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

Optimal. Leaf size=298 \[ \frac{d x \left (a+b x^3\right )^{m+1} \left (28 a^2 d^2-a b c d (15 m+92)+b^2 c^2 \left (9 m^2+60 m+118\right )\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (28 a^3 d^3-12 a^2 b c d^2 (3 m+10)+3 a b^2 c^2 d \left (9 m^2+51 m+70\right )-b^3 c^3 \left (27 m^3+189 m^2+414 m+280\right )\right ) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac{d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1} (7 a d-b c (3 m+16))}{b^2 (3 m+7) (3 m+10)}+\frac{d x \left (c+d x^3\right )^2 \left (a+b x^3\right )^{m+1}}{b (3 m+10)} \]

[Out]

(d*(28*a^2*d^2 - a*b*c*d*(92 + 15*m) + b^2*c^2*(118 + 60*m + 9*m^2))*x*(a + b*x^
3)^(1 + m))/(b^3*(4 + 3*m)*(7 + 3*m)*(10 + 3*m)) - (d*(7*a*d - b*c*(16 + 3*m))*x
*(a + b*x^3)^(1 + m)*(c + d*x^3))/(b^2*(7 + 3*m)*(10 + 3*m)) + (d*x*(a + b*x^3)^
(1 + m)*(c + d*x^3)^2)/(b*(10 + 3*m)) - ((28*a^3*d^3 - 12*a^2*b*c*d^2*(10 + 3*m)
+ 3*a*b^2*c^2*d*(70 + 51*m + 9*m^2) - b^3*c^3*(280 + 414*m + 189*m^2 + 27*m^3))
*x*(a + b*x^3)^m*Hypergeometric2F1[1/3, -m, 4/3, -((b*x^3)/a)])/(b^3*(4 + 3*m)*(
7 + 3*m)*(10 + 3*m)*(1 + (b*x^3)/a)^m)

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Rubi [A]  time = 0.689545, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{d x \left (a+b x^3\right )^{m+1} \left (28 a^2 d^2-a b c d (15 m+92)+b^2 c^2 \left (9 m^2+60 m+118\right )\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (28 a^3 d^3-12 a^2 b c d^2 (3 m+10)+3 a b^2 c^2 d \left (9 m^2+51 m+70\right )-b^3 c^3 \left (27 m^3+189 m^2+414 m+280\right )\right ) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac{d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1} (7 a d-b c (3 m+16))}{b^2 (3 m+7) (3 m+10)}+\frac{d x \left (c+d x^3\right )^2 \left (a+b x^3\right )^{m+1}}{b (3 m+10)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(28*a^2*d^2 - a*b*c*d*(92 + 15*m) + b^2*c^2*(118 + 60*m + 9*m^2))*x*(a + b*x^
3)^(1 + m))/(b^3*(4 + 3*m)*(7 + 3*m)*(10 + 3*m)) - (d*(7*a*d - b*c*(16 + 3*m))*x
*(a + b*x^3)^(1 + m)*(c + d*x^3))/(b^2*(7 + 3*m)*(10 + 3*m)) + (d*x*(a + b*x^3)^
(1 + m)*(c + d*x^3)^2)/(b*(10 + 3*m)) - ((28*a^3*d^3 - 12*a^2*b*c*d^2*(10 + 3*m)
+ 3*a*b^2*c^2*d*(70 + 51*m + 9*m^2) - b^3*c^3*(280 + 414*m + 189*m^2 + 27*m^3))
*x*(a + b*x^3)^m*Hypergeometric2F1[1/3, -m, 4/3, -((b*x^3)/a)])/(b^3*(4 + 3*m)*(
7 + 3*m)*(10 + 3*m)*(1 + (b*x^3)/a)^m)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.103983, size = 137, normalized size = 0.46 \[ \frac{1}{140} x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (140 c^3 \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )+d x^3 \left (105 c^2 \, _2F_1\left (\frac{4}{3},-m;\frac{7}{3};-\frac{b x^3}{a}\right )+2 d x^3 \left (30 c \, _2F_1\left (\frac{7}{3},-m;\frac{10}{3};-\frac{b x^3}{a}\right )+7 d x^3 \, _2F_1\left (\frac{10}{3},-m;\frac{13}{3};-\frac{b x^3}{a}\right )\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^3)^m*(140*c^3*Hypergeometric2F1[1/3, -m, 4/3, -((b*x^3)/a)] + d*x^3*
(105*c^2*Hypergeometric2F1[4/3, -m, 7/3, -((b*x^3)/a)] + 2*d*x^3*(30*c*Hypergeom
etric2F1[7/3, -m, 10/3, -((b*x^3)/a)] + 7*d*x^3*Hypergeometric2F1[10/3, -m, 13/3
, -((b*x^3)/a)]))))/(140*(1 + (b*x^3)/a)^m)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^m*(d*x^3+c)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d^{3} x^{9} + 3 \, c d^{2} x^{6} + 3 \, c^{2} d x^{3} + c^{3}\right )}{\left (b x^{3} + a\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d^3*x^9 + 3*c*d^2*x^6 + 3*c^2*d*x^3 + c^3)*(b*x^3 + a)^m, 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)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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