Optimal. Leaf size=267 \[ \frac{x \left (a+b x^3\right )^{2/3} (6 b c-5 a d)}{18 c^2 \left (c+d x^3\right ) (b c-a d)}+\frac{a (6 b c-5 a d) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{4/3}}-\frac{a (6 b c-5 a d) \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{4/3}}+\frac{a (6 b c-5 a d) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} c^{8/3} (b c-a d)^{4/3}}-\frac{d x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2 (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.243439, antiderivative size = 326, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {382, 378, 377, 200, 31, 634, 617, 204, 628} \[ \frac{x \left (a+b x^3\right )^{2/3} (6 b c-5 a d)}{18 c^2 \left (c+d x^3\right ) (b c-a d)}-\frac{a (6 b c-5 a d) \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)^{4/3}}+\frac{a (6 b c-5 a d) \log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{54 c^{8/3} (b c-a d)^{4/3}}+\frac{a (6 b c-5 a d) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} (b c-a d)^{4/3}}-\frac{d x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 382
Rule 378
Rule 377
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx &=-\frac{d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac{(6 b c-5 a d) \int \frac{\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^2} \, dx}{6 c (b c-a d)}\\ &=-\frac{d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac{(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}+\frac{(a (6 b c-5 a d)) \int \frac{1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{9 c^2 (b c-a d)}\\ &=-\frac{d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac{(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}+\frac{(a (6 b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 c^2 (b c-a d)}\\ &=-\frac{d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac{(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}+\frac{(a (6 b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{c}-\sqrt [3]{b c-a d} x} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)}+\frac{(a (6 b c-5 a d)) \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{c}+\sqrt [3]{b c-a d} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)}\\ &=-\frac{d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac{(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}-\frac{a (6 b c-5 a d) \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)^{4/3}}+\frac{(a (6 b c-5 a d)) \operatorname{Subst}\left (\int \frac{\sqrt [3]{c} \sqrt [3]{b c-a d}+2 (b c-a d)^{2/3} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{54 c^{8/3} (b c-a d)^{4/3}}+\frac{(a (6 b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{18 c^{7/3} (b c-a d)}\\ &=-\frac{d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac{(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}-\frac{a (6 b c-5 a d) \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)^{4/3}}+\frac{a (6 b c-5 a d) \log \left (c^{2/3}+\frac{(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{54 c^{8/3} (b c-a d)^{4/3}}-\frac{(a (6 b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{9 c^{8/3} (b c-a d)^{4/3}}\\ &=-\frac{d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac{(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}+\frac{a (6 b c-5 a d) \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{9 \sqrt{3} c^{8/3} (b c-a d)^{4/3}}-\frac{a (6 b c-5 a d) \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)^{4/3}}+\frac{a (6 b c-5 a d) \log \left (c^{2/3}+\frac{(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{54 c^{8/3} (b c-a d)^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.160942, size = 153, normalized size = 0.57 \[ \frac{x \left (c \left (-a^2 d \left (8 c+5 d x^3\right )+a b \left (6 c^2-5 c d x^3-5 d^2 x^6\right )+3 b^2 c x^3 \left (2 c+d x^3\right )\right )-2 a \left (c+d x^3\right )^2 (5 a d-6 b c) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )\right )}{18 c^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.432, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, 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{2}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]