3.42 \(\int \frac{1}{(c+d x)^3 (a+b x^3)^{2/3}} \, dx\)
Optimal. Leaf size=1357 \[ \text{result too large to display} \]
[Out]
(3*c^4*d^2*(a + b*x^3)^(1/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) + (3*c^2*d^4*x^2*(a + b*x^3)^(1/3))/(2*(b*
c^3 - a*d^3)*(c^3 + d^3*x^3)^2) + (5*b*c^4*d^2*(a + b*x^3)^(1/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (c*d
^2*(b*c^3 - 6*a*d^3)*(a + b*x^3)^(1/3))/(6*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (d^4*(9*b*c^3 - 4*a*d^3)*x^2*(
a + b*x^3)^(1/3))/(6*c*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (d^4*(3*b*c^3 + 2*a*d^3)*x^2*(a + b*x^3)^(1/3))/(3
*c*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (x*(1 + (b*x^3)/a)^(2/3)*AppellF1[1/3, 2/3, 3, 4/3, -((b*x^3)/a), -((d
^3*x^3)/c^3)])/(c^3*(a + b*x^3)^(2/3)) - (7*d^3*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 3, 7/3, -((b*x^3)
/a), -((d^3*x^3)/c^3)])/(4*c^6*(a + b*x^3)^(2/3)) + (d^6*x^7*(1 + (b*x^3)/a)^(2/3)*AppellF1[7/3, 2/3, 3, 10/3,
-((b*x^3)/a), -((d^3*x^3)/c^3)])/(7*c^9*(a + b*x^3)^(2/3)) + (2*a*d^4*(6*b*c^3 - a*d^3)*ArcTan[(1 + (2*(b*c^3
- a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c^2*(b*c^3 - a*d^3)^(8/3)) + (d*(9*b^2*c^6 - 6*
a*b*c^3*d^3 + 2*a^2*d^6)*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c
^2*(b*c^3 - a*d^3)^(8/3)) - (10*b^2*c^4*d*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]])
/(3*Sqrt[3]*(b*c^3 - a*d^3)^(8/3)) + (b*c*d*(b*c^3 - 6*a*d^3)*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d
^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*(b*c^3 - a*d^3)^(8/3)) - (5*b^2*c^4*d*Log[c^3 + d^3*x^3])/(9*(b*c^3 - a*d^3)^(
8/3)) + (b*c*d*(b*c^3 - 6*a*d^3)*Log[c^3 + d^3*x^3])/(18*(b*c^3 - a*d^3)^(8/3)) - (a*d^4*(6*b*c^3 - a*d^3)*Log
[c^3 + d^3*x^3])/(9*c^2*(b*c^3 - a*d^3)^(8/3)) - (d*(9*b^2*c^6 - 6*a*b*c^3*d^3 + 2*a^2*d^6)*Log[c^3 + d^3*x^3]
)/(18*c^2*(b*c^3 - a*d^3)^(8/3)) + (a*d^4*(6*b*c^3 - a*d^3)*Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3
)])/(3*c^2*(b*c^3 - a*d^3)^(8/3)) + (d*(9*b^2*c^6 - 6*a*b*c^3*d^3 + 2*a^2*d^6)*Log[((b*c^3 - a*d^3)^(1/3)*x)/c
- (a + b*x^3)^(1/3)])/(6*c^2*(b*c^3 - a*d^3)^(8/3)) + (5*b^2*c^4*d*Log[(b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^
(1/3)])/(3*(b*c^3 - a*d^3)^(8/3)) - (b*c*d*(b*c^3 - 6*a*d^3)*Log[(b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)])
/(6*(b*c^3 - a*d^3)^(8/3))
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Rubi [F] time = 0.0831781, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((c + d*x)^3*(a + b*x^3)^(2/3)),x]
[Out]
Defer[Int][1/((c + d*x)^3*(a + b*x^3)^(2/3)), x]
Rubi steps
\begin{align*} \int \frac{1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx &=\int \frac{1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx\\ \end{align*}
Mathematica [F] time = 0.43289, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/((c + d*x)^3*(a + b*x^3)^(2/3)),x]
[Out]
Integrate[1/((c + d*x)^3*(a + b*x^3)^(2/3)), x]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{3}} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(d*x+c)^3/(b*x^3+a)^(2/3),x)
[Out]
int(1/(d*x+c)^3/(b*x^3+a)^(2/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)^3/(b*x^3+a)^(2/3),x, algorithm="maxima")
[Out]
integrate(1/((b*x^3 + a)^(2/3)*(d*x + c)^3), x)
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)^3/(b*x^3+a)^(2/3),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{3}\right )^{\frac{2}{3}} \left (c + d x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)**3/(b*x**3+a)**(2/3),x)
[Out]
Integral(1/((a + b*x**3)**(2/3)*(c + d*x)**3), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)^3/(b*x^3+a)^(2/3),x, algorithm="giac")
[Out]
integrate(1/((b*x^3 + a)^(2/3)*(d*x + c)^3), x)