Optimal. Leaf size=201 \[ \frac{(a B+2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{5/3}}+\frac{x^2 (a B+2 A b)}{9 a^2 b \left (a+b x^3\right )}+\frac{x^2 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11535, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {457, 290, 292, 31, 634, 617, 204, 628} \[ \frac{(a B+2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{5/3}}+\frac{x^2 (a B+2 A b)}{9 a^2 b \left (a+b x^3\right )}+\frac{x^2 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 457
Rule 290
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac{(4 A b+2 a B) \int \frac{x}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}+\frac{(2 A b+a B) \int \frac{x}{a+b x^3} \, dx}{9 a^2 b}\\ &=\frac{(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}-\frac{(2 A b+a B) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} b^{4/3}}+\frac{(2 A b+a B) \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{7/3} b^{4/3}}\\ &=\frac{(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}-\frac{(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}+\frac{(2 A b+a B) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{7/3} b^{5/3}}+\frac{(2 A b+a B) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^2 b^{4/3}}\\ &=\frac{(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}-\frac{(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}+\frac{(2 A b+a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{5/3}}+\frac{(2 A b+a B) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{7/3} b^{5/3}}\\ &=\frac{(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac{(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}-\frac{(2 A b+a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{5/3}}-\frac{(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}+\frac{(2 A b+a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.129181, size = 178, normalized size = 0.89 \[ \frac{-\frac{9 a^{4/3} b^{2/3} x^2 (a B-A b)}{\left (a+b x^3\right )^2}+(a B+2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{6 \sqrt [3]{a} b^{2/3} x^2 (a B+2 A b)}{a+b x^3}-2 (a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} (a B+2 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^{7/3} b^{5/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 251, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{ \left ( 2\,Ab+Ba \right ){x}^{5}}{9\,{a}^{2}}}+{\frac{ \left ( 7\,Ab-Ba \right ){x}^{2}}{18\,ab}} \right ) }-{\frac{2\,A}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{A}{27\,{a}^{2}b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{54\,a{b}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,\sqrt{3}A}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}B}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.77298, size = 1648, normalized size = 8.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.19226, size = 153, normalized size = 0.76 \begin{align*} \frac{x^{5} \left (4 A b^{2} + 2 B a b\right ) + x^{2} \left (7 A a b - B a^{2}\right )}{18 a^{4} b + 36 a^{3} b^{2} x^{3} + 18 a^{2} b^{3} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{7} b^{5} + 8 A^{3} b^{3} + 12 A^{2} B a b^{2} + 6 A B^{2} a^{2} b + B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a^{5} b^{3}}{4 A^{2} b^{2} + 4 A B a b + B^{2} a^{2}} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12806, size = 301, normalized size = 1.5 \begin{align*} -\frac{{\left (B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{3}} + \frac{2 \, B a b x^{5} + 4 \, A b^{2} x^{5} - B a^{2} x^{2} + 7 \, A a b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]