3.36 \(\int \left (a+b x^3\right ) \left (c+d x^3\right )^q \, dx\)

Optimal. Leaf size=84 \[ \frac{b x \left (c+d x^3\right )^{q+1}}{d (3 q+4)}-\frac{x \left (c+d x^3\right )^{q+1} (b c-a d (3 q+4)) \, _2F_1\left (1,q+\frac{4}{3};\frac{4}{3};-\frac{d x^3}{c}\right )}{c d (3 q+4)} \]

[Out]

(b*x*(c + d*x^3)^(1 + q))/(d*(4 + 3*q)) - ((b*c - a*d*(4 + 3*q))*x*(c + d*x^3)^(
1 + q)*Hypergeometric2F1[1, 4/3 + q, 4/3, -((d*x^3)/c)])/(c*d*(4 + 3*q))

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Rubi [A]  time = 0.0984101, antiderivative size = 85, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} \left (a-\frac{b c}{3 d q+4 d}\right ) \, _2F_1\left (\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right )+\frac{b x \left (c+d x^3\right )^{q+1}}{d (3 q+4)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)*(c + d*x^3)^q,x]

[Out]

(b*x*(c + d*x^3)^(1 + q))/(d*(4 + 3*q)) + ((a - (b*c)/(4*d + 3*d*q))*x*(c + d*x^
3)^q*Hypergeometric2F1[1/3, -q, 4/3, -((d*x^3)/c)])/(1 + (d*x^3)/c)^q

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Rubi in Sympy [A]  time = 11.049, size = 73, normalized size = 0.87 \[ \frac{b x \left (c + d x^{3}\right )^{q + 1}}{d \left (3 q + 4\right )} - \frac{x \left (1 + \frac{d x^{3}}{c}\right )^{- q} \left (c + d x^{3}\right )^{q} \left (- a d \left (3 q + 4\right ) + b c\right ){{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{d x^{3}}{c}} \right )}}{d \left (3 q + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)*(d*x**3+c)**q,x)

[Out]

b*x*(c + d*x**3)**(q + 1)/(d*(3*q + 4)) - x*(1 + d*x**3/c)**(-q)*(c + d*x**3)**q
*(-a*d*(3*q + 4) + b*c)*hyper((-q, 1/3), (4/3,), -d*x**3/c)/(d*(3*q + 4))

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Mathematica [A]  time = 0.0312364, size = 75, normalized size = 0.89 \[ \frac{1}{4} x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} \left (4 a \, _2F_1\left (\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right )+b x^3 \, _2F_1\left (\frac{4}{3},-q;\frac{7}{3};-\frac{d x^3}{c}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)*(c + d*x^3)^q,x]

[Out]

(x*(c + d*x^3)^q*(4*a*Hypergeometric2F1[1/3, -q, 4/3, -((d*x^3)/c)] + b*x^3*Hype
rgeometric2F1[4/3, -q, 7/3, -((d*x^3)/c)]))/(4*(1 + (d*x^3)/c)^q)

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) \left ( d{x}^{3}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(d*x^3+c)^q,x)

[Out]

int((b*x^3+a)*(d*x^3+c)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)*(d*x^3 + c)^q,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)*(d*x^3 + c)^q, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{q}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)*(d*x^3 + c)^q,x, algorithm="fricas")

[Out]

integral((b*x^3 + a)*(d*x^3 + c)^q, x)

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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((b*x**3+a)*(d*x**3+c)**q,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)*(d*x^3 + c)^q,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)*(d*x^3 + c)^q, x)