3.85 \(\int \frac {(a+b x^3)^3}{(c+d x^3)^{13/3}} \, dx\)

Optimal. Leaf size=109 \[ \frac {81 a^3 x}{140 c^4 \sqrt [3]{c+d x^3}}+\frac {27 a^2 x \left (a+b x^3\right )}{140 c^3 \left (c+d x^3\right )^{4/3}}+\frac {9 a x \left (a+b x^3\right )^2}{70 c^2 \left (c+d x^3\right )^{7/3}}+\frac {x \left (a+b x^3\right )^3}{10 c \left (c+d x^3\right )^{10/3}} \]

[Out]

1/10*x*(b*x^3+a)^3/c/(d*x^3+c)^(10/3)+9/70*a*x*(b*x^3+a)^2/c^2/(d*x^3+c)^(7/3)+27/140*a^2*x*(b*x^3+a)/c^3/(d*x
^3+c)^(4/3)+81/140*a^3*x/c^4/(d*x^3+c)^(1/3)

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Rubi [A]  time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {378, 191} \[ \frac {27 a^2 x \left (a+b x^3\right )}{140 c^3 \left (c+d x^3\right )^{4/3}}+\frac {81 a^3 x}{140 c^4 \sqrt [3]{c+d x^3}}+\frac {9 a x \left (a+b x^3\right )^2}{70 c^2 \left (c+d x^3\right )^{7/3}}+\frac {x \left (a+b x^3\right )^3}{10 c \left (c+d x^3\right )^{10/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/(c + d*x^3)^(13/3),x]

[Out]

(x*(a + b*x^3)^3)/(10*c*(c + d*x^3)^(10/3)) + (9*a*x*(a + b*x^3)^2)/(70*c^2*(c + d*x^3)^(7/3)) + (27*a^2*x*(a
+ b*x^3))/(140*c^3*(c + d*x^3)^(4/3)) + (81*a^3*x)/(140*c^4*(c + d*x^3)^(1/3))

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 (a+b x^3\right )^3}{\left (c+d x^3\right )^{13/3}} \, dx &=\frac {x \left (a+b x^3\right )^3}{10 c \left (c+d x^3\right )^{10/3}}+\frac {(9 a) \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^{10/3}} \, dx}{10 c}\\ &=\frac {x \left (a+b x^3\right )^3}{10 c \left (c+d x^3\right )^{10/3}}+\frac {9 a x \left (a+b x^3\right )^2}{70 c^2 \left (c+d x^3\right )^{7/3}}+\frac {\left (27 a^2\right ) \int \frac {a+b x^3}{\left (c+d x^3\right )^{7/3}} \, dx}{35 c^2}\\ &=\frac {x \left (a+b x^3\right )^3}{10 c \left (c+d x^3\right )^{10/3}}+\frac {9 a x \left (a+b x^3\right )^2}{70 c^2 \left (c+d x^3\right )^{7/3}}+\frac {27 a^2 x \left (a+b x^3\right )}{140 c^3 \left (c+d x^3\right )^{4/3}}+\frac {\left (81 a^3\right ) \int \frac {1}{\left (c+d x^3\right )^{4/3}} \, dx}{140 c^3}\\ &=\frac {x \left (a+b x^3\right )^3}{10 c \left (c+d x^3\right )^{10/3}}+\frac {9 a x \left (a+b x^3\right )^2}{70 c^2 \left (c+d x^3\right )^{7/3}}+\frac {27 a^2 x \left (a+b x^3\right )}{140 c^3 \left (c+d x^3\right )^{4/3}}+\frac {81 a^3 x}{140 c^4 \sqrt [3]{c+d x^3}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 120, normalized size = 1.10 \[ \frac {x \left (a^3 \left (140 c^3+315 c^2 d x^3+270 c d^2 x^6+81 d^3 x^9\right )+3 a^2 b c x^3 \left (35 c^2+30 c d x^3+9 d^2 x^6\right )+6 a b^2 c^2 x^6 \left (10 c+3 d x^3\right )+14 b^3 c^3 x^9\right )}{140 c^4 \left (c+d x^3\right )^{10/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/(c + d*x^3)^(13/3),x]

