3.417 \(\int \frac{\left (c+d x^3\right )^{3/2}}{x^7 \left (8 c-d x^3\right )^2} \, dx\)

Optimal. Leaf size=161 \[ \frac{15 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2048 c^{5/2}}-\frac{17 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{5/2}}+\frac{7 d^2 \sqrt{c+d x^3}}{512 c^2 \left (8 c-d x^3\right )}-\frac{23 d \sqrt{c+d x^3}}{384 c x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 x^6 \left (8 c-d x^3\right )} \]

[Out]

(7*d^2*Sqrt[c + d*x^3])/(512*c^2*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(48*x^6*(8*c -
d*x^3)) - (23*d*Sqrt[c + d*x^3])/(384*c*x^3*(8*c - d*x^3)) + (15*d^2*ArcTanh[Sq
rt[c + d*x^3]/(3*Sqrt[c])])/(2048*c^(5/2)) - (17*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqr
t[c]])/(2048*c^(5/2))

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Rubi [A]  time = 0.498108, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{15 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2048 c^{5/2}}-\frac{17 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{5/2}}+\frac{7 d^2 \sqrt{c+d x^3}}{512 c^2 \left (8 c-d x^3\right )}-\frac{23 d \sqrt{c+d x^3}}{384 c x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 x^6 \left (8 c-d x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3)^(3/2)/(x^7*(8*c - d*x^3)^2),x]

[Out]

(7*d^2*Sqrt[c + d*x^3])/(512*c^2*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(48*x^6*(8*c -
d*x^3)) - (23*d*Sqrt[c + d*x^3])/(384*c*x^3*(8*c - d*x^3)) + (15*d^2*ArcTanh[Sq
rt[c + d*x^3]/(3*Sqrt[c])])/(2048*c^(5/2)) - (17*d^2*ArcTanh[Sqrt[c + d*x^3]/Sqr
t[c]])/(2048*c^(5/2))

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Rubi in Sympy [A]  time = 76.5204, size = 124, normalized size = 0.77 \[ \frac{3 \sqrt{c + d x^{3}}}{8 x^{6} \left (8 c - d x^{3}\right )} - \frac{19 \sqrt{c + d x^{3}}}{384 c x^{6}} - \frac{7 d \sqrt{c + d x^{3}}}{512 c^{2} x^{3}} + \frac{15 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{2048 c^{\frac{5}{2}}} - \frac{17 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{2048 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+c)**(3/2)/x**7/(-d*x**3+8*c)**2,x)

[Out]

3*sqrt(c + d*x**3)/(8*x**6*(8*c - d*x**3)) - 19*sqrt(c + d*x**3)/(384*c*x**6) -
7*d*sqrt(c + d*x**3)/(512*c**2*x**3) + 15*d**2*atanh(sqrt(c + d*x**3)/(3*sqrt(c)
))/(2048*c**(5/2)) - 17*d**2*atanh(sqrt(c + d*x**3)/sqrt(c))/(2048*c**(5/2))

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Mathematica [C]  time = 0.498654, size = 349, normalized size = 2.17 \[ \frac{\frac{\frac{170 c d^3 x^9 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}+32 c^3+124 c^2 d x^3+71 c d^2 x^6-21 d^3 x^9}{d x^3-8 c}+\frac{168 c d^3 x^9 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}}{1536 c^2 x^6 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(c + d*x^3)^(3/2)/(x^7*(8*c - d*x^3)^2),x]

[Out]

((168*c*d^3*x^9*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), (d*x^3)/(8*c)])/((8*c - d*x
^3)*(16*c*AppellF1[1, 1/2, 1, 2, -((d*x^3)/c), (d*x^3)/(8*c)] + d*x^3*(AppellF1[
2, 1/2, 2, 3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[2, 3/2, 1, 3, -((d*x^3)/
c), (d*x^3)/(8*c)]))) + (32*c^3 + 124*c^2*d*x^3 + 71*c*d^2*x^6 - 21*d^3*x^9 + (1
70*c*d^3*x^9*AppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), (8*c)/(d*x^3)])/(5*d*x^3*A
ppellF1[3/2, 1/2, 1, 5/2, -(c/(d*x^3)), (8*c)/(d*x^3)] + 16*c*AppellF1[5/2, 1/2,
2, 7/2, -(c/(d*x^3)), (8*c)/(d*x^3)] - c*AppellF1[5/2, 3/2, 1, 7/2, -(c/(d*x^3)
), (8*c)/(d*x^3)]))/(-8*c + d*x^3))/(1536*c^2*x^6*Sqrt[c + d*x^3])

