3.75 \(\int \frac {(c+d x^3)^2}{(a+b x^3)^{10/3}} \, dx\)

Optimal. Leaf size=78 \[ \frac {9 c^2 x}{14 a^3 \sqrt [3]{a+b x^3}}+\frac {3 c x \left (c+d x^3\right )}{14 a^2 \left (a+b x^3\right )^{4/3}}+\frac {x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}} \]

[Out]

9/14*c^2*x/a^3/(b*x^3+a)^(1/3)+3/14*c*x*(d*x^3+c)/a^2/(b*x^3+a)^(4/3)+1/7*x*(d*x^3+c)^2/a/(b*x^3+a)^(7/3)

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Rubi [A]  time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {378, 191} \[ \frac {9 c^2 x}{14 a^3 \sqrt [3]{a+b x^3}}+\frac {3 c x \left (c+d x^3\right )}{14 a^2 \left (a+b x^3\right )^{4/3}}+\frac {x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^2/(a + b*x^3)^(10/3),x]

[Out]

(9*c^2*x)/(14*a^3*(a + b*x^3)^(1/3)) + (3*c*x*(c + d*x^3))/(14*a^2*(a + b*x^3)^(4/3)) + (x*(c + d*x^3)^2)/(7*a
*(a + b*x^3)^(7/3))

Rule 191

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

Rule 378

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

Rubi steps

\begin {align*} \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{10/3}} \, dx &=\frac {x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}+\frac {(6 c) \int \frac {c+d x^3}{\left (a+b x^3\right )^{7/3}} \, dx}{7 a}\\ &=\frac {3 c x \left (c+d x^3\right )}{14 a^2 \left (a+b x^3\right )^{4/3}}+\frac {x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}+\frac {\left (9 c^2\right ) \int \frac {1}{\left (a+b x^3\right )^{4/3}} \, dx}{14 a^2}\\ &=\frac {9 c^2 x}{14 a^3 \sqrt [3]{a+b x^3}}+\frac {3 c x \left (c+d x^3\right )}{14 a^2 \left (a+b x^3\right )^{4/3}}+\frac {x \left (c+d x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 126, normalized size = 1.62 \[ \frac {x \sqrt [3]{\frac {b x^3}{a}+1} \left (a^2 \left (14 c^2+7 c d x^3+2 d^2 x^6\right )+3 a b c x^3 \left (7 c+d x^3\right )+9 b^2 c^2 x^6\right )}{14 a^3 \left (a+b x^3\right )^{7/3} \sqrt [3]{\frac {d x^3}{c}+1} \sqrt [3]{\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x^3)^2/(a + b*x^3)^(10/3),x]

[Out]

(x*(1 + (b*x^3)/a)^(1/3)*(9*b^2*c^2*x^6 + 3*a*b*c*x^3*(7*c + d*x^3) + a^2*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)))/(
14*a^3*(a + b*x^3)^(7/3)*((c*(a + b*x^3))/(a*(c + d*x^3)))^(1/3)*(1 + (d*x^3)/c)^(1/3))

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fricas [A]  time = 0.62, size = 103, normalized size = 1.32 \[ \frac {{\left ({\left (9 \, b^{2} c^{2} + 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{7} + 14 \, a^{2} c^{2} x + 7 \, {\left (3 \, a b c^{2} + a^{2} c d\right )} x^{4}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{14 \, {\left (a^{3} b^{3} x^{9} + 3 \, a^{4} b^{2} x^{6} + 3 \, a^{5} b x^{3} + a^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(10/3),x, algorithm="fricas")

[Out]

1/14*((9*b^2*c^2 + 3*a*b*c*d + 2*a^2*d^2)*x^7 + 14*a^2*c^2*x + 7*(3*a*b*c^2 + a^2*c*d)*x^4)*(b*x^3 + a)^(2/3)/
(a^3*b^3*x^9 + 3*a^4*b^2*x^6 + 3*a^5*b*x^3 + a^6)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {10}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(10/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^2/(b*x^3 + a)^(10/3), x)

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maple [A]  time = 0.05, size = 76, normalized size = 0.97 \[ \frac {\left (2 a^{2} d^{2} x^{6}+3 a b c d \,x^{6}+9 b^{2} c^{2} x^{6}+7 a^{2} c d \,x^{3}+21 a b \,c^{2} x^{3}+14 a^{2} c^{2}\right ) x}{14 \left (b \,x^{3}+a \right )^{\frac {7}{3}} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^2/(b*x^3+a)^(10/3),x)

[Out]

1/14*x*(2*a^2*d^2*x^6+3*a*b*c*d*x^6+9*b^2*c^2*x^6+7*a^2*c*d*x^3+21*a*b*c^2*x^3+14*a^2*c^2)/(b*x^3+a)^(7/3)/a^3

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maxima [A]  time = 0.51, size = 109, normalized size = 1.40 \[ -\frac {{\left (4 \, b - \frac {7 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} c d x^{7}}{14 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a^{2}} + \frac {d^{2} x^{7}}{7 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a} + \frac {{\left (2 \, b^{2} - \frac {7 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {14 \, {\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} c^{2} x^{7}}{14 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(10/3),x, algorithm="maxima")

[Out]

-1/14*(4*b - 7*(b*x^3 + a)/x^3)*c*d*x^7/((b*x^3 + a)^(7/3)*a^2) + 1/7*d^2*x^7/((b*x^3 + a)^(7/3)*a) + 1/14*(2*
b^2 - 7*(b*x^3 + a)*b/x^3 + 14*(b*x^3 + a)^2/x^6)*c^2*x^7/((b*x^3 + a)^(7/3)*a^3)

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mupad [B]  time = 1.43, size = 148, normalized size = 1.90 \[ \frac {2\,a^4\,d^2\,x+2\,a^2\,d^2\,x\,{\left (b\,x^3+a\right )}^2+9\,b^2\,c^2\,x\,{\left (b\,x^3+a\right )}^2+2\,a^2\,b^2\,c^2\,x-4\,a^3\,d^2\,x\,\left (b\,x^3+a\right )+3\,a\,b^2\,c^2\,x\,\left (b\,x^3+a\right )-4\,a^3\,b\,c\,d\,x+3\,a\,b\,c\,d\,x\,{\left (b\,x^3+a\right )}^2+a^2\,b\,c\,d\,x\,\left (b\,x^3+a\right )}{14\,a^3\,b^2\,{\left (b\,x^3+a\right )}^{7/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)^2/(a + b*x^3)^(10/3),x)

[Out]

(2*a^4*d^2*x + 2*a^2*d^2*x*(a + b*x^3)^2 + 9*b^2*c^2*x*(a + b*x^3)^2 + 2*a^2*b^2*c^2*x - 4*a^3*d^2*x*(a + b*x^
3) + 3*a*b^2*c^2*x*(a + b*x^3) - 4*a^3*b*c*d*x + 3*a*b*c*d*x*(a + b*x^3)^2 + a^2*b*c*d*x*(a + b*x^3))/(14*a^3*
b^2*(a + b*x^3)^(7/3))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)**2/(b*x**3+a)**(10/3),x)

[Out]

Timed out

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