[Out]

(x*(14*b^3*c^3*x^9 + 6*a*b^2*c^2*x^6*(10*c + 3*d*x^3) + 3*a^2*b*c*x^3*(35*c^2 + 30*c*d*x^3 + 9*d^2*x^6) + a^3*
(140*c^3 + 315*c^2*d*x^3 + 270*c*d^2*x^6 + 81*d^3*x^9)))/(140*c^4*(c + d*x^3)^(10/3))

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fricas [A]  time = 0.67, size = 166, normalized size = 1.52 \[ \frac {{\left ({\left (14 \, b^{3} c^{3} + 18 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} + 81 \, a^{3} d^{3}\right )} x^{10} + 30 \, {\left (2 \, a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + 9 \, a^{3} c d^{2}\right )} x^{7} + 140 \, a^{3} c^{3} x + 105 \, {\left (a^{2} b c^{3} + 3 \, a^{3} c^{2} d\right )} x^{4}\right )} {\left (d x^{3} + c\right )}^{\frac {2}{3}}}{140 \, {\left (c^{4} d^{4} x^{12} + 4 \, c^{5} d^{3} x^{9} + 6 \, c^{6} d^{2} x^{6} + 4 \, c^{7} d x^{3} + c^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/(d*x^3+c)^(13/3),x, algorithm="fricas")

[Out]

1/140*((14*b^3*c^3 + 18*a*b^2*c^2*d + 27*a^2*b*c*d^2 + 81*a^3*d^3)*x^10 + 30*(2*a*b^2*c^3 + 3*a^2*b*c^2*d + 9*
a^3*c*d^2)*x^7 + 140*a^3*c^3*x + 105*(a^2*b*c^3 + 3*a^3*c^2*d)*x^4)*(d*x^3 + c)^(2/3)/(c^4*d^4*x^12 + 4*c^5*d^
3*x^9 + 6*c^6*d^2*x^6 + 4*c^7*d*x^3 + c^8)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{3}}{{\left (d x^{3} + c\right )}^{\frac {13}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/(d*x^3+c)^(13/3),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^3/(d*x^3 + c)^(13/3), x)

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maple [A]  time = 0.05, size = 134, normalized size = 1.23 \[ \frac {\left (81 a^{3} d^{3} x^{9}+27 a^{2} b c \,d^{2} x^{9}+18 a \,b^{2} c^{2} d \,x^{9}+14 b^{3} c^{3} x^{9}+270 a^{3} c \,d^{2} x^{6}+90 a^{2} b \,c^{2} d \,x^{6}+60 a \,b^{2} c^{3} x^{6}+315 a^{3} c^{2} d \,x^{3}+105 a^{2} b \,c^{3} x^{3}+140 a^{3} c^{3}\right ) x}{140 \left (d \,x^{3}+c \right )^{\frac {10}{3}} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3/(d*x^3+c)^(13/3),x)

[Out]

1/140*x*(81*a^3*d^3*x^9+27*a^2*b*c*d^2*x^9+18*a*b^2*c^2*d*x^9+14*b^3*c^3*x^9+270*a^3*c*d^2*x^6+90*a^2*b*c^2*d*
x^6+60*a*b^2*c^3*x^6+315*a^3*c^2*d*x^3+105*a^2*b*c^3*x^3+140*a^3*c^3)/(d*x^3+c)^(10/3)/c^4

