3.604 \(\int \frac {(a+b x^3)^{2/3}}{x^5 (a d-b d x^3)} \, dx\)

Optimal. Leaf size=512 \[ \frac {b^{4/3} \log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{2} a^{5/3} d}-\frac {2^{2/3} b^{4/3} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 a^{5/3} d}-\frac {b^{4/3} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} a^{5/3} d}+\frac {2^{2/3} b^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^{5/3} d}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a^{5/3} d}+\frac {b^{4/3} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} a^{5/3} d}+\frac {3 b^2 x^2 \sqrt [3]{\frac {b x^3}{a}+1} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{4 a^2 d \sqrt [3]{a+b x^3}}-\frac {3 b \left (a+b x^3\right )^{2/3}}{2 a^2 d x}-\frac {\left (a+b x^3\right )^{2/3}}{4 a d x^4} \]

[Out]

-1/4*(b*x^3+a)^(2/3)/a/d/x^4-3/2*b*(b*x^3+a)^(2/3)/a^2/d/x+3/4*b^2*x^2*(1+b*x^3/a)^(1/3)*hypergeom([1/3, 2/3],
[5/3],-b*x^3/a)/a^2/d/(b*x^3+a)^(1/3)+1/12*b^(4/3)*ln((a^(1/3)-b^(1/3)*x)^2*(a^(1/3)+b^(1/3)*x)/a)*2^(2/3)/a^(
5/3)/d+1/6*b^(4/3)*ln(1+2^(2/3)*(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^(2/3)-2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1
/3))*2^(2/3)/a^(5/3)/d-1/3*2^(2/3)*b^(4/3)*ln(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))/a^(5/3)/d-1/4*b^(
4/3)*ln(b^(1/3)*(a^(1/3)+b^(1/3)*x)/a^(1/3)-2^(2/3)*b^(1/3)*(b*x^3+a)^(1/3)/a^(1/3))*2^(2/3)/a^(5/3)/d+1/3*2^(
2/3)*b^(4/3)*arctan(1/3*(1-2*2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))/a^(5/3)/d*3^(1/2)+1/6*b^(4/
3)*arctan(1/3*(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*2^(2/3)/a^(5/3)/d*3^(1/2)

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Rubi [C]  time = 0.07, antiderivative size = 66, normalized size of antiderivative = 0.13, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {511, 510} \[ -\frac {\left (a+b x^3\right )^{2/3} F_1\left (-\frac {4}{3};-\frac {2}{3},1;-\frac {1}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{4 a d x^4 \left (\frac {b x^3}{a}+1\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(2/3)/(x^5*(a*d - b*d*x^3)),x]

[Out]

-((a + b*x^3)^(2/3)*AppellF1[-4/3, -2/3, 1, -1/3, -((b*x^3)/a), (b*x^3)/a])/(4*a*d*x^4*(1 + (b*x^3)/a)^(2/3))

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^5 \left (a d-b 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^5 \left (a d-b d x^3\right )} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {\left (a+b x^3\right )^{2/3} F_1\left (-\frac {4}{3};-\frac {2}{3},1;-\frac {1}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{4 a d x^4 \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.09, size = 148, normalized size = 0.29 \[ \frac {-5 a \left (a^2+7 a b x^3+6 b^2 x^6\right )-6 b^3 x^9 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )+35 a b^2 x^6 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{20 a^3 d x^4 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(2/3)/(x^5*(a*d - b*d*x^3)),x]

[Out]

(-5*a*(a^2 + 7*a*b*x^3 + 6*b^2*x^6) + 35*a*b^2*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/
a), (b*x^3)/a] - 6*b^3*x^9*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a])/(20*a^3*
d*x^4*(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^5/(-b*d*x^3+a*d),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^5/(-b*d*x^3+a*d),x, algorithm="giac")

[Out]

integrate(-(b*x^3 + a)^(2/3)/((b*d*x^3 - a*d)*x^5), x)

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maple [F]  time = 0.57, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (-b d \,x^{3}+a d \right ) x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(2/3)/x^5/(-b*d*x^3+a*d),x)

[Out]

int((b*x^3+a)^(2/3)/x^5/(-b*d*x^3+a*d),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^5/(-b*d*x^3+a*d),x, algorithm="maxima")

[Out]

-integrate((b*x^3 + a)^(2/3)/((b*d*x^3 - a*d)*x^5), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^5\,\left (a\,d-b\,d\,x^3\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(2/3)/(x^5*(a*d - b*d*x^3)),x)

[Out]

int((a + b*x^3)^(2/3)/(x^5*(a*d - b*d*x^3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{- a x^{5} + b x^{8}}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(2/3)/x**5/(-b*d*x**3+a*d),x)

[Out]

-Integral((a + b*x**3)**(2/3)/(-a*x**5 + b*x**8), x)/d

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