(b*x^3+a)^(1/2)/x^6-1/24/a*b^2*(b*x^3+a)^(1
/2)/x^3+1/24/a^(3/2)*b^3*arctanh((b*x^3+a)^(1/2)/a^(1/2)))+f*(-1/8*a*(b*x^3+a)^(
1/2)/x^8-19/80*b*(b*x^3+a)^(1/2)/x^5-27/320/a*b^2*(b*x^3+a)^(1/2)/x^2+9/320*I/a*
b^2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(
1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(
1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^
(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*Elliptic
F(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)
*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*
I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)))+g*(-1/7*a*(b*x^3+a)^(1/2)/x^7-17/56*b*(b*x^
3+a)^(1/2)/x^4-27/112/a*b^2*(b*x^3+a)^(1/2)/x-9/112*I/a*b^2*3^(1/2)*(-a*b^2)^(1/
3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)
^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a
*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*
3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^
(1/2)/b*(-a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3
^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(
1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)
^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)
^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*
b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^12,x, algorithm="maxima")

[Out]

integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^12, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b g x^{7} + b f x^{6} + b e x^{5} +{\left (b d + a g\right )} x^{4} + a e x^{2} +{\left (b c + a f\right )} x^{3} + a d x + a c\right )} \sqrt{b x^{3} + a}}{x^{12}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^12,x, algorithm="fricas")

[Out]

integral((b*g*x^7 + b*f*x^6 + b*e*x^5 + (b*d + a*g)*x^4 + a*e*x^2 + (b*c + a*f)*
x^3 + a*d*x + a*c)*sqrt(b*x^3 + a)/x^12, x)

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Sympy [A]  time = 38.6223, size = 541, normalized size = 0.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**12,x)

[Out]

a**(3/2)*c*gamma(-11/3)*hyper((-11/3, -1/2), (-8/3,), b*x**3*exp_polar(I*pi)/a)/
(3*x**11*gamma(-8/3)) + a**(3/2)*d*gamma(-10/3)*hyper((-10/3, -1/2), (-7/3,), b*
x**3*exp_polar(I*pi)/a)/(3*x**10*gamma(-7/3)) + a**(3/2)*f*gamma(-8/3)*hyper((-8
/3, -1/2), (-5/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**8*gamma(-5/3)) + a**(3/2)*g*
gamma(-7/3)*hyper((-7/3, -1/2), (-4/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**7*gamma
(-4/3)) + sqrt(a)*b*c*gamma(-8/3)*hyper((-8/3, -1/2), (-5/3,), b*x**3*exp_polar(
I*pi)/a)/(3*x**8*gamma(-5/3)) + sqrt(a)*b*d*gamma(-7/3)*hyper((-7/3, -1/2), (-4/
3,), b*x**3*exp_polar(I*pi)/a)/(3*x**7*gamma(-4/3)) + sqrt(a)*b*f*gamma(-5/3)*hy
per((-5/3, -1/2), (-2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(-2/3)) + sqrt
(a)*b*g*gamma(-4/3)*hyper((-4/3, -1/2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x*
*4*gamma(-1/3)) - a**2*e/(9*sqrt(b)*x**(21/2)*sqrt(a/(b*x**3) + 1)) - 11*a*sqrt(
b)*e/(36*x**(15/2)*sqrt(a/(b*x**3) + 1)) - 17*b**(3/2)*e/(72*x**(9/2)*sqrt(a/(b*
x**3) + 1)) - b**(5/2)*e/(24*a*x**(3/2)*sqrt(a/(b*x**3) + 1)) + b**3*e*asinh(sqr
t(a)/(sqrt(b)*x**(3/2)))/(24*a**(3/2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^12,x, algorithm="giac")

[Out]

integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^12, x)