Optimal. Leaf size=348 \[ \frac {d x \left (b e \left (-3 c^2 f^2-10 c d e f+105 d^2 e^2\right )-a f \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )}{48 e^3 f^4}-\frac {x \left (c+d x^2\right ) \left (b e \left (-3 c^2 f^2-8 c d e f+35 d^2 e^2\right )-a f \left (15 c^2 f^2+4 c d e f+5 d^2 e^2\right )\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^3 f^3-3 c^2 d e f^2-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-a f (5 c f+d e))}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{6 e f \left (e+f x^2\right )^3} \]
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Rubi [A] time = 0.45, antiderivative size = 348, normalized size of antiderivative = 1.00, 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 = {526, 388, 205} \[ -\frac {x \left (c+d x^2\right ) \left (b e \left (-3 c^2 f^2-8 c d e f+35 d^2 e^2\right )-a f \left (15 c^2 f^2+4 c d e f+5 d^2 e^2\right )\right )}{48 e^3 f^3 \left (e+f x^2\right )}+\frac {d x \left (b e \left (-3 c^2 f^2-10 c d e f+105 d^2 e^2\right )-a f \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )}{48 e^3 f^4}-\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-3 c^2 d e f^2-c^3 f^3-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (3 c^2 d e f^2+5 c^3 f^3+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-a f (5 c f+d e))}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{6 e f \left (e+f x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 526
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 )^4} \, dx &=-\frac {(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac {\int \frac {\left (c+d x^2\right )^2 \left (-c (b e+5 a f)-d (7 b e-a f) x^2\right )}{\left (e+f x^2\right )^3} \, dx}{6 e f}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (c+d x^2\right )^2}{24 e^2 f^2 \left (e+f x^2\right )^2}+\frac {\int \frac {\left (c+d x^2\right ) \left (c (d e (7 b e-a f)+3 c f (b e+5 a f))+d (b e (35 d e-c f)-5 a f (d e+c f)) x^2\right )}{\left (e+f x^2\right )^2} \, dx}{24 e^2 f^2}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (c+d x^2\right )^2}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {\left (b e \left (35 d^2 e^2-8 c d e f-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d e f+15 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac {\int \frac {c \left (a f \left (5 d^2 e^2+6 c d e f-15 c^2 f^2\right )-b e \left (35 d^2 e^2+6 c d e f+3 c^2 f^2\right )\right )-d \left (b e \left (105 d^2 e^2-10 c d e f-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right ) x^2}{e+f x^2} \, dx}{48 e^3 f^3}\\ &=\frac {d \left (b e \left (105 d^2 e^2-10 c d e f-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right ) x}{48 e^3 f^4}-\frac {(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (c+d x^2\right )^2}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {\left (b e \left (35 d^2 e^2-8 c d e f-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d e f+15 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac {\left (b e \left (35 d^3 e^3-15 c d^2 e^2 f-3 c^2 d e f^2-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \int \frac {1}{e+f x^2} \, dx}{16 e^3 f^4}\\ &=\frac {d \left (b e \left (105 d^2 e^2-10 c d e f-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right ) x}{48 e^3 f^4}-\frac {(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac {(b e (7 d e-c f)-a f (d e+5 c f)) x \left (c+d x^2\right )^2}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac {\left (b e \left (35 