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Maple [C]  time = 0.02, size = 1075, normalized size = 6.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^(3/2)/x^7/(-d*x^3+8*c)^2,x)

[Out]

1/64/c^2*(-1/6*c*(d*x^3+c)^(1/2)/x^6-5/12*d*(d*x^3+c)^(1/2)/x^3-1/4*d^2*arctanh(
(d*x^3+c)^(1/2)/c^(1/2))/c^(1/2))+3/4096/c^4*d^2*(2/9*d*x^3*(d*x^3+c)^(1/2)+8/9*
c*(d*x^3+c)^(1/2)-2/3*c^(3/2)*arctanh((d*x^3+c)^(1/2)/c^(1/2)))+1/256/c^3*d*(-1/
3*c*(d*x^3+c)^(1/2)/x^3+2/3*d*(d*x^3+c)^(1/2)-c^(1/2)*d*arctanh((d*x^3+c)^(1/2)/
c^(1/2)))+1/512*d^3/c^3*(-3*c/d*(d*x^3+c)^(1/2)/(d*x^3-8*c)+2/3*(d*x^3+c)^(1/2)/
d+1/2*I/d^3*2^(1/2)*sum((-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1
/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2
)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^
(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_
alpha*3^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-
(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/
d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*_alpha^2*(-c*d^2)
^(1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)
^(2/3)-3*c*d)/c,(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)
/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d-8*c)))-3/4096*d^3/c^4*(2/9*x^3*(
d*x^3+c)^(1/2)+56/9*c*(d*x^3+c)^(1/2)/d+3*I*c/d^3*2^(1/2)*sum((-c*d^2)^(1/3)*(1/
2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)
*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(
-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/
2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(
-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*
(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3)
)^(1/2),-1/18/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3
^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/c,(I*3^(1/2)/d*(-c*d^2)^(1/3
)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_
Z^3*d-8*c)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)^2*x^7),x, algorithm="maxima")

[Out]

integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)^2*x^7), x)

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Fricas [A]  time = 0.237815, size = 1, normalized size = 0.01 \[ \left [-\frac{8 \,{\left (21 \, d^{2} x^{6} - 92 \, c d x^{3} - 32 \, c^{2}\right )} \sqrt{d x^{3} + c} \sqrt{c} - 45 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 51 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{12288 \,{\left (c^{2} d x^{9} - 8 \, c^{3} x^{6}\right )} \sqrt{c}}, -\frac{4 \,{\left (21 \, d^{2} x^{6} - 92 \, c d x^{3} - 32 \, c^{2}\right )} \sqrt{d x^{3} + c} \sqrt{-c} + 45 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 51 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )}{6144 \,{\left (c^{2} d x^{9} - 8 \, c^{3} x^{6}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)^2*x^7),x, algorithm="fricas")

[Out]

[-1/12288*(8*(21*d^2*x^6 - 92*c*d*x^3 - 32*c^2)*sqrt(d*x^3 + c)*sqrt(c) - 45*(d^
3*x^9 - 8*c*d^2*x^6)*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c)/(d*x^3 -
8*c)) - 51*(d^3*x^9 - 8*c*d^2*x^6)*log(((d*x^3 + 2*c)*sqrt(c) - 2*sqrt(d*x^3 +
c)*c)/x^3))/((c^2*d*x^9 - 8*c^3*x^6)*sqrt(c)), -1/6144*(4*(21*d^2*x^6 - 92*c*d*x
^3 - 32*c^2)*sqrt(d*x^3 + c)*sqrt(-c) + 45*(d^3*x^9 - 8*c*d^2*x^6)*arctan(3*c/(s
qrt(d*x^3 + c)*sqrt(-c))) - 51*(d^3*x^9 - 8*c*d^2*x^6)*arctan(c/(sqrt(d*x^3 + c)
*sqrt(-c))))/((c^2*d*x^9 - 8*c^3*x^6)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**3+c)**(3/2)/x**7/(-d*x**3+8*c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220481, size = 161, normalized size = 1. \[ \frac{1}{6144} \, d^{2}{\left (\frac{51 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{45 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{36 \, \sqrt{d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} c^{2}} - \frac{16 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 2 \, \sqrt{d x^{3} + c} c\right )}}{c^{2} d^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)^2*x^7),x, algorithm="giac")

[Out]

1/6144*d^2*(51*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) - 45*arctan(1/3*s
qrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^2) - 36*sqrt(d*x^3 + c)/((d*x^3 - 8*c)*c^2)
- 16*(3*(d*x^3 + c)^(3/2) - 2*sqrt(d*x^3 + c)*c)/(c^2*d^2*x^6))