Optimal. Leaf size=348 \[ \frac{x^7 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{7 b^4}-\frac{a x^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 b^5}+\frac{a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{19/3}}+\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}-\frac{a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{19/3}}+\frac{a^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{19/3}}+\frac{x^{10} \left (a^2 f-a b e+b^2 d\right )}{10 b^3}+\frac{x^{13} (b e-a f)}{13 b^2}+\frac{f x^{16}}{16 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.332909, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {1836, 1488, 200, 31, 634, 617, 204, 628} \[ \frac{x^7 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{7 b^4}-\frac{a x^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 b^5}+\frac{a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{19/3}}+\frac{a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}-\frac{a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{19/3}}+\frac{a^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} b^{19/3}}+\frac{x^{10} \left (a^2 f-a b e+b^2 d\right )}{10 b^3}+\frac{x^{13} (b e-a f)}{13 b^2}+\frac{f x^{16}}{16 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1836
Rule 1488
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^9 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac{f x^{16}}{16 b}+\frac{\int \frac{x^9 \left (16 b c+16 b d x^3+16 (b e-a f) x^6\right )}{a+b x^3} \, dx}{16 b}\\ &=\frac{f x^{16}}{16 b}+\frac{\int \left (\frac{16 a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{b^5}-\frac{16 a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{b^4}+\frac{16 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{b^3}+\frac{16 \left (b^2 d-a b e+a^2 f\right ) x^9}{b^2}+\frac{16 (b e-a f) x^{12}}{b}+\frac{16 \left (-a^3 b^3 c+a^4 b^2 d-a^5 b e+a^6 f\right )}{b^5 \left (a+b x^3\right )}\right ) \, dx}{16 b}\\ &=\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^7}{7 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^{10}}{10 b^3}+\frac{(b e-a f) x^{13}}{13 b^2}+\frac{f x^{16}}{16 b}-\frac{\left (a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{a+b x^3} \, dx}{b^6}\\ &=\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^7}{7 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^{10}}{10 b^3}+\frac{(b e-a f) x^{13}}{13 b^2}+\frac{f x^{16}}{16 b}-\frac{\left (a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^6}-\frac{\left (a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 b^6}\\ &=\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^7}{7 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^{10}}{10 b^3}+\frac{(b e-a f) x^{13}}{13 b^2}+\frac{f x^{16}}{16 b}-\frac{a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{19/3}}+\frac{\left (a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \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}{6 b^{19/3}}-\frac{\left (a^{8/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^6}\\ &=\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^7}{7 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^{10}}{10 b^3}+\frac{(b e-a f) x^{13}}{13 b^2}+\frac{f x^{16}}{16 b}-\frac{a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{19/3}}+\frac{a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{19/3}}-\frac{\left (a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{19/3}}\\ &=\frac{a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{4 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^7}{7 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^{10}}{10 b^3}+\frac{(b e-a f) x^{13}}{13 b^2}+\frac{f x^{16}}{16 b}+\frac{a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{19/3}}-\frac{a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{19/3}}+\frac{a^{7/3} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{19/3}}\\ \end{align*}
Mathematica [A] time = 0.0786584, size = 351, normalized size = 1.01 \[ \frac{x^7 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{7 b^4}+\frac{a x^4 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{4 b^5}-\frac{a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{6 b^{19/3}}-\frac{a^2 x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{b^6}+\frac{a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{3 b^{19/3}}+\frac{a^{7/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\sqrt{3} b^{19/3}}+\frac{x^{10} \left (a^2 f-a b e+b^2 d\right )}{10 b^3}+\frac{x^{13} (b e-a f)}{13 b^2}+\frac{f x^{16}}{16 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 592, normalized size = 1.7 \begin{align*} \text{result too large to display} \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 [A] time = 1.39803, size = 767, normalized size = 2.2 \begin{align*} \frac{1365 \, b^{5} f x^{16} + 1680 \,{\left (b^{5} e - a b^{4} f\right )} x^{13} + 2184 \,{\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{10} + 3120 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{7} - 5460 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} - 7280 \, \sqrt{3}{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) + 3640 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 7280 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 21840 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{21840 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.87413, size = 450, normalized size = 1.29 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{19} - a^{16} f^{3} + 3 a^{15} b e f^{2} - 3 a^{14} b^{2} d f^{2} - 3 a^{14} b^{2} e^{2} f + 3 a^{13} b^{3} c f^{2} + 6 a^{13} b^{3} d e f + a^{13} b^{3} e^{3} - 6 a^{12} b^{4} c e f - 3 a^{12} b^{4} d^{2} f - 3 a^{12} b^{4} d e^{2} + 6 a^{11} b^{5} c d f + 3 a^{11} b^{5} c e^{2} + 3 a^{11} b^{5} d^{2} e - 3 a^{10} b^{6} c^{2} f - 6 a^{10} b^{6} c d e - a^{10} b^{6} d^{3} + 3 a^{9} b^{7} c^{2} e + 3 a^{9} b^{7} c d^{2} - 3 a^{8} b^{8} c^{2} d + a^{7} b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t b^{6}}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )} \right )\right )} + \frac{f x^{16}}{16 b} - \frac{x^{13} \left (a f - b e\right )}{13 b^{2}} + \frac{x^{10} \left (a^{2} f - a b e + b^{2} d\right )}{10 b^{3}} - \frac{x^{7} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{7 b^{4}} + \frac{x^{4} \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{4 b^{5}} - \frac{x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0624, size = 613, normalized size = 1.76 \begin{align*} -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{5} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} b e\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 )}{3 \, b^{7}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{5} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} b e\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{7}} + \frac{{\left (a^{3} b^{13} c - a^{4} b^{12} d - a^{6} b^{10} f + a^{5} b^{11} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{16}} + \frac{455 \, b^{15} f x^{16} - 560 \, a b^{14} f x^{13} + 560 \, b^{15} x^{13} e + 728 \, b^{15} d x^{10} + 728 \, a^{2} b^{13} f x^{10} - 728 \, a b^{14} x^{10} e + 1040 \, b^{15} c x^{7} - 1040 \, a b^{14} d x^{7} - 1040 \, a^{3} b^{12} f x^{7} + 1040 \, a^{2} b^{13} x^{7} e - 1820 \, a b^{14} c x^{4} + 1820 \, a^{2} b^{13} d x^{4} + 1820 \, a^{4} b^{11} f x^{4} - 1820 \, a^{3} b^{12} x^{4} e + 7280 \, a^{2} b^{13} c x - 7280 \, a^{3} b^{12} d x - 7280 \, a^{5} b^{10} f x + 7280 \, a^{4} b^{11} x e}{7280 \, b^{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]