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maxima [A]  time = 0.63, size = 182, normalized size = 1.67 \[ \frac {b^{3} x^{10}}{10 \, {\left (d x^{3} + c\right )}^{\frac {10}{3}} c} - \frac {3 \, a b^{2} {\left (7 \, d - \frac {10 \, {\left (d x^{3} + c\right )}}{x^{3}}\right )} x^{10}}{70 \, {\left (d x^{3} + c\right )}^{\frac {10}{3}} c^{2}} + \frac {3 \, {\left (14 \, d^{2} - \frac {40 \, {\left (d x^{3} + c\right )} d}{x^{3}} + \frac {35 \, {\left (d x^{3} + c\right )}^{2}}{x^{6}}\right )} a^{2} b x^{10}}{140 \, {\left (d x^{3} + c\right )}^{\frac {10}{3}} c^{3}} - \frac {{\left (14 \, d^{3} - \frac {60 \, {\left (d x^{3} + c\right )} d^{2}}{x^{3}} + \frac {105 \, {\left (d x^{3} + c\right )}^{2} d}{x^{6}} - \frac {140 \, {\left (d x^{3} + c\right )}^{3}}{x^{9}}\right )} a^{3} x^{10}}{140 \, {\left (d x^{3} + c\right )}^{\frac {10}{3}} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/(d*x^3+c)^(13/3),x, algorithm="maxima")

[Out]

1/10*b^3*x^10/((d*x^3 + c)^(10/3)*c) - 3/70*a*b^2*(7*d - 10*(d*x^3 + c)/x^3)*x^10/((d*x^3 + c)^(10/3)*c^2) + 3
/140*(14*d^2 - 40*(d*x^3 + c)*d/x^3 + 35*(d*x^3 + c)^2/x^6)*a^2*b*x^10/((d*x^3 + c)^(10/3)*c^3) - 1/140*(14*d^
3 - 60*(d*x^3 + c)*d^2/x^3 + 105*(d*x^3 + c)^2*d/x^6 - 140*(d*x^3 + c)^3/x^9)*a^3*x^10/((d*x^3 + c)^(10/3)*c^4
)

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mupad [B]  time = 1.56, size = 271, normalized size = 2.49 \[ \frac {x\,\left (\frac {a^3}{10\,c}-\frac {c\,\left (\frac {c\,\left (\frac {b^3}{10\,d}-\frac {3\,a\,b^2}{10\,c}\right )}{d}+\frac {3\,a^2\,b}{10\,c}\right )}{d}\right )}{{\left (d\,x^3+c\right )}^{10/3}}-\frac {x\,\left (\frac {b^3}{4\,d^3}-\frac {27\,a^3\,d^3+9\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-7\,b^3\,c^3}{140\,c^3\,d^3}\right )}{{\left (d\,x^3+c\right )}^{4/3}}+\frac {x\,\left (\frac {c\,\left (\frac {b^3}{7\,d^2}-\frac {b^2\,\left (3\,a\,d-b\,c\right )}{7\,c\,d^2}\right )}{d}+\frac {9\,a^3\,d^3+3\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d+b^3\,c^3}{70\,c^2\,d^3}\right )}{{\left (d\,x^3+c\right )}^{7/3}}+\frac {x\,\left (81\,a^3\,d^3+27\,a^2\,b\,c\,d^2+18\,a\,b^2\,c^2\,d+14\,b^3\,c^3\right )}{140\,c^4\,d^3\,{\left (d\,x^3+c\right )}^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^3/(c + d*x^3)^(13/3),x)

[Out]

(x*(a^3/(10*c) - (c*((c*(b^3/(10*d) - (3*a*b^2)/(10*c)))/d + (3*a^2*b)/(10*c)))/d))/(c + d*x^3)^(10/3) - (x*(b
^3/(4*d^3) - (27*a^3*d^3 - 7*b^3*c^3 + 6*a*b^2*c^2*d + 9*a^2*b*c*d^2)/(140*c^3*d^3)))/(c + d*x^3)^(4/3) + (x*(
(c*(b^3/(7*d^2) - (b^2*(3*a*d - b*c))/(7*c*d^2)))/d + (9*a^3*d^3 + b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2)/(7
0*c^2*d^3)))/(c + d*x^3)^(7/3) + (x*(81*a^3*d^3 + 14*b^3*c^3 + 18*a*b^2*c^2*d + 27*a^2*b*c*d^2))/(140*c^4*d^3*
(c + d*x^3)^(1/3))

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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((b*x**3+a)**3/(d*x**3+c)**(13/3),x)

[Out]

Timed out

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