d^2 e^2-8 c d e f-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d e f+15 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac {\left (b e \left (35 d^3 e^3-15 c d^2 e^2 f-3 c^2 d e f^2-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 295, normalized size = 0.85 \[ \frac {x (d e-c f) \left (b e \left (-c^2 f^2-4 c d e f+29 d^2 e^2\right )-a f \left (5 c^2 f^2+8 c d e f+11 d^2 e^2\right )\right )}{16 e^3 f^4 \left (e+f x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^3 f^3-3 c^2 d e f^2-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}-\frac {x (d e-c f)^2 (b e (19 d e-c f)-a f (5 c f+13 d e))}{24 e^2 f^4 \left (e+f x^2\right )^2}+\frac {x (b e-a f) (d e-c f)^3}{6 e f^4 \left (e+f x^2\right )^3}+\frac {b d^3 x}{f^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.97, size = 1422, normalized size = 4.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 447, normalized size = 1.28 \[ \frac {b d^{3} x}{f^{4}} + \frac {{\left (5 \, a c^{3} f^{4} + b c^{3} f^{3} e + 3 \, a c^{2} d f^{3} e + 3 \, b c^{2} d f^{2} e^{2} + 3 \, a c d^{2} f^{2} e^{2} + 15 \, b c d^{2} f e^{3} + 5 \, a d^{3} f e^{3} - 35 \, b d^{3} e^{4}\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {7}{2}\right )}}{16 \, f^{\frac {9}{2}}} + \frac {{\left (15 \, a c^{3} f^{6} x^{5} + 3 \, b c^{3} f^{5} x^{5} e + 9 \, a c^{2} d f^{5} x^{5} e + 9 \, b c^{2} d f^{4} x^{5} e^{2} + 9 \, a c d^{2} f^{4} x^{5} e^{2} - 99 \, b c d^{2} f^{3} x^{5} e^{3} - 33 \, a d^{3} f^{3} x^{5} e^{3} + 40 \, a c^{3} f^{5} x^{3} e + 87 \, b d^{3} f^{2} x^{5} e^{4} + 8 \, b c^{3} f^{4} x^{3} e^{2} + 24 \, a c^{2} d f^{4} x^{3} e^{2} - 24 \, b c^{2} d f^{3} x^{3} e^{3} - 24 \, a c d^{2} f^{3} x^{3} e^{3} - 120 \, b c d^{2} f^{2} x^{3} e^{4} - 40 \, a d^{3} f^{2} x^{3} e^{4} + 33 \, a c^{3} f^{4} x e^{2} + 136 \, b d^{3} f x^{3} e^{5} - 3 \, b c^{3} f^{3} x e^{3} - 9 \, a c^{2} d f^{3} x e^{3} - 9 \, b c^{2} d f^{2} x e^{4} - 9 \, a c d^{2} f^{2} x e^{4} - 45 \, b c d^{2} f x e^{5} - 15 \, a d^{3} f x e^{5} + 57 \, b d^{3} x e^{6}\right )} e^{\left (-3\right )}}{48 \, {\left (f x^{2} + e\right )}^{3} f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 735, normalized size = 2.11 \[ \frac {5 a \,c^{3} f^{2} x^{5}}{16 \left (f \,x^{2}+e \right )^{3} e^{3}}+\frac {3 a \,c^{2} d f \,x^{5}}{16 \left (f \,x^{2}+e \right )^{3} e^{2}}+\frac {3 a c \,d^{2} x^{5}}{16 \left (f \,x^{2}+e \right )^{3} e}-\frac {11 a \,d^{3} x^{5}}{16 \left (f \,x^{2}+e \right )^{3} f}+\frac {b \,c^{3} f \,x^{5}}{16 \left (f \,x^{2}+e \right )^{3} e^{2}}+\frac {3 b \,c^{2} d \,x^{5}}{16 \left (f \,x^{2}+e \right )^{3} e}-\frac {33 b c \,d^{2} x^{5}}{16 \left (f \,x^{2}+e \right )^{3} f}+\frac {29 b \,d^{3} e \,x^{5}}{16 \left (f \,x^{2}+e \right )^{3} f^{2}}+\frac {5 a \,c^{3} f \,x^{3}}{6 \left (f \,x^{2}+e \right )^{3} e^{2}}+\frac {a \,c^{2} d \,x^{3}}{2 \left (f \,x^{2}+e \right )^{3} e}-\frac {a c \,d^{2} x^{3}}{2 \left (f \,x^{2}+e \right )^{3} f}-\frac {5 a \,d^{3} e \,x^{3}}{6 \left (f \,x^{2}+e \right )^{3} f^{2}}+\frac {b \,c^{3} x^{3}}{6 \left (f \,x^{2}+e \right )^{3} e}-\frac {b \,c^{2} d \,x^{3}}{2 \left (f \,x^{2}+e \right )^{3} f}-\frac {5 b c \,d^{2} e \,x^{3}}{2 \left (f \,x^{2}+e \right )^{3} f^{2}}+\frac {17 b \,d^{3} e^{2} x^{3}}{6 \left (f \,x^{2}+e \right )^{3} f^{3}}+\frac {11 a \,c^{3} x}{16 \left (f \,x^{2}+e \right )^{3} e}-\frac {3 a \,c^{2} d x}{16 \left (f \,x^{2}+e \right )^{3} f}-\frac {3 a c \,d^{2} e x}{16 \left (f \,x^{2}+e \right )^{3} f^{2}}-\frac {5 a \,d^{3} e^{2} x}{16 \left (f \,x^{2}+e \right )^{3} f^{3}}-\frac {b \,c^{3} x}{16 \left (f \,x^{2}+e \right )^{3} f}-\frac {3 b \,c^{2} d e x}{16 \left (f \,x^{2}+e \right )^{3} f^{2}}-\frac {15 b c \,d^{2} e^{2} x}{16 \left (f \,x^{2}+e \right )^{3} f^{3}}+\frac {19 b \,d^{3} e^{3} x}{16 \left (f \,x^{2}+e \right )^{3} f^{4}}+\frac {5 a \,c^{3} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, e^{3}}+\frac {3 a \,c^{2} d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, e^{2} f}+\frac {3 a c \,d^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, e \,f^{2}}+\frac {5 a \,d^{3} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, f^{3}}+\frac {b \,c^{3} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, e^{2} f}+\frac {3 b \,c^{2} d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, e \,f^{2}}+\frac {15 b c \,d^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, f^{3}}-\frac {35 b \,d^{3} e \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \sqrt {e f}\, f^{4}}+\frac {b \,d^{3} x}{f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 416, normalized size = 1.20 \[ \frac {b d^{3} x}{f^{4}} + \frac {3 \, {\left (29 \, b d^{3} e^{4} f^{2} + 5 \, a c^{3} f^{6} - 11 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f^{3} + 3 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{4} + {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{5}\right )} x^{5} + 8 \, {\left (17 \, b d^{3} e^{5} f + 5 \, a c^{3} e f^{5} - 5 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{4} f^{2} - 3 \, {\left (b c^{2} d + a c d^{2}\right )} e^{3} f^{3} + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{2} f^{4}\right )} x^{3} + 3 \, {\left (19 \, b d^{3} e^{6} + 11 \, a c^{3} e^{2} f^{4} - 5 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{5} f - 3 \, {\left (b c^{2} d + a c d^{2}\right )} e^{4} f^{2} - {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{3} f^{3}\right )} x}{48 \, {\left (e^{3} f^{7} x^{6} + 3 \, e^{4} f^{6} x^{4} + 3 \, e^{5} f^{5} x^{2} + e^{6} f^{4}\right )}} - \frac {{\left (35 \, b d^{3} e^{4} - 5 \, a c^{3} f^{4} - 5 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f - 3 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{2} - {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{3}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \, \sqrt {e f} e^{3} f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 444, normalized size = 1.28 \[ \frac {\frac {x^3\,\left (b\,c^3\,e\,f^4+5\,a\,c^3\,f^5-3\,b\,c^2\,d\,e^2\,f^3+3\,a\,c^2\,d\,e\,f^4-15\,b\,c\,d^2\,e^3\,f^2-3\,a\,c\,d^2\,e^2\,f^3+17\,b\,d^3\,e^4\,f-5\,a\,d^3\,e^3\,f^2\right )}{6\,e^2}+\frac {x^5\,\left (b\,c^3\,e\,f^5+5\,a\,c^3\,f^6+3\,b\,c^2\,d\,e^2\,f^4+3\,a\,c^2\,d\,e\,f^5-33\,b\,c\,d^2\,e^3\,f^3+3\,a\,c\,d^2\,e^2\,f^4+29\,b\,d^3\,e^4\,f^2-11\,a\,d^3\,e^3\,f^3\right )}{16\,e^3}-\frac {x\,\left (b\,c^3\,e\,f^3-11\,a\,c^3\,f^4+3\,b\,c^2\,d\,e^2\,f^2+3\,a\,c^2\,d\,e\,f^3+15\,b\,c\,d^2\,e^3\,f+3\,a\,c\,d^2\,e^2\,f^2-19\,b\,d^3\,e^4+5\,a\,d^3\,e^3\,f\right )}{16\,e}}{e^3\,f^4+3\,e^2\,f^5\,x^2+3\,e\,f^6\,x^4+f^7\,x^6}+\frac {b\,d^3\,x}{f^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (b\,c^3\,e\,f^3+5\,a\,c^3\,f^4+3\,b\,c^2\,d\,e^2\,f^2+3\,a\,c^2\,d\,e\,f^3+15\,b\,c\,d^2\,e^3\,f+3\,a\,c\,d^2\,e^2\,f^2-35\,b\,d^3\,e^4+5\,a\,d^3\,e^3\,f\right )}{16\,e^{7/2}\,f^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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