Optimal. Leaf size=291 \[ \frac{d x \left (3 a f \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )-b e \left (3 c^2 f^2-100 c d e f+105 d^2 e^2\right )\right )}{24 e^2 f^4}+\frac{(d e-c f) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-c^2 f^2-10 c d e f+35 d^2 e^2\right )-3 a f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{8 e^{5/2} f^{9/2}}-\frac{x \left (c+d x^2\right )^2 (b e (7 d e-c f)-3 a f (c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{d x \left (c+d x^2\right ) (b e (35 d e-3 c f)-3 a f (3 c f+5 d e))}{24 e^2 f^3}-\frac{x \left (c+d x^2\right )^3 (b e-a f)}{4 e f \left (e+f x^2\right )^2} \]
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Rubi [A] time = 0.415631, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {526, 528, 388, 205} \[ \frac{d x \left (3 a f \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )-b e \left (3 c^2 f^2-100 c d e f+105 d^2 e^2\right )\right )}{24 e^2 f^4}+\frac{(d e-c f) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-c^2 f^2-10 c d e f+35 d^2 e^2\right )-3 a f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{8 e^{5/2} f^{9/2}}-\frac{x \left (c+d x^2\right )^2 (b e (7 d e-c f)-3 a f (c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{d x \left (c+d x^2\right ) (b e (35 d e-3 c f)-3 a f (3 c f+5 d e))}{24 e^2 f^3}-\frac{x \left (c+d x^2\right )^3 (b e-a f)}{4 e f \left (e+f x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 526
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}{\left (e+f x^2\right )^3} \, dx &=-\frac{(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac{\int \frac{\left (c+d x^2\right )^2 \left (-c (b e+3 a f)-d (7 b e-3 a f) x^2\right )}{\left (e+f x^2\right )^2} \, dx}{4 e f}\\ &=-\frac{(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac{(b e (7 d e-c f)-3 a f (d e+c f)) x \left (c+d x^2\right )^2}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{\int \frac{\left (c+d x^2\right ) \left (-c (3 a f (d e-c f)-b e (7 d e+c f))+d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{e+f x^2} \, dx}{8 e^2 f^2}\\ &=\frac{d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (c+d x^2\right )}{24 e^2 f^3}-\frac{(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac{(b e (7 d e-c f)-3 a f (d e+c f)) x \left (c+d x^2\right )^2}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{\int \frac{-c \left (b e \left (35 d^2 e^2-24 c d e f-3 c^2 f^2\right )-3 a f \left (5 d^2 e^2+3 c^2 f^2\right )\right )+d \left (3 a f \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d e f+3 c^2 f^2\right )\right ) x^2}{e+f x^2} \, dx}{24 e^2 f^3}\\ &=\frac{d \left (3 a f \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d e f+3 c^2 f^2\right )\right ) x}{24 e^2 f^4}+\frac{d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (c+d x^2\right )}{24 e^2 f^3}-\frac{(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac{(b e (7 d e-c f)-3 a f (d e+c f)) x \left (c+d x^2\right )^2}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{\left ((d e-c f) \left (b e \left (35 d^2 e^2-10 c d e f-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right ) \int \frac{1}{e+f x^2} \, dx}{8 e^2 f^4}\\ &=\frac{d \left (3 a f \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d e f+3 c^2 f^2\right )\right ) x}{24 e^2 f^4}+\frac{d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (c+d x^2\right )}{24 e^2 f^3}-\frac{(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac{(b e (7 d e-c f)-3 