3.689 \(\int \frac {(a+b x^3)^{2/3}}{x^9 (c+d x^3)} \, dx\)

Optimal. Leaf size=257 \[ \frac {\left (a+b x^3\right )^{2/3} \left (-20 a^2 d^2+8 a b c d+3 b^2 c^2\right )}{40 a^2 c^3 x^2}+\frac {d^2 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{11/3}}-\frac {d^2 (b c-a d)^{2/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{11/3}}+\frac {d^2 (b c-a d)^{2/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{11/3}}-\frac {\left (a+b x^3\right )^{2/3} (b c-4 a d)}{20 a c^2 x^5}-\frac {\left (a+b x^3\right )^{2/3}}{8 c x^8} \]

[Out]

-1/8*(b*x^3+a)^(2/3)/c/x^8-1/20*(-4*a*d+b*c)*(b*x^3+a)^(2/3)/a/c^2/x^5+1/40*(-20*a^2*d^2+8*a*b*c*d+3*b^2*c^2)*
(b*x^3+a)^(2/3)/a^2/c^3/x^2+1/6*d^2*(-a*d+b*c)^(2/3)*ln(d*x^3+c)/c^(11/3)-1/2*d^2*(-a*d+b*c)^(2/3)*ln((-a*d+b*
c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(11/3)+1/3*d^2*(-a*d+b*c)^(2/3)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/
3)/(b*x^3+a)^(1/3))*3^(1/2))/c^(11/3)*3^(1/2)

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Rubi [C]  time = 0.97, antiderivative size = 451, normalized size of antiderivative = 1.75, 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} \[ -\frac {-9 x^3 \left (c+d x^3\right )^2 (b c-a d) \, _3F_2\left (\frac {1}{3},2,2;1,\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-2 x^3 \left (5 c^2-6 c d x^3+9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-6 b c^2 d x^6 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+21 b c^3 x^3 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-21 a c^2 d x^3 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+27 a d^3 x^9 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-27 b c d^2 x^9 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 a c d^2 x^6 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-6 a c^2 d x^3+5 a c^3+9 a c d^2 x^6-6 b c^2 d x^6+5 b c^3 x^3+9 b c d^2 x^9}{40 c^4 x^8 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(2/3)/(x^9*(c + d*x^3)),x]

[Out]

-(5*a*c^3 + 5*b*c^3*x^3 - 6*a*c^2*d*x^3 - 6*b*c^2*d*x^6 + 9*a*c*d^2*x^6 + 9*b*c*d^2*x^9 - 2*(b*c - a*d)*x^3*(5
*c^2 - 6*c*d*x^3 + 9*d^2*x^6)*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 21*b*c^3*x^3
*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 21*a*c^2*d*x^3*Hypergeometric2F1[1/3, 2,
4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 6*b*c^2*d*x^6*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a
+ b*x^3))] + 6*a*c*d^2*x^6*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 27*b*c*d^2*x^9
*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*a*d^3*x^9*Hypergeometric2F1[1/3, 2, 4/
3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 9*(b*c - a*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))])/(40*c^4*x^8*(a + b*x^3)^(1/3))

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])

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])

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^9 \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^9 \left (c+d x^3\right )} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {5 a c^3+5 b c^3 x^3-6 a c^2 d x^3-6 b c^2 d x^6+9 a c d^2 x^6+9 b c d^2 x^9-2 (b c-a d) x^3 \left (5 c^2-6 c d x^3+9 d^2 x^6\right ) \, _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 )+21 b c^3 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 )-21 a c^2 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 )-6 b c^2 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 )+6 a c 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 )-27 b c d^2 x^9 \, _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 a d^3 x^9 \, _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 )-9 (b c-a d) x^3 \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 )}{40 c^4 x^8 \sqrt [3]{a+b x^3}}\\ \end {align*}

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Mathematica [C]  time = 2.20, size = 451, normalized size = 1.75 \[ -\frac {-9 x^3 \left (c+d x^3\right )^2 (b c-a d) \, _3F_2\left (\frac {1}{3},2,2;1,\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+21 b c^3 x^3 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-2 x^3 \left (5 c^2-6 c d x^3+9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-21 a c^2 d x^3 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-6 b c^2 d x^6 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+27 a d^3 x^9 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-27 b c d^2 x^9 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 a c d^2 x^6 \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+5 a c^3-6 a c^2 d x^3+9 a c d^2 x^6+5 b c^3 x^3-6 b c^2 d x^6+9 b c d^2 x^9}{40 c^4 x^8 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(2/3)/(x^9*(c + d*x^3)),x]

[Out]

-1/40*(5*a*c^3 + 5*b*c^3*x^3 - 6*a*c^2*d*x^3 - 6*b*c^2*d*x^6 + 9*a*c*d^2*x^6 + 9*b*c*d^2*x^9 - 2*(b*c - a*d)*x
^3*(5*c^2 - 6*c*d*x^3 + 9*d^2*x^6)*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 21*b*c^
3*x^3*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 21*a*c^2*d*x^3*Hypergeometric2F1[1/3
, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 6*b*c^2*d*x^6*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/
(c*(a + b*x^3))] + 6*a*c*d^2*x^6*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 27*b*c*d^
2*x^9*Hypergeometric2F1[1/3, 2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*a*d^3*x^9*Hypergeometric2F1[1/3,
2, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 9*(b*c - a*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))])/(c^4*x^8*(a + b*x^3)^(1/3))

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fricas [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)^(2/3)/x^9/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^9/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^9), x)

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maple [F]  time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (d \,x^{3}+c \right ) x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(2/3)/x^9/(d*x^3+c),x)

[Out]

int((b*x^3+a)^(2/3)/x^9/(d*x^3+c),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^9/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^9), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^9\,\left (d\,x^3+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(2/3)/(x^9*(c + d*x^3)),x)

[Out]

int((a + b*x^3)^(2/3)/(x^9*(c + d*x^3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{x^{9} \left (c + d x^{3}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(2/3)/x**9/(d*x**3+c),x)

[Out]

Integral((a + b*x**3)**(2/3)/(x**9*(c + d*x**3)), x)

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