Optimal. Leaf size=227 \[ -\frac{x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{105 f^3}+\frac{x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (231 c^2 d e f^2-48 c^3 f^3-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{105 f^4}-\frac{x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{35 f^2}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{9/2}}+\frac{b x \left (c+d x^2\right )^3}{7 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.371285, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {528, 388, 205} \[ -\frac{x \left (c+d x^2\right ) \left (7 a d f (5 d e-9 c f)-b \left (24 c^2 f^2-63 c d e f+35 d^2 e^2\right )\right )}{105 f^3}+\frac{x \left (7 a d f \left (33 c^2 f^2-40 c d e f+15 d^2 e^2\right )-b \left (231 c^2 d e f^2-48 c^3 f^3-280 c d^2 e^2 f+105 d^3 e^3\right )\right )}{105 f^4}-\frac{x \left (c+d x^2\right )^2 (-7 a d f-6 b c f+7 b d e)}{35 f^2}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{9/2}}+\frac{b x \left (c+d x^2\right )^3}{7 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 528
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (c+d x^2\right )^3}{e+f x^2} \, dx &=\frac{b x \left (c+d x^2\right )^3}{7 f}+\frac{\int \frac{\left (c+d x^2\right )^2 \left (-c (b e-7 a f)+(-7 b d e+6 b c f+7 a d f) x^2\right )}{e+f x^2} \, dx}{7 f}\\ &=-\frac{(7 b d e-6 b c f-7 a d f) x \left (c+d x^2\right )^2}{35 f^2}+\frac{b x \left (c+d x^2\right )^3}{7 f}+\frac{\int \frac{\left (c+d x^2\right ) \left (c (b e (7 d e-11 c f)-7 a f (d e-5 c f))+\left (-7 a d f (5 d e-9 c f)+b \left (35 d^2 e^2-63 c d e f+24 c^2 f^2\right )\right ) x^2\right )}{e+f x^2} \, dx}{35 f^2}\\ &=-\frac{\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d e f+24 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{105 f^3}-\frac{(7 b d e-6 b c f-7 a d f) x \left (c+d x^2\right )^2}{35 f^2}+\frac{b x \left (c+d x^2\right )^3}{7 f}+\frac{\int \frac{c \left (7 a f \left (5 d^2 e^2-12 c d e f+15 c^2 f^2\right )-b e \left (35 d^2 e^2-84 c d e f+57 c^2 f^2\right )\right )+\left (7 a d f \left (15 d^2 e^2-40 c d e f+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 e^2 f+231 c^2 d e f^2-48 c^3 f^3\right )\right ) x^2}{e+f x^2} \, dx}{105 f^3}\\ &=\frac{\left (7 a d f \left (15 d^2 e^2-40 c d e f+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 e^2 f+231 c^2 d e f^2-48 c^3 f^3\right )\right ) x}{105 f^4}-\frac{\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d e f+24 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{105 f^3}-\frac{(7 b d e-6 b c f-7 a d f) x \left (c+d x^2\right )^2}{35 f^2}+\frac{b x \left (c+d x^2\right )^3}{7 f}+\frac{\left ((b e-a f) (d e-c f)^3\right ) \int \frac{1}{e+f x^2} \, dx}{f^4}\\ &=\frac{\left (7 a d f \left (15 d^2 e^2-40 c d e f+33 c^2 f^2\right )-b \left (105 d^3 e^3-280 c d^2 e^2 f+231 c^2 d e f^2-48 c^3 f^3\right )\right ) x}{105 f^4}-\frac{\left (7 a d f (5 d e-9 c f)-b \left (35 d^2 e^2-63 c d e f+24 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{105 f^3}-\frac{(7 b d e-6 b c f-7 a d f) x \left (c+d x^2\right )^2}{35 f^2}+\frac{b x \left (c+d x^2\right )^3}{7 f}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0932308, size = 179, normalized size = 0.79 \[ \frac{d x^3 \left (a d f (3 c f-d e)+b \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{3 f^3}+\frac{x \left (a d f \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )-b (d e-c f)^3\right )}{f^4}+\frac{d^2 x^5 (a d f+3 b c f-b d e)}{5 f^2}+\frac{(b e-a f) (d e-c f)^3 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{9/2}}+\frac{b d^3 x^7}{7 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 401, normalized size = 1.8 \begin{align*}{\frac{b{d}^{3}{x}^{7}}{7\,f}}+{\frac{{x}^{5}a{d}^{3}}{5\,f}}+{\frac{3\,{x}^{5}bc{d}^{2}}{5\,f}}-{\frac{{x}^{5}b{d}^{3}e}{5\,{f}^{2}}}+{\frac{{x}^{3}ac{d}^{2}}{f}}-{\frac{{x}^{3}a{d}^{3}e}{3\,{f}^{2}}}+{\frac{{x}^{3}b{c}^{2}d}{f}}-{\frac{{x}^{3}bc{d}^{2}e}{{f}^{2}}}+{\frac{{x}^{3}b{d}^{3}{e}^{2}}{3\,{f}^{3}}}+3\,{\frac{a{c}^{2}dx}{f}}-3\,{\frac{ac{d}^{2}ex}{{f}^{2}}}+{\frac{a{d}^{3}{e}^{2}x}{{f}^{3}}}+{\frac{b{c}^{3}x}{f}}-3\,{\frac{b{c}^{2}dex}{{f}^{2}}}+3\,{\frac{bc{d}^{2}{e}^{2}x}{{f}^{3}}}-{\frac{b{d}^{3}{e}^{3}x}{{f}^{4}}}+{a{c}^{3}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-3\,{\frac{a{c}^{2}de}{f\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }+3\,{\frac{ac{d}^{2}{e}^{2}}{{f}^{2}\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }-{\frac{a{d}^{3}{e}^{3}}{{f}^{3}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{b{c}^{3}e}{f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+3\,{\frac{b{c}^{2}d{e}^{2}}{{f}^{2}\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }-3\,{\frac{bc{d}^{2}{e}^{3}}{{f}^{3}\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }+{\frac{b{d}^{3}{e}^{4}}{{f}^{4}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}} \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 = 1.55086, size = 1216, normalized size = 5.36 \begin{align*} \left [\frac{30 \, b d^{3} e f^{4} x^{7} - 42 \,{\left (b d^{3} e^{2} f^{3} -{\left (3 \, b c d^{2} + a d^{3}\right )} e f^{4}\right )} x^{5} + 70 \,{\left (b d^{3} e^{3} f^{2} -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f^{3} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e f^{4}\right )} x^{3} - 105 \,{\left (b d^{3} e^{4} + a c^{3} f^{4} -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{2} -{\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{3}\right )} \sqrt{-e f} \log \left (\frac{f x^{2} - 2 \, \sqrt{-e f} x - e}{f x^{2} + e}\right ) - 210 \,{\left (b d^{3} e^{4} f -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f^{2} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{3} -{\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{4}\right )} x}{210 \, e f^{5}}, \frac{15 \, b d^{3} e f^{4} x^{7} - 21 \,{\left (b d^{3} e^{2} f^{3} -{\left (3 \, b c d^{2} + a d^{3}\right )} e f^{4}\right )} x^{5} + 35 \,{\left (b d^{3} e^{3} f^{2} -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f^{3} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e f^{4}\right )} x^{3} + 105 \,{\left (b d^{3} e^{4} + a c^{3} f^{4} -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{2} -{\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{3}\right )} \sqrt{e f} \arctan \left (\frac{\sqrt{e f} x}{e}\right ) - 105 \,{\left (b d^{3} e^{4} f -{\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f^{2} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{3} -{\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{4}\right )} x}{105 \, e f^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 2.00094, size = 508, normalized size = 2.24 \begin{align*} \frac{b d^{3} x^{7}}{7 f} - \frac{\sqrt{- \frac{1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3} \log{\left (- \frac{e f^{4} \sqrt{- \frac{1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3}}{a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3} \log{\left (\frac{e f^{4} \sqrt{- \frac{1}{e f^{9}}} \left (a f - b e\right ) \left (c f - d e\right )^{3}}{a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}} + x \right )}}{2} + \frac{x^{5} \left (a d^{3} f + 3 b c d^{2} f - b d^{3} e\right )}{5 f^{2}} + \frac{x^{3} \left (3 a c d^{2} f^{2} - a d^{3} e f + 3 b c^{2} d f^{2} - 3 b c d^{2} e f + b d^{3} e^{2}\right )}{3 f^{3}} + \frac{x \left (3 a c^{2} d f^{3} - 3 a c d^{2} e f^{2} + a d^{3} e^{2} f + b c^{3} f^{3} - 3 b c^{2} d e f^{2} + 3 b c d^{2} e^{2} f - b d^{3} e^{3}\right )}{f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14847, size = 414, normalized size = 1.82 \begin{align*} \frac{{\left (a c^{3} f^{4} - b c^{3} f^{3} e - 3 \, a c^{2} d f^{3} e + 3 \, b c^{2} d f^{2} e^{2} + 3 \, a c d^{2} f^{2} e^{2} - 3 \, b c d^{2} f e^{3} - a d^{3} f e^{3} + b d^{3} e^{4}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{1}{2}\right )}}{f^{\frac{9}{2}}} + \frac{15 \, b d^{3} f^{6} x^{7} + 63 \, b c d^{2} f^{6} x^{5} + 21 \, a d^{3} f^{6} x^{5} - 21 \, b d^{3} f^{5} x^{5} e + 105 \, b c^{2} d f^{6} x^{3} + 105 \, a c d^{2} f^{6} x^{3} - 105 \, b c d^{2} f^{5} x^{3} e - 35 \, a d^{3} f^{5} x^{3} e + 35 \, b d^{3} f^{4} x^{3} e^{2} + 105 \, b c^{3} f^{6} x + 315 \, a c^{2} d f^{6} x - 315 \, b c^{2} d f^{5} x e - 315 \, a c d^{2} f^{5} x e + 315 \, b c d^{2} f^{4} x e^{2} + 105 \, a d^{3} f^{4} x e^{2} - 105 \, b d^{3} f^{3} x e^{3}}{105 \, f^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]