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

Optimal. Leaf size=164 \[ \frac{23 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{18432 c^{7/2}}-\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{7/2}}+\frac{5 d^2 \sqrt{c+d x^3}}{1536 c^3 \left (8 c-d x^3\right )}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )} \]

[Out]

(5*d^2*Sqrt[c + d*x^3])/(1536*c^3*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(48*c*x^6*(8*
c - d*x^3)) - (7*d*Sqrt[c + d*x^3])/(384*c^2*x^3*(8*c - d*x^3)) + (23*d^2*ArcTan
h[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(18432*c^(7/2)) - (d^2*ArcTanh[Sqrt[c + d*x^3]/S
qrt[c]])/(2048*c^(7/2))

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Rubi [A]  time = 0.502961, antiderivative size = 164, 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 \[ \frac{23 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{18432 c^{7/2}}-\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{7/2}}+\frac{5 d^2 \sqrt{c+d x^3}}{1536 c^3 \left (8 c-d x^3\right )}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(5*d^2*Sqrt[c + d*x^3])/(1536*c^3*(8*c - d*x^3)) - Sqrt[c + d*x^3]/(48*c*x^6*(8*
c - d*x^3)) - (7*d*Sqrt[c + d*x^3])/(384*c^2*x^3*(8*c - d*x^3)) + (23*d^2*ArcTan
h[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(18432*c^(7/2)) - (d^2*ArcTanh[Sqrt[c + d*x^3]/S
qrt[c]])/(2048*c^(7/2))

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Rubi in Sympy [A]  time = 70.0886, size = 122, normalized size = 0.74 \[ \frac{\sqrt{c + d x^{3}}}{24 c x^{6} \left (8 c - d x^{3}\right )} - \frac{\sqrt{c + d x^{3}}}{128 c^{2} x^{6}} - \frac{5 d \sqrt{c + d x^{3}}}{1536 c^{3} x^{3}} + \frac{23 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{18432 c^{\frac{7}{2}}} - \frac{d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{c}} \right )}}{2048 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+c)**(1/2)/x**7/(-d*x**3+8*c)**2,x)

[Out]

sqrt(c + d*x**3)/(24*c*x**6*(8*c - d*x**3)) - sqrt(c + d*x**3)/(128*c**2*x**6) -
5*d*sqrt(c + d*x**3)/(1536*c**3*x**3) + 23*d**2*atanh(sqrt(c + d*x**3)/(3*sqrt(
c)))/(18432*c**(7/2)) - d**2*atanh(sqrt(c + d*x**3)/sqrt(c))/(2048*c**(7/2))

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Mathematica [C]  time = 0.460603, size = 349, normalized size = 2.13 \[ \frac{\frac{\frac{10 c d^3 x^9 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}{5 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )+16 c F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )-c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^3},\frac{8 c}{d x^3}\right )}+32 c^3+60 c^2 d x^3+23 c d^2 x^6-5 d^3 x^9}{d x^3-8 c}+\frac{40 c d^3 x^9 F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (d x^3 \left (F_1\left (2;\frac{1}{2},2;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (2;\frac{3}{2},1;3;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+16 c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}}{1536 c^3 x^6 \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((40*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 + 60*c^2*d*x^3 + 23*c*d^2*x^6 - 5*d^3*x^9 + (10*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*Appel
lF1[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^3*x^6*Sqrt[c + d*x^3])

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Maple [C]  time = 0.02, size = 1020, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^(1/2)/x^7/(-d*x^3+8*c)^2,x)

[Out]

1/64/c^2*(-1/6*(d*x^3+c)^(1/2)/x^6-1/12*d*(d*x^3+c)^(1/2)/c/x^3+1/12*d^2*arctanh
((d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))+1/256/c^3*d*(-1/3*(d*x^3+c)^(1/2)/x^3-1/3*d*a
rctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(1/2))+3/4096/c^4*d^2*(2/3*(d*x^3+c)^(1/2)-2/3
*arctanh((d*x^3+c)^(1/2)/c^(1/2))*c^(1/2))+1/512*d^3/c^3*(-1/3/d*(d*x^3+c)^(1/2)
/(d*x^3-8*c)+1/54*I/d^3/c*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*_al
pha^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*(2/3*(d*x^3+c)^(1/2)/d+1/3*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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232993, size = 1, normalized size = 0.01 \[ \left [-\frac{24 \,{\left (5 \, d^{2} x^{6} - 28 \, c d x^{3} - 32 \, c^{2}\right )} \sqrt{d x^{3} + c} \sqrt{c} - 23 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 9 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \log \left (\frac{{\left (d x^{3} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{3} + c} c}{x^{3}}\right )}{36864 \,{\left (c^{3} d x^{9} - 8 \, c^{4} x^{6}\right )} \sqrt{c}}, -\frac{12 \,{\left (5 \, d^{2} x^{6} - 28 \, c d x^{3} - 32 \, c^{2}\right )} \sqrt{d x^{3} + c} \sqrt{-c} + 23 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 9 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \arctan \left (\frac{c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right )}{18432 \,{\left (c^{3} d x^{9} - 8 \, c^{4} x^{6}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/36864*(24*(5*d^2*x^6 - 28*c*d*x^3 - 32*c^2)*sqrt(d*x^3 + c)*sqrt(c) - 23*(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)) - 9*(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^3*d*x^9 - 8*c^4*x^6)*sqrt(c)), -1/18432*(12*(5*d^2*x^6 - 28*c*d*x
^3 - 32*c^2)*sqrt(d*x^3 + c)*sqrt(-c) + 23*(d^3*x^9 - 8*c*d^2*x^6)*arctan(3*c/(s
qrt(d*x^3 + c)*sqrt(-c))) - 9*(d^3*x^9 - 8*c*d^2*x^6)*arctan(c/(sqrt(d*x^3 + c)*
sqrt(-c))))/((c^3*d*x^9 - 8*c^4*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)**(1/2)/x**7/(-d*x**3+8*c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225429, size = 140, normalized size = 0.85 \[ \frac{1}{18432} \, d^{2}{\left (\frac{9 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} - \frac{23 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{3}} - \frac{12 \, \sqrt{d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} c^{3}} - \frac{48 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{c^{3} d^{2} x^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18432*d^2*(9*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) - 23*arctan(1/3*s
qrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^3) - 12*sqrt(d*x^3 + c)/((d*x^3 - 8*c)*c^3)
- 48*(d*x^3 + c)^(3/2)/(c^3*d^2*x^6))