Optimal. Leaf size=268 \[ \frac{a^2 \log \left (a d-b d x^3\right )}{3\ 2^{2/3} b^{8/3} d}+\frac{11 a^2 \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{8/3} d}-\frac{a^2 \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^{8/3} d}+\frac{11 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3} d}-\frac{\sqrt [3]{2} a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{8/3} d}-\frac{7 a x^2 \sqrt [3]{a+b x^3}}{18 b^2 d}-\frac{x^5 \sqrt [3]{a+b x^3}}{6 b d} \]
[Out]
________________________________________________________________________________________
Rubi [C] time = 0.0596524, antiderivative size = 66, normalized size of antiderivative = 0.25, 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} \[ \frac{x^8 \sqrt [3]{a+b x^3} F_1\left (\frac{8}{3};-\frac{1}{3},1;\frac{11}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{8 a d \sqrt [3]{\frac{b x^3}{a}+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x^7 \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{x^7 \sqrt [3]{1+\frac{b x^3}{a}}}{a d-b d x^3} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{x^8 \sqrt [3]{a+b x^3} F_1\left (\frac{8}{3};-\frac{1}{3},1;\frac{11}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{8 a d \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [C] time = 0.191141, size = 177, normalized size = 0.66 \[ \frac{22 a b x^5 \left (1-\frac{b^2 x^6}{a^2}\right )^{2/3} F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )-5 x^2 \left (\left (1-\frac{b x^3}{a}\right )^{2/3} \left (7 a^2+10 a b x^3+3 b^2 x^6\right )-7 a^2 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{2 b x^3}{a-b x^3}\right )\right )}{90 b^2 d \left (a+b x^3\right )^{2/3} \left (1-\frac{b x^3}{a}\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{7}}{-bd{x}^{3}+ad}\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{7}}{b d x^{3} - a d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.42261, size = 980, normalized size = 3.66 \begin{align*} -\frac{18 \, \sqrt{3} 2^{\frac{1}{3}} a^{2} b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b \left (-\frac{1}{b^{2}}\right )^{\frac{2}{3}} + \sqrt{3} x}{3 \, x}\right ) - 18 \cdot 2^{\frac{1}{3}} a^{2} b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{1}{3}} b x \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) + 9 \cdot 2^{\frac{1}{3}} a^{2} b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{2}{3}} b^{2} x^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{2}{3}} - 2^{\frac{1}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b x \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 22 \, \sqrt{3} a^{2}{\left (b^{2}\right )}^{\frac{1}{6}} b \arctan \left (\frac{{\left (\sqrt{3}{\left (b^{2}\right )}^{\frac{1}{3}} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}}\right )}{\left (b^{2}\right )}^{\frac{1}{6}}}{3 \, b^{2} x}\right ) - 22 \, a^{2}{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (-\frac{{\left (b^{2}\right )}^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b}{x}\right ) + 11 \, a^{2}{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (\frac{{\left (b^{2}\right )}^{\frac{1}{3}} b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} b}{x^{2}}\right ) + 3 \,{\left (3 \, b^{3} x^{5} + 7 \, a b^{2} x^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{54 \, b^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{x^{7} \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{7}}{b d x^{3} - a d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]