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

Optimal. Leaf size=176 \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (4 a^2 d^2-2 a b c d (3 m+7)+b^2 c^2 \left (9 m^2+33 m+28\right )\right ) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )}{b^2 (3 m+4) (3 m+7)}-\frac{d x \left (a+b x^3\right )^{m+1} (4 a d-b c (3 m+10))}{b^2 (3 m+4) (3 m+7)}+\frac{d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1}}{b (3 m+7)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.28185, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (4 a^2 d^2-2 a b c d (3 m+7)+b^2 c^2 \left (9 m^2+33 m+28\right )\right ) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )}{b^2 (3 m+4) (3 m+7)}-\frac{d x \left (a+b x^3\right )^{m+1} (4 a d-b c (3 m+10))}{b^2 (3 m+4) (3 m+7)}+\frac{d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1}}{b (3 m+7)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.2166, size = 153, normalized size = 0.87 \[ \frac{d x \left (a + b x^{3}\right )^{m + 1} \left (c + d x^{3}\right )}{b \left (3 m + 7\right )} - \frac{d x \left (a + b x^{3}\right )^{m + 1} \left (4 a d - b c \left (3 m + 10\right )\right )}{b^{2} \left (3 m + 4\right ) \left (3 m + 7\right )} + \frac{x \left (1 + \frac{b x^{3}}{a}\right )^{- m} \left (a + b x^{3}\right )^{m} \left (a d \left (4 a d - b c \left (3 m + 10\right )\right ) - b c \left (3 m + 4\right ) \left (a d - b c \left (3 m + 7\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{b^{2} \left (3 m + 4\right ) \left (3 m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0568683, size = 106, normalized size = 0.6 \[ \frac{1}{14} x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (14 c^2 \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )+d x^3 \left (7 c \, _2F_1\left (\frac{4}{3},-m;\frac{7}{3};-\frac{b x^3}{a}\right )+2 d x^3 \, _2F_1\left (\frac{7}{3},-m;\frac{10}{3};-\frac{b x^3}{a}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{m} \left ( d{x}^{3}+c \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^m*(d*x^3+c)^2,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)**m*(d*x**3+c)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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