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

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*Hyp
ergeometric2F1[1/3, -m, 4/3, -((b*x^3)/a)])/(b^2*(4 + 3*m)*(7 + 3*m)*(1 + (b*x^3)/a)^m)

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Rubi [A]  time = 0.125731, 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, Rules used = {416, 388, 246, 245} \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (4 a^2 d^2-2 a b c d (3 m+7)+b^2 c^2 \left (9 m^2+33 m+28\right )\right ) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )}{b^2 (3 m+4) (3 m+7)}-\frac{d x \left (a+b x^3\right )^{m+1} (4 a d-b c (3 m+10))}{b^2 (3 m+4) (3 m+7)}+\frac{d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1}}{b (3 m+7)} \]

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*Hyp
ergeometric2F1[1/3, -m, 4/3, -((b*x^3)/a)])/(b^2*(4 + 3*m)*(7 + 3*m)*(1 + (b*x^3)/a)^m)

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (a+b x^3\right )^m \left (c+d x^3\right )^2 \, dx &=\frac{d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b (7+3 m)}+\frac{\int \left (a+b x^3\right )^m \left (-c (a d-b c (7+3 m))-d (4 a d-b c (10+3 m)) x^3\right ) \, dx}{b (7+3 m)}\\ &=-\frac{d (4 a d-b c (10+3 m)) x \left (a+b x^3\right )^{1+m}}{b^2 (4+3 m) (7+3 m)}+\frac{d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b (7+3 m)}+\frac{\left (4 a^2 d^2-2 a b c d (7+3 m)+b^2 c^2 \left (28+33 m+9 m^2\right )\right ) \int \left (a+b x^3\right )^m \, dx}{b^2 (4+3 m) (7+3 m)}\\ &=-\frac{d (4 a d-b c (10+3 m)) x \left (a+b x^3\right )^{1+m}}{b^2 (4+3 m) (7+3 m)}+\frac{d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b (7+3 m)}+\frac{\left (\left (4 a^2 d^2-2 a b c d (7+3 m)+b^2 c^2 \left (28+33 m+9 m^2\right )\right ) \left (a+b x^3\right )^m \left (1+\frac{b x^3}{a}\right )^{-m}\right ) \int \left (1+\frac{b x^3}{a}\right )^m \, dx}{b^2 (4+3 m) (7+3 m)}\\ &=-\frac{d (4 a d-b c (10+3 m)) x \left (a+b x^3\right )^{1+m}}{b^2 (4+3 m) (7+3 m)}+\frac{d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b (7+3 m)}+\frac{\left (4 a^2 d^2-2 a b c d (7+3 m)+b^2 c^2 \left (28+33 m+9 m^2\right )\right ) x \left (a+b x^3\right )^m \left (1+\frac{b x^3}{a}\right )^{-m} \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )}{b^2 (4+3 m) (7+3 m)}\\ \end{align*}

Mathematica [A]  time = 5.04246, 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)

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Maple [F]  time = 0.398, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{3}+a \right ) ^{m} \left ( d{x}^{3}+c \right ) ^{2}\, dx \end{align*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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