a f (d e+c f)) x \left (c+d x^2\right )^2}{8 e^2 f^2 \left (e+f x^2\right )}+\frac{(d e-c f) \left (b e \left (35 d^2 e^2-10 c d e f-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{8 e^{5/2} f^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.157301, size = 219, normalized size = 0.75 \[ \frac{(d e-c f) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-c^2 f^2-10 c d e f+35 d^2 e^2\right )-3 a f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{8 e^{5/2} f^{9/2}}+\frac{d^2 x (a d f+3 b c f-3 b d e)}{f^4}-\frac{x (d e-c f)^2 (b e (13 d e-c f)-3 a f (c f+3 d e))}{8 e^2 f^4 \left (e+f x^2\right )}+\frac{x (b e-a f) (d e-c f)^3}{4 e f^4 \left (e+f x^2\right )^2}+\frac{b d^3 x^3}{3 f^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 589, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.6465, size = 2291, normalized size = 7.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 59.853, size = 862, normalized size = 2.96 \begin{align*} \frac{b d^{3} x^{3}}{3 f^{3}} - \frac{\sqrt{- \frac{1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right ) \log{\left (- \frac{e^{3} f^{4} \sqrt{- \frac{1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right )}{3 a c^{3} f^{4} + 3 a c^{2} d e f^{3} + 9 a c d^{2} e^{2} f^{2} - 15 a d^{3} e^{3} f + b c^{3} e f^{3} + 9 b c^{2} d e^{2} f^{2} - 45 b c d^{2} e^{3} f + 35 b d^{3} e^{4}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right ) \log{\left (\frac{e^{3} f^{4} \sqrt{- \frac{1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right )}{3 a c^{3} f^{4} + 3 a c^{2} d e f^{3} + 9 a c d^{2} e^{2} f^{2} - 15 a d^{3} e^{3} f + b c^{3} e f^{3} + 9 b c^{2} d e^{2} f^{2} - 45 b c d^{2} e^{3} f + 35 b d^{3} e^{4}} + x \right )}}{16} + \frac{x^{3} \left (3 a c^{3} f^{5} + 3 a c^{2} d e f^{4} - 15 a c d^{2} e^{2} f^{3} + 9 a d^{3} e^{3} f^{2} + b c^{3} e f^{4} - 15 b c^{2} d e^{2} f^{3} + 27 b c d^{2} e^{3} f^{2} - 13 b d^{3} e^{4} f\right ) + x \left (5 a c^{3} e f^{4} - 3 a c^{2} d e^{2} f^{3} - 9 a c d^{2} e^{3} f^{2} + 7 a d^{3} e^{4} f - b c^{3} e^{2} f^{3} - 9 b c^{2} d e^{3} f^{2} + 21 b c d^{2} e^{4} f - 11 b d^{3} e^{5}\right )}{8 e^{4} f^{4} + 16 e^{3} f^{5} x^{2} + 8 e^{2} f^{6} x^{4}} + \frac{x \left (a d^{3} f + 3 b c d^{2} f - 3 b d^{3} e\right )}{f^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1868, size = 501, normalized size = 1.72 \begin{align*} \frac{{\left (3 \, a c^{3} f^{4} + b c^{3} f^{3} e + 3 \, a c^{2} d f^{3} e + 9 \, b c^{2} d f^{2} e^{2} + 9 \, a c d^{2} f^{2} e^{2} - 45 \, b c d^{2} f e^{3} - 15 \, a d^{3} f e^{3} + 35 \, b d^{3} e^{4}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{5}{2}\right )}}{8 \, f^{\frac{9}{2}}} + \frac{{\left (3 \, a c^{3} f^{5} x^{3} + b c^{3} f^{4} x^{3} e + 3 \, a c^{2} d f^{4} x^{3} e - 15 \, b c^{2} d f^{3} x^{3} e^{2} - 15 \, a c d^{2} f^{3} x^{3} e^{2} + 27 \, b c d^{2} f^{2} x^{3} e^{3} + 9 \, a d^{3} f^{2} x^{3} e^{3} + 5 \, a c^{3} f^{4} x e - 13 \, b d^{3} f x^{3} e^{4} - b c^{3} f^{3} x e^{2} - 3 \, a c^{2} d f^{3} x e^{2} - 9 \, b c^{2} d f^{2} x e^{3} - 9 \, a c d^{2} f^{2} x e^{3} + 21 \, b c d^{2} f x e^{4} + 7 \, a d^{3} f x e^{4} - 11 \, b d^{3} x e^{5}\right )} e^{\left (-2\right )}}{8 \,{\left (f x^{2} + e\right )}^{2} f^{4}} + \frac{b d^{3} f^{6} x^{3} + 9 \, b c d^{2} f^{6} x + 3 \, a d^{3} f^{6} x - 9 \, b d^{3} f^{5} x e}{3 \, f^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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