3.706 \(\int \frac{(a+b x^3)^{4/3}}{x^8 (c+d x^3)} \, dx\)

Optimal. Leaf size=250 \[ -\frac{\sqrt [3]{a+b x^3} \left (28 a^2 d^2-35 a b c d+4 b^2 c^2\right )}{28 a c^3 x}-\frac{\sqrt [3]{a+b x^3} (8 b c-7 a d)}{28 c^2 x^4}-\frac{d (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^{10/3}}+\frac{d (b c-a d)^{4/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{10/3}}+\frac{d (b c-a d)^{4/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{10/3}}-\frac{a \sqrt [3]{a+b x^3}}{7 c x^7} \]

[Out]

-(a*(a + b*x^3)^(1/3))/(7*c*x^7) - ((8*b*c - 7*a*d)*(a + b*x^3)^(1/3))/(28*c^2*x^4) - ((4*b^2*c^2 - 35*a*b*c*d
+ 28*a^2*d^2)*(a + b*x^3)^(1/3))/(28*a*c^3*x) + (d*(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^(10/3)) - (d*(b*c - a*d)^(4/3)*Log[c + d*x^3])/(6*c^(10/3)) + (d
*(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^(10/3))

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Rubi [C]  time = 0.506586, antiderivative size = 169, normalized size of antiderivative = 0.68, 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} \[ \frac{12 c x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d) \, _2F_1\left (-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-\left (4 c-3 d x^3\right ) \left (c \left (a+b x^3\right ) \left (a \left (c-4 d x^3\right )+5 b c x^3\right )-2 x^6 (b c-a d)^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )\right )}{28 c^4 x^7 \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(4/3)/(x^8*(c + d*x^3)),x]

[Out]

(12*c*(b*c - a*d)*x^3*(a + b*x^3)*(c + d*x^3)*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3)
)] - (4*c - 3*d*x^3)*(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))]))/(28*c^4*x^7*(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^8 \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^8 \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{12 c (b c-a d) x^3 \left (a+b x^3\right ) \left (c+d x^3\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 (4 c-3 d x^3\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 )}{28 c^4 x^7 \left (a+b x^3\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.422977, size = 179, normalized size = 0.72 \[ -\frac{a \left (\frac{b x^3}{a}+1\right ) \left (12 c x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) (a d-b c) \, _2F_1\left (-\frac{1}{3},2;\frac{2}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+\left (4 c-3 d x^3\right ) \left (c \left (a+b x^3\right ) \left (a \left (c-4 d x^3\right )+5 b c x^3\right )-2 x^6 (b c-a d)^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )\right )\right )}{28 c^4 x^7 \left (a+b x^3\right )^{5/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(4/3)/(x^8*(c + d*x^3)),x]

[Out]

-(a*(1 + (b*x^3)/a)*(12*c*(-(b*c) + a*d)*x^3*(a + b*x^3)*(c + d*x^3)*Hypergeometric2F1[-1/3, 2, 2/3, ((b*c - a
*d)*x^3)/(c*(a + b*x^3))] + (4*c - 3*d*x^3)*(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))])))/(28*c^4*x^7*(a + b*x^3)^(5/3))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{8} \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^8/(d*x^3+c),x)

[Out]

int((b*x^3+a)^(4/3)/x^8/(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^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(4/3)/x^8/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^8), 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^8/(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**8/(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^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(4/3)/x^8/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^8), x)