3.690 \(\int \frac{(a+b x^3)^{2/3}}{x^{12} (c+d x^3)} \, dx\)

Optimal. Leaf size=320 \[ -\frac{\left (a+b x^3\right )^{2/3} \left (88 a^2 b c d^2-220 a^3 d^3+33 a b^2 c^2 d+18 b^3 c^3\right )}{440 a^3 c^4 x^2}+\frac{\left (a+b x^3\right )^{2/3} \left (-44 a^2 d^2+11 a b c d+6 b^2 c^2\right )}{220 a^2 c^3 x^5}-\frac{d^3 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{14/3}}+\frac{d^3 (b c-a d)^{2/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{14/3}}-\frac{d^3 (b c-a d)^{2/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{14/3}}-\frac{\left (a+b x^3\right )^{2/3} (2 b c-11 a d)}{88 a c^2 x^8}-\frac{\left (a+b x^3\right )^{2/3}}{11 c x^{11}} \]

[Out]

-(a + b*x^3)^(2/3)/(11*c*x^11) - ((2*b*c - 11*a*d)*(a + b*x^3)^(2/3))/(88*a*c^2*x^8) + ((6*b^2*c^2 + 11*a*b*c*
d - 44*a^2*d^2)*(a + b*x^3)^(2/3))/(220*a^2*c^3*x^5) - ((18*b^3*c^3 + 33*a*b^2*c^2*d + 88*a^2*b*c*d^2 - 220*a^
3*d^3)*(a + b*x^3)^(2/3))/(440*a^3*c^4*x^2) - (d^3*(b*c - a*d)^(2/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1
/3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*c^(14/3)) - (d^3*(b*c - a*d)^(2/3)*Log[c + d*x^3])/(6*c^(14/3)) + (
d^3*(b*c - a*d)^(2/3)*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2*c^(14/3))

________________________________________________________________________________________

Rubi [C]  time = 2.62553, antiderivative size = 819, normalized size of antiderivative = 2.56, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac{-81 b c d^3 x^{12}-162 a d^4 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+162 b c d^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-297 a d^4 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+297 b c d^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-81 a c d^3 x^9+54 b c^2 d^2 x^9+108 a c d^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-108 b c^2 d^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+54 a c^2 d^2 x^6-45 b c^3 d x^6-90 a c^2 d^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+90 b c^3 d \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+99 a c^2 d^2 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-99 b c^3 d \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+40 b c^4 x^3-45 a c^3 d x^3-80 b c^4 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+80 a c^3 d \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+198 b c^4 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-198 a c^3 d \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-27 (b c-a d) \left (5 c-6 d x^3\right ) \left (d x^3+c\right )^2 \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+27 (b c-a d) \left (d x^3+c\right )^3 \, _4F_3\left (\frac{1}{3},2,2,2;1,1,\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+40 a c^4}{440 c^5 x^{11} \sqrt [3]{b x^3+a}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(2/3)/(x^12*(c + d*x^3)),x]

[Out]

-(40*a*c^4 + 40*b*c^4*x^3 - 45*a*c^3*d*x^3 - 45*b*c^3*d*x^6 + 54*a*c^2*d^2*x^6 + 54*b*c^2*d^2*x^9 - 81*a*c*d^3
*x^9 - 81*b*c*d^3*x^12 - 80*b*c^4*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 80*a
*c^3*d*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 90*b*c^3*d*x^6*Hypergeometric2F
1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 90*a*c^2*d^2*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*
d)*x^3)/(c*(a + b*x^3))] - 108*b*c^2*d^2*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]
+ 108*a*c*d^3*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 162*b*c*d^3*x^12*Hyperg
eometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 162*a*d^4*x^12*Hypergeometric2F1[1/3, 1, 4/3, ((
b*c - a*d)*x^3)/(c*(a + b*x^3))] + 198*b*c^4*x^3*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^
3))] - 198*a*c^3*d*x^3*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 99*b*c^3*d*x^6*Hype
rgeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 99*a*c^2*d^2*x^6*Hypergeometric2F1[1/3, 2, 4/3
, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 297*b*c*d^3*x^12*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a
+ b*x^3))] - 297*a*d^4*x^12*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 27*(b*c - a*d
)*x^3*(5*c - 6*d*x^3)*(c + d*x^3)^2*HypergeometricPFQ[{1/3, 2, 2}, {1, 4/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))
] + 27*(b*c - a*d)*x^3*(c + d*x^3)^3*HypergeometricPFQ[{1/3, 2, 2, 2}, {1, 1, 4/3}, ((b*c - a*d)*x^3)/(c*(a +
b*x^3))])/(440*c^5*x^11*(a + b*x^3)^(1/3))

