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

Optimal. Leaf size=79 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (c+d x^3\right )^p \left (\frac{d x^3}{c}+1\right )^{-p} F_1\left (\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right ) \]

[Out]

(x*(a + b*x^3)^m*(c + d*x^3)^p*AppellF1[1/3, -m, -p, 4/3, -((b*x^3)/a), -((d*x^3
)/c)])/((1 + (b*x^3)/a)^m*(1 + (d*x^3)/c)^p)

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Rubi [A]  time = 0.106559, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (c+d x^3\right )^p \left (\frac{d x^3}{c}+1\right )^{-p} F_1\left (\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^3)^m*(c + d*x^3)^p*AppellF1[1/3, -m, -p, 4/3, -((b*x^3)/a), -((d*x^3
)/c)])/((1 + (b*x^3)/a)^m*(1 + (d*x^3)/c)^p)

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Rubi in Sympy [A]  time = 21.1481, size = 61, normalized size = 0.77 \[ x \left (1 + \frac{b x^{3}}{a}\right )^{- m} \left (1 + \frac{d x^{3}}{c}\right )^{- p} \left (a + b x^{3}\right )^{m} \left (c + d x^{3}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{3},- m,- p,\frac{4}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + b*x**3/a)**(-m)*(1 + d*x**3/c)**(-p)*(a + b*x**3)**m*(c + d*x**3)**p*appe
llf1(1/3, -m, -p, 4/3, -b*x**3/a, -d*x**3/c)

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Mathematica [B]  time = 0.355429, size = 172, normalized size = 2.18 \[ \frac{4 a c x \left (a+b x^3\right )^m \left (c+d x^3\right )^p F_1\left (\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{3 x^3 \left (b c m F_1\left (\frac{4}{3};1-m,-p;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+a d p F_1\left (\frac{4}{3};-m,1-p;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )+4 a c F_1\left (\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(4*a*c*x*(a + b*x^3)^m*(c + d*x^3)^p*AppellF1[1/3, -m, -p, 4/3, -((b*x^3)/a), -(
(d*x^3)/c)])/(4*a*c*AppellF1[1/3, -m, -p, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 3*x
^3*(b*c*m*AppellF1[4/3, 1 - m, -p, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + a*d*p*Appe
llF1[4/3, -m, 1 - p, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))

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Maple [F]  time = 0.114, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{m} \left ( d{x}^{3}+c \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^m*(d*x^3+c)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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