3.35 \(\int \frac{1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx\)
Optimal. Leaf size=1513 \[ \text{result too large to display} \]
[Out]
(3*c^4*d^2*(a + b*x^3)^(2/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) - (3*c^3*d^3*x*(a + b*x^3)^(2/3))/(2*(b*c^
3 - a*d^3)*(c^3 + d^3*x^3)^2) + (4*b*c^4*d^2*(a + b*x^3)^(2/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (c*d^2
*(b*c^3 - 3*a*d^3)*(a + b*x^3)^(2/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (d^3*(3*b*c^3 - 7*a*d^3)*x*(a +
b*x^3)^(2/3))/(18*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (d^3*(9*b*c^3 - 5*a*d^3)*x*(a + b*x^3)^(2/3))/(18*(b*c^
3 - a*d^3)^2*(c^3 + d^3*x^3)) - (7*d^3*(3*b*c^3 + a*d^3)*x*(a + b*x^3)^(2/3))/(18*(b*c^3 - a*d^3)^2*(c^3 + d^3
*x^3)) - (3*d*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 3, 5/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(2*c^4*(a
+ b*x^3)^(1/3)) + (6*d^4*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 3, 8/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])
/(5*c^7*(a + b*x^3)^(1/3)) + (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]
])/(9*Sqrt[3]*c^2*(b*c^3 - a*d^3)^(7/3)) + (7*a*d^3*(3*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]])/(9*Sqrt[3]*c^2*(b*c^3 - a*d^3)^(7/3)) + ((9*b^2*c^6 - 12*a*b*c^3*d^3 + 5*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]])/(9*Sqrt[3]*c^2*(b*c^3 - a*d^3)^(
7/3)) - (4*b^2*c^4*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)^(7/3)) + (b*c*(b*c^3 - 3*a*d^3)*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]])/(3*S
qrt[3]*(b*c^3 - a*d^3)^(7/3)) + (2*b^2*c^4*Log[c^3 + d^3*x^3])/(9*(b*c^3 - a*d^3)^(7/3)) + (a^2*d^6*Log[c^3 +
d^3*x^3])/(27*c^2*(b*c^3 - a*d^3)^(7/3)) - (b*c*(b*c^3 - 3*a*d^3)*Log[c^3 + d^3*x^3])/(18*(b*c^3 - a*d^3)^(7/3
)) + (7*a*d^3*(3*b*c^3 - a*d^3)*Log[c^3 + d^3*x^3])/(54*c^2*(b*c^3 - a*d^3)^(7/3)) + ((9*b^2*c^6 - 12*a*b*c^3*
d^3 + 5*a^2*d^6)*Log[c^3 + d^3*x^3])/(54*c^2*(b*c^3 - a*d^3)^(7/3)) - (a^2*d^6*Log[((b*c^3 - a*d^3)^(1/3)*x)/c
- (a + b*x^3)^(1/3)])/(9*c^2*(b*c^3 - a*d^3)^(7/3)) - (7*a*d^3*(3*b*c^3 - a*d^3)*Log[((b*c^3 - a*d^3)^(1/3)*x
)/c - (a + b*x^3)^(1/3)])/(18*c^2*(b*c^3 - a*d^3)^(7/3)) - ((9*b^2*c^6 - 12*a*b*c^3*d^3 + 5*a^2*d^6)*Log[((b*c
^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)])/(18*c^2*(b*c^3 - a*d^3)^(7/3)) - (2*b^2*c^4*Log[(b*c^3 - a*d^3)^(
1/3) + d*(a + b*x^3)^(1/3)])/(3*(b*c^3 - a*d^3)^(7/3)) + (b*c*(b*c^3 - 3*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)^(7/3))
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Rubi [F] time = 0.0825825, 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 = {}
\[ \int \frac{1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((c + d*x)^3*(a + b*x^3)^(1/3)),x]
[Out]
Defer[Int][1/((c + d*x)^3*(a + b*x^3)^(1/3)), x]
Rubi steps
\begin{align*} \int \frac{1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx &=\int \frac{1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx\\ \end{align*}
Mathematica [F] time = 0.475202, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/((c + d*x)^3*(a + b*x^3)^(1/3)),x]
[Out]
Integrate[1/((c + d*x)^3*(a + b*x^3)^(1/3)), 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}}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(d*x+c)^3/(b*x^3+a)^(1/3),x)
[Out]
int(1/(d*x+c)^3/(b*x^3+a)^(1/3),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{1}{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)^(1/3),x, algorithm="maxima")
[Out]
integrate(1/((b*x^3 + a)^(1/3)*(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)^(1/3),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{a + b x^{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)**(1/3),x)
[Out]
Integral(1/((a + b*x**3)**(1/3)*(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{1}{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)^(1/3),x, algorithm="giac")
[Out]
integrate(1/((b*x^3 + a)^(1/3)*(d*x + c)^3), x)