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx &=\frac{\left (a+b x^3\right )^{2/3} \int \frac{\left (1+\frac{b x^3}{a}\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=-\frac{40 a c^4+40 b c^4 x^3-45 a c^3 d x^3-45 b c^3 d x^6+54 a c^2 d^2 x^6+54 b c^2 d^2 x^9-81 a c d^3 x^9-81 b c d^3 x^{12}-80 b c^4 x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+80 a c^3 d x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+90 b c^3 d x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-90 a c^2 d^2 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-108 b c^2 d^2 x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+108 a c d^3 x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+162 b c d^3 x^{12} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-162 a d^4 x^{12} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+198 b c^4 x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-198 a c^3 d x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-99 b c^3 d x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+99 a c^2 d^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+297 b c d^3 x^{12} \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-297 a d^4 x^{12} \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-27 (b c-a d) x^3 \left (5 c-6 d x^3\right ) \left (c+d x^3\right )^2 \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+27 (b c-a d) x^3 \left (c+d x^3\right )^3 \, _4F_3\left (\frac{1}{3},2,2,2;1,1,\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{440 c^5 x^{11} \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [C]  time = 3.91969, size = 819, normalized size = 2.56 \[ -\frac{-81 b c d^3 x^{12}-162 a d^4 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+162 b c d^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-297 a d^4 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+297 b c d^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-81 a c d^3 x^9+54 b c^2 d^2 x^9+108 a c d^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-108 b c^2 d^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+54 a c^2 d^2 x^6-45 b c^3 d x^6-90 a c^2 d^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+90 b c^3 d \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+99 a c^2 d^2 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-99 b c^3 d \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+40 b c^4 x^3-45 a c^3 d x^3-80 b c^4 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+80 a c^3 d \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+198 b c^4 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-198 a c^3 d \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+27 (b c-a d) \left (d x^3+c\right )^2 \left (6 d x^3-5 c\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{3},2,2\right \},\left \{1,\frac{4}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+27 (b c-a d) \left (d x^3+c\right )^3 \text{HypergeometricPFQ}\left (\left \{\frac{1}{3},2,2,2\right \},\left \{1,1,\frac{4}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+40 a c^4}{440 c^5 x^{11} \sqrt [3]{b x^3+a}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(2/3)/(x^12*(c + d*x^3)),x]

[Out]

-(40*a*c^4 + 40*b*c^4*x^3 - 45*a*c^3*d*x^3 - 45*b*c^3*d*x^6 + 54*a*c^2*d^2*x^6 + 54*b*c^2*d^2*x^9 - 81*a*c*d^3
*x^9 - 81*b*c*d^3*x^12 - 80*b*c^4*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 80*a
*c^3*d*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 90*b*c^3*d*x^6*Hypergeometric2F
1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 90*a*c^2*d^2*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*
d)*x^3)/(c*(a + b*x^3))] - 108*b*c^2*d^2*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]
+ 108*a*c*d^3*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 162*b*c*d^3*x^12*Hyperg
eometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 162*a*d^4*x^12*Hypergeometric2F1[1/3, 1, 4/3, ((
b*c - a*d)*x^3)/(c*(a + b*x^3))] + 198*b*c^4*x^3*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^
3))] - 198*a*c^3*d*x^3*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 99*b*c^3*d*x^6*Hype
rgeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 99*a*c^2*d^2*x^6*Hypergeometric2F1[1/3, 2, 4/3
, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 297*b*c*d^3*x^12*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a
+ b*x^3))] - 297*a*d^4*x^12*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*(b*c - a*d
)*x^3*(c + d*x^3)^2*(-5*c + 6*d*x^3)*HypergeometricPFQ[{1/3, 2, 2}, {1, 4/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3)
)] + 27*(b*c - a*d)*x^3*(c + d*x^3)^3*HypergeometricPFQ[{1/3, 2, 2, 2}, {1, 1, 4/3}, ((b*c - a*d)*x^3)/(c*(a +
b*x^3))])/(440*c^5*x^11*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{12} \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(2/3)/x^12/(d*x^3+c),x)

[Out]

int((b*x^3+a)^(2/3)/x^12/(d*x^3+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (d x^{3} + c\right )} x^{12}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^12/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^12), x)

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Fricas [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)^(2/3)/x^12/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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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)**(2/3)/x**12/(d*x**3+c),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (d x^{3} + c\right )} x^{12}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^12/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^12), x)