Optimal. Leaf size=162 \[ \frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{a} f+\sqrt{b} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}-\frac{c \log \left (a-b x^4\right )}{4 b}-\frac{d x}{b}-\frac{e x^2}{2 b}-\frac{f x^3}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.202951, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {1831, 1252, 774, 635, 208, 260, 1280, 1167, 205} \[ \frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{a} f+\sqrt{b} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}-\frac{c \log \left (a-b x^4\right )}{4 b}-\frac{d x}{b}-\frac{e x^2}{2 b}-\frac{f x^3}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1831
Rule 1252
Rule 774
Rule 635
Rule 208
Rule 260
Rule 1280
Rule 1167
Rule 205
Rubi steps
\begin{align*} \int \frac{x^3 \left (c+d x+e x^2+f x^3\right )}{a-b x^4} \, dx &=\int \left (\frac{x^3 \left (c+e x^2\right )}{a-b x^4}+\frac{x^4 \left (d+f x^2\right )}{a-b x^4}\right ) \, dx\\ &=\int \frac{x^3 \left (c+e x^2\right )}{a-b x^4} \, dx+\int \frac{x^4 \left (d+f x^2\right )}{a-b x^4} \, dx\\ &=-\frac{f x^3}{3 b}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (c+e x)}{a-b x^2} \, dx,x,x^2\right )+\frac{\int \frac{x^2 \left (3 a f+3 b d x^2\right )}{a-b x^4} \, dx}{3 b}\\ &=-\frac{d x}{b}-\frac{e x^2}{2 b}-\frac{f x^3}{3 b}+\frac{\int \frac{3 a b d+3 a b f x^2}{a-b x^4} \, dx}{3 b^2}-\frac{\operatorname{Subst}\left (\int \frac{-a e-b c x}{a-b x^2} \, dx,x,x^2\right )}{2 b}\\ &=-\frac{d x}{b}-\frac{e x^2}{2 b}-\frac{f x^3}{3 b}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{x}{a-b x^2} \, dx,x,x^2\right )+\frac{(a e) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{2 b}-\frac{\left (\sqrt{a} \left (\sqrt{b} d-\sqrt{a} f\right )\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{2 b}+\frac{\left (\sqrt{a} \left (\sqrt{b} d+\sqrt{a} f\right )\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{2 b}\\ &=-\frac{d x}{b}-\frac{e x^2}{2 b}-\frac{f x^3}{3 b}+\frac{\sqrt [4]{a} \left (\sqrt{b} d-\sqrt{a} f\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt [4]{a} \left (\sqrt{b} d+\sqrt{a} f\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 b^{7/4}}+\frac{\sqrt{a} e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 b^{3/2}}-\frac{c \log \left (a-b x^4\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0842981, size = 221, normalized size = 1.36 \[ \frac{-3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{3/4} f+\sqrt [4]{a} \sqrt{b} d+\sqrt{a} \sqrt [4]{b} e\right )+3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (a^{3/4} f+\sqrt [4]{a} \sqrt{b} d-\sqrt{a} \sqrt [4]{b} e\right )+6 \left (\sqrt [4]{a} \sqrt{b} d-a^{3/4} f\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-3 b^{3/4} c \log \left (a-b x^4\right )+3 \sqrt{a} \sqrt [4]{b} e \log \left (\sqrt{a}+\sqrt{b} x^2\right )-12 b^{3/4} d x-6 b^{3/4} e x^2-4 b^{3/4} f x^3}{12 b^{7/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 208, normalized size = 1.3 \begin{align*} -{\frac{f{x}^{3}}{3\,b}}-{\frac{e{x}^{2}}{2\,b}}-{\frac{dx}{b}}+{\frac{d}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{d}{4\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{ae}{4\,b}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{af}{2\,{b}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{af}{4\,{b}^{2}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{c\ln \left ( b{x}^{4}-a \right ) }{4\,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 [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 [B] time = 12.5476, size = 887, normalized size = 5.48 \begin{align*} - \operatorname{RootSum}{\left (256 t^{4} b^{7} - 256 t^{3} b^{6} c + t^{2} \left (- 64 a b^{4} d f - 32 a b^{4} e^{2} + 96 b^{5} c^{2}\right ) + t \left (- 16 a^{2} b^{2} e f^{2} + 32 a b^{3} c d f + 16 a b^{3} c e^{2} - 16 a b^{3} d^{2} e - 16 b^{4} c^{3}\right ) - a^{3} f^{4} + 4 a^{2} b c e f^{2} + 2 a^{2} b d^{2} f^{2} - 4 a^{2} b d e^{2} f + a^{2} b e^{4} - 4 a b^{2} c^{2} d f - 2 a b^{2} c^{2} e^{2} + 4 a b^{2} c d^{2} e - a b^{2} d^{4} + b^{3} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a b^{5} f^{3} - 64 t^{3} b^{6} d^{2} f + 128 t^{3} b^{6} d e^{2} + 48 t^{2} a b^{4} c f^{3} + 48 t^{2} a b^{4} d e f^{2} - 32 t^{2} a b^{4} e^{3} f + 48 t^{2} b^{5} c d^{2} f - 96 t^{2} b^{5} c d e^{2} - 16 t^{2} b^{5} d^{3} e + 12 t a^{2} b^{2} d f^{4} + 12 t a^{2} b^{2} e^{2} f^{3} - 12 t a b^{3} c^{2} f^{3} - 24 t a b^{3} c d e f^{2} + 16 t a b^{3} c e^{3} f + 16 t a b^{3} d^{3} f^{2} - 36 t a b^{3} d^{2} e^{2} f - 8 t a b^{3} d e^{4} - 12 t b^{4} c^{2} d^{2} f + 24 t b^{4} c^{2} d e^{2} + 8 t b^{4} c d^{3} e + 4 t b^{4} d^{5} + 3 a^{3} e f^{5} - 3 a^{2} b c d f^{4} - 3 a^{2} b c e^{2} f^{3} - 5 a^{2} b d e^{3} f^{2} + 2 a^{2} b e^{5} f + a b^{2} c^{3} f^{3} + 3 a b^{2} c^{2} d e f^{2} - 2 a b^{2} c^{2} e^{3} f - 4 a b^{2} c d^{3} f^{2} + 9 a b^{2} c d^{2} e^{2} f + 2 a b^{2} c d e^{4} + 5 a b^{2} d^{4} e f - 5 a b^{2} d^{3} e^{3} + b^{3} c^{3} d^{2} f - 2 b^{3} c^{3} d e^{2} - b^{3} c^{2} d^{3} e - b^{3} c d^{5}}{a^{3} f^{6} + a^{2} b d^{2} f^{4} - 8 a^{2} b d e^{2} f^{3} + 4 a^{2} b e^{4} f^{2} - a b^{2} d^{4} f^{2} + 8 a b^{2} d^{3} e^{2} f - 4 a b^{2} d^{2} e^{4} - b^{3} d^{6}} \right )} \right )\right )} - \frac{d x}{b} - \frac{e x^{2}}{2 b} - \frac{f x^{3}}{3 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.07629, size = 443, normalized size = 2.73 \begin{align*} -\frac{c \log \left ({\left | b x^{4} - a \right |}\right )}{4 \, b} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b^{2} e - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, b^{4}} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b^{2} e - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, b^{4}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac{3}{4}} f\right )} \log \left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, b^{4}} - \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (-a b^{3}\right )^{\frac{3}{4}} f\right )} \log \left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, b^{4}} - \frac{2 \, b^{2} f x^{3} + 3 \, b^{2} x^{2} e + 6 \, b^{2} d x}{6 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]