3.35 \(\int \left (a+b x^3\right )^2 \left (c+d x^3\right )^q \, dx\)

Optimal. Leaf size=167 \[ \frac{x \left (c+d x^3\right )^{q+1} \left (a^2 d^2 \left (9 q^2+33 q+28\right )-2 a b c d (3 q+7)+4 b^2 c^2\right ) \, _2F_1\left (1,q+\frac{4}{3};\frac{4}{3};-\frac{d x^3}{c}\right )}{c d^2 (3 q+4) (3 q+7)}-\frac{b x \left (c+d x^3\right )^{q+1} (4 b c-a d (3 q+10))}{d^2 (3 q+4) (3 q+7)}+\frac{b x \left (a+b x^3\right ) \left (c+d x^3\right )^{q+1}}{d (3 q+7)} \]

[Out]

-((b*(4*b*c - a*d*(10 + 3*q))*x*(c + d*x^3)^(1 + q))/(d^2*(4 + 3*q)*(7 + 3*q)))
+ (b*x*(a + b*x^3)*(c + d*x^3)^(1 + q))/(d*(7 + 3*q)) + ((4*b^2*c^2 - 2*a*b*c*d*
(7 + 3*q) + a^2*d^2*(28 + 33*q + 9*q^2))*x*(c + d*x^3)^(1 + q)*Hypergeometric2F1
[1, 4/3 + q, 4/3, -((d*x^3)/c)])/(c*d^2*(4 + 3*q)*(7 + 3*q))

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Rubi [A]  time = 0.278101, antiderivative size = 176, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} \left (a^2 d^2 \left (9 q^2+33 q+28\right )-2 a b c d (3 q+7)+4 b^2 c^2\right ) \, _2F_1\left (\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right )}{d^2 (3 q+4) (3 q+7)}-\frac{b x \left (c+d x^3\right )^{q+1} (4 b c-a d (3 q+10))}{d^2 (3 q+4) (3 q+7)}+\frac{b x \left (a+b x^3\right ) \left (c+d x^3\right )^{q+1}}{d (3 q+7)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^2*(c + d*x^3)^q,x]

[Out]

-((b*(4*b*c - a*d*(10 + 3*q))*x*(c + d*x^3)^(1 + q))/(d^2*(4 + 3*q)*(7 + 3*q)))
+ (b*x*(a + b*x^3)*(c + d*x^3)^(1 + q))/(d*(7 + 3*q)) + ((4*b^2*c^2 - 2*a*b*c*d*
(7 + 3*q) + a^2*d^2*(28 + 33*q + 9*q^2))*x*(c + d*x^3)^q*Hypergeometric2F1[1/3,
-q, 4/3, -((d*x^3)/c)])/(d^2*(4 + 3*q)*(7 + 3*q)*(1 + (d*x^3)/c)^q)

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Rubi in Sympy [A]  time = 33.0944, size = 153, normalized size = 0.92 \[ \frac{b x \left (a + b x^{3}\right ) \left (c + d x^{3}\right )^{q + 1}}{d \left (3 q + 7\right )} - \frac{b x \left (c + d x^{3}\right )^{q + 1} \left (- a d \left (3 q + 10\right ) + 4 b c\right )}{d^{2} \left (3 q + 4\right ) \left (3 q + 7\right )} + \frac{x \left (1 + \frac{d x^{3}}{c}\right )^{- q} \left (c + d x^{3}\right )^{q} \left (- a d \left (3 q + 4\right ) \left (- a d \left (3 q + 7\right ) + b c\right ) + b c \left (- a d \left (3 q + 10\right ) + 4 b c\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{d x^{3}}{c}} \right )}}{d^{2} \left (3 q + 4\right ) \left (3 q + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**2*(d*x**3+c)**q,x)

[Out]

b*x*(a + b*x**3)*(c + d*x**3)**(q + 1)/(d*(3*q + 7)) - b*x*(c + d*x**3)**(q + 1)
*(-a*d*(3*q + 10) + 4*b*c)/(d**2*(3*q + 4)*(3*q + 7)) + x*(1 + d*x**3/c)**(-q)*(
c + d*x**3)**q*(-a*d*(3*q + 4)*(-a*d*(3*q + 7) + b*c) + b*c*(-a*d*(3*q + 10) + 4
*b*c))*hyper((-q, 1/3), (4/3,), -d*x**3/c)/(d**2*(3*q + 4)*(3*q + 7))

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Mathematica [A]  time = 0.0799186, size = 106, normalized size = 0.63 \[ \frac{1}{14} x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} \left (14 a^2 \, _2F_1\left (\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right )+b x^3 \left (7 a \, _2F_1\left (\frac{4}{3},-q;\frac{7}{3};-\frac{d x^3}{c}\right )+2 b x^3 \, _2F_1\left (\frac{7}{3},-q;\frac{10}{3};-\frac{d x^3}{c}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^2*(c + d*x^3)^q,x]

[Out]

(x*(c + d*x^3)^q*(14*a^2*Hypergeometric2F1[1/3, -q, 4/3, -((d*x^3)/c)] + b*x^3*(
7*a*Hypergeometric2F1[4/3, -q, 7/3, -((d*x^3)/c)] + 2*b*x^3*Hypergeometric2F1[7/
3, -q, 10/3, -((d*x^3)/c)])))/(14*(1 + (d*x^3)/c)^q)

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{2} \left ( d{x}^{3}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2*(d*x^3+c)^q,x)

[Out]

int((b*x^3+a)^2*(d*x^3+c)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{2}{\left (d x^{3} + c\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*(d*x^3 + c)^q,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^2*(d*x^3 + c)^q, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}{\left (d x^{3} + c\right )}^{q}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*(d*x^3 + c)^q,x, algorithm="fricas")

[Out]

integral((b^2*x^6 + 2*a*b*x^3 + a^2)*(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)**2*(d*x**3+c)**q,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{2}{\left (d x^{3} + c\right )}^{q}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*(d*x^3 + c)^q,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^2*(d*x^3 + c)^q, x)