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

Optimal. Leaf size=93 \[ \frac{d x \left (a+b x^3\right )^{m+1}}{b (3 m+4)}-\frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} (a d-b c (3 m+4)) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )}{b (3 m+4)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.099899, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (c-\frac{a d}{3 b m+4 b}\right ) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )+\frac{d x \left (a+b x^3\right )^{m+1}}{b (3 m+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.7156, size = 73, normalized size = 0.78 \[ \frac{d x \left (a + b x^{3}\right )^{m + 1}}{b \left (3 m + 4\right )} - \frac{x \left (1 + \frac{b x^{3}}{a}\right )^{- m} \left (a + b x^{3}\right )^{m} \left (a d - b c \left (3 m + 4\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{b \left (3 m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0298675, size = 75, normalized size = 0.81 \[ \frac{1}{4} x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (4 c \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )+d x^3 \, _2F_1\left (\frac{4}{3},-m;\frac{7}{3};-\frac{b x^3}{a}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{m} \left ( d{x}^{3}+c \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^m*(d*x^3+c),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x^3 + c)*(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),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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