3.707 \(\int \frac{(a+b x^3)^{4/3}}{x^{11} (c+d x^3)} \, dx\)

Optimal. Leaf size=318 \[ \frac{\sqrt [3]{a+b x^3} \left (-175 a^2 b c d^2+140 a^3 d^3+20 a b^2 c^2 d+6 b^3 c^3\right )}{140 a^2 c^4 x}-\frac{\sqrt [3]{a+b x^3} \left (35 a^2 d^2-40 a b c d+2 b^2 c^2\right )}{140 a c^3 x^4}+\frac{d^2 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^{13/3}}-\frac{d^2 (b c-a d)^{4/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{13/3}}-\frac{d^2 (b c-a d)^{4/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{13/3}}-\frac{\sqrt [3]{a+b x^3} (11 b c-10 a d)}{70 c^2 x^7}-\frac{a \sqrt [3]{a+b x^3}}{10 c x^{10}} \]

[Out]

-(a*(a + b*x^3)^(1/3))/(10*c*x^10) - ((11*b*c - 10*a*d)*(a + b*x^3)^(1/3))/(70*c^2*x^7) - ((2*b^2*c^2 - 40*a*b
*c*d + 35*a^2*d^2)*(a + b*x^3)^(1/3))/(140*a*c^3*x^4) + ((6*b^3*c^3 + 20*a*b^2*c^2*d - 175*a^2*b*c*d^2 + 140*a
^3*d^3)*(a + b*x^3)^(1/3))/(140*a^2*c^4*x) - (d^2*(b*c - a*d)^(4/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/
3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*c^(13/3)) + (d^2*(b*c - a*d)^(4/3)*Log[c + d*x^3])/(6*c^(13/3)) - (d
^2*(b*c - a*d)^(4/3)*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2*c^(13/3))

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Rubi [C]  time = 1.80114, antiderivative size = 260, normalized size of antiderivative = 0.82, 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} \[ \frac{-18 c x^3 \left (a+b x^3\right ) \left (c+d x^3\right )^2 (b c-a d) \, _3F_2\left (-\frac{1}{3},2,2;\frac{2}{3},1;\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 c x^3 \left (a+b x^3\right ) \left (11 c^2+2 c d x^3-9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-\left (14 c^2-12 c d x^3+9 d^2 x^6\right ) \left (c \left (a+b x^3\right ) \left (a \left (c-4 d x^3\right )+5 b c x^3\right )-2 x^6 (b c-a d)^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )\right )}{140 c^5 x^{10} \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(4/3)/(x^11*(c + d*x^3)),x]

[Out]

(6*c*(b*c - a*d)*x^3*(a + b*x^3)*(11*c^2 + 2*c*d*x^3 - 9*d^2*x^6)*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)
*x^3)/(c*(a + b*x^3))] - (14*c^2 - 12*c*d*x^3 + 9*d^2*x^6)*(c*(a + b*x^3)*(5*b*c*x^3 + a*(c - 4*d*x^3)) - 2*(b
*c - a*d)^2*x^6*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]) - 18*c*(b*c - a*d)*x^3*(a +
b*x^3)*(c + d*x^3)^2*HypergeometricPFQ[{-1/3, 2, 2}, {2/3, 1}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(140*c^5*x
^10*(a + b*x^3)^(2/3))

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 )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx &=\frac{\left (a \sqrt [3]{a+b x^3}\right ) \int \frac{\left (1+\frac{b x^3}{a}\right )^{4/3}}{x^{11} \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{6 c (b c-a d) x^3 \left (a+b x^3\right ) \left (11 c^2+2 c d x^3-9 d^2 x^6\right ) \, _2F_1\left (-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-\left (14 c^2-12 c d x^3+9 d^2 x^6\right ) \left (c \left (a+b x^3\right ) \left (5 b c x^3+a \left (c-4 d x^3\right )\right )-2 (b c-a d)^2 x^6 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )\right )-18 c (b c-a d) x^3 \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, _3F_2\left (-\frac{1}{3},2,2;\frac{2}{3},1;\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{140 c^5 x^{10} \left (a+b x^3\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 1.65054, size = 270, normalized size = 0.85 \[ -\frac{a \left (\frac{b x^3}{a}+1\right ) \left (18 c x^3 \left (a+b x^3\right ) \left (c+d x^3\right )^2 (b c-a d) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{3},2,2\right \},\left \{\frac{2}{3},1\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )+6 c x^3 \left (a+b x^3\right ) \left (-11 c^2-2 c d x^3+9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+\left (14 c^2-12 c d x^3+9 d^2 x^6\right ) \left (c \left (a+b x^3\right ) \left (a \left (c-4 d x^3\right )+5 b c x^3\right )-2 x^6 (b c-a d)^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )\right )\right )}{140 c^5 x^{10} \left (a+b x^3\right )^{5/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(4/3)/(x^11*(c + d*x^3)),x]

[Out]

-(a*(1 + (b*x^3)/a)*(6*c*(b*c - a*d)*x^3*(a + b*x^3)*(-11*c^2 - 2*c*d*x^3 + 9*d^2*x^6)*Hypergeometric2F1[-1/3,
2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + (14*c^2 - 12*c*d*x^3 + 9*d^2*x^6)*(c*(a + b*x^3)*(5*b*c*x^3 + a*
(c - 4*d*x^3)) - 2*(b*c - a*d)^2*x^6*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]) + 18*c
*(b*c - a*d)*x^3*(a + b*x^3)*(c + d*x^3)^2*HypergeometricPFQ[{-1/3, 2, 2}, {2/3, 1}, ((b*c - a*d)*x^3)/(c*(a +
b*x^3))]))/(140*c^5*x^10*(a + b*x^3)^(5/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{11} \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(4/3)/x^11/(d*x^3+c),x)

[Out]

int((b*x^3+a)^(4/3)/x^11/(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{4}{3}}}{{\left (d x^{3} + c\right )} x^{11}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(4/3)/x^11/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^11), 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)^(4/3)/x^11/(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)**(4/3)/x**11/(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{4}{3}}}{{\left (d x^{3} + c\right )} x^{11}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(4/3)/x^11/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^11), x)