Optimal. Leaf size=291 \[ \frac {d x \left (3 a f \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )-b e \left (3 c^2 f^2-100 c d e f+105 d^2 e^2\right )\right )}{24 e^2 f^4}+\frac {(d e-c f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-10 c d e f+35 d^2 e^2\right )-3 a f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{8 e^{5/2} f^{9/2}}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-3 c f)-3 a f (3 c f+5 d e))}{24 e^2 f^3}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-3 a f (c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{4 e f \left (e+f x^2\right )^2} \]
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Rubi [A] time = 0.42, antiderivative size = 291, normalized size of antiderivative = 1.00, 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 205
Rule 388
Rule 526
Rule 528
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*}
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Mathematica [A] time = 0.17, 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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fricas [A] time = 0.84, size = 1102, normalized size = 3.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 371, normalized size = 1.27 \[ \frac {{\left (3 \, a c^{3} f^{4} + b c^{3} f^{3} e + 3 \, a c^{2} d f^{3} e + 9 \, b c^{2} d f^{2} e^{2} + 9 \, a c d^{2} f^{2} e^{2} - 45 \, b c d^{2} f e^{3} - 15 \, a d^{3} f e^{3} + 35 \, b d^{3} e^{4}\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {5}{2}\right )}}{8 \, f^{\frac {9}{2}}} + \frac {{\left (3 \, a c^{3} f^{5} x^{3} + b c^{3} f^{4} x^{3} e + 3 \, a c^{2} d f^{4} x^{3} e - 15 \, b c^{2} d f^{3} x^{3} e^{2} - 15 \, a c d^{2} f^{3} x^{3} e^{2} + 27 \, b c d^{2} f^{2} x^{3} e^{3} + 9 \, a d^{3} f^{2} x^{3} e^{3} + 5 \, a c^{3} f^{4} x e - 13 \, b d^{3} f x^{3} e^{4} - b c^{3} f^{3} x e^{2} - 3 \, a c^{2} d f^{3} x e^{2} - 9 \, b c^{2} d f^{2} x e^{3} - 9 \, a c d^{2} f^{2} x e^{3} + 21 \, b c d^{2} f x e^{4} + 7 \, a d^{3} f x e^{4} - 11 \, b d^{3} x e^{5}\right )} e^{\left (-2\right )}}{8 \, {\left (f x^{2} + e\right )}^{2} f^{4}} + \frac {b d^{3} f^{6} x^{3} + 9 \, b c d^{2} f^{6} x + 3 \, a d^{3} f^{6} x - 9 \, b d^{3} f^{5} x e}{3 \, f^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 589, normalized size = 2.02 \[ \frac {3 a \,c^{3} f \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} e^{2}}+\frac {3 a \,c^{2} d \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} e}-\frac {15 a c \,d^{2} x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f}+\frac {9 a \,d^{3} e \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}+\frac {b \,c^{3} x^{3}}{8 \left (f \,x^{2}+e \right )^{2} e}-\frac {15 b \,c^{2} d \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f}+\frac {27 b c \,d^{2} e \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}-\frac {13 b \,d^{3} e^{2} x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f^{3}}+\frac {5 a \,c^{3} x}{8 \left (f \,x^{2}+e \right )^{2} e}-\frac {3 a \,c^{2} d x}{8 \left (f \,x^{2}+e \right )^{2} f}-\frac {9 a c \,d^{2} e x}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}+\frac {7 a \,d^{3} e^{2} x}{8 \left (f \,x^{2}+e \right )^{2} f^{3}}-\frac {b \,c^{3} x}{8 \left (f \,x^{2}+e \right )^{2} f}-\frac {9 b \,c^{2} d e x}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}+\frac {21 b c \,d^{2} e^{2} x}{8 \left (f \,x^{2}+e \right )^{2} f^{3}}-\frac {11 b \,d^{3} e^{3} x}{8 \left (f \,x^{2}+e \right )^{2} f^{4}}+\frac {b \,d^{3} x^{3}}{3 f^{3}}+\frac {3 a \,c^{3} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e^{2}}+\frac {3 a \,c^{2} d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e f}+\frac {9 a c \,d^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{2}}-\frac {15 a \,d^{3} e \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{3}}+\frac {b \,c^{3} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e f}+\frac {9 b \,c^{2} d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{2}}-\frac {45 b c \,d^{2} e \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{3}}+\frac {35 b \,d^{3} e^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{4}}+\frac {a \,d^{3} x}{f^{3}}+\frac {3 b c \,d^{2} x}{f^{3}}-\frac {3 b \,d^{3} e x}{f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.94, size = 343, normalized size = 1.18 \[ -\frac {{\left (13 \, b d^{3} e^{4} f - 3 \, a c^{3} f^{5} - 9 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f^{2} + 15 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{3} - {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{4}\right )} x^{3} + {\left (11 \, b d^{3} e^{5} - 5 \, a c^{3} e f^{4} - 7 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{4} f + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e^{3} f^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{2} f^{3}\right )} x}{8 \, {\left (e^{2} f^{6} x^{4} + 2 \, e^{3} f^{5} x^{2} + e^{4} f^{4}\right )}} + \frac {b d^{3} f x^{3} - 3 \, {\left (3 \, b d^{3} e - {\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} x}{3 \, f^{4}} + \frac {{\left (35 \, b d^{3} e^{4} + 3 \, a c^{3} f^{4} - 15 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{3}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, \sqrt {e f} e^{2} f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 495, normalized size = 1.70 \[ \frac {\frac {x^3\,\left (b\,c^3\,e\,f^4+3\,a\,c^3\,f^5-15\,b\,c^2\,d\,e^2\,f^3+3\,a\,c^2\,d\,e\,f^4+27\,b\,c\,d^2\,e^3\,f^2-15\,a\,c\,d^2\,e^2\,f^3-13\,b\,d^3\,e^4\,f+9\,a\,d^3\,e^3\,f^2\right )}{8\,e^2}-\frac {x\,\left (b\,c^3\,e\,f^3-5\,a\,c^3\,f^4+9\,b\,c^2\,d\,e^2\,f^2+3\,a\,c^2\,d\,e\,f^3-21\,b\,c\,d^2\,e^3\,f+9\,a\,c\,d^2\,e^2\,f^2+11\,b\,d^3\,e^4-7\,a\,d^3\,e^3\,f\right )}{8\,e}}{e^2\,f^4+2\,e\,f^5\,x^2+f^6\,x^4}+x\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{f^3}-\frac {3\,b\,d^3\,e}{f^4}\right )+\frac {b\,d^3\,x^3}{3\,f^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,\left (c\,f-d\,e\right )\,\left (b\,c^2\,e\,f^2+3\,a\,c^2\,f^3+10\,b\,c\,d\,e^2\,f+6\,a\,c\,d\,e\,f^2-35\,b\,d^2\,e^3+15\,a\,d^2\,e^2\,f\right )}{\sqrt {e}\,\left (b\,c^3\,e\,f^3+3\,a\,c^3\,f^4+9\,b\,c^2\,d\,e^2\,f^2+3\,a\,c^2\,d\,e\,f^3-45\,b\,c\,d^2\,e^3\,f+9\,a\,c\,d^2\,e^2\,f^2+35\,b\,d^3\,e^4-15\,a\,d^3\,e^3\,f\right )}\right )\,\left (c\,f-d\,e\right )\,\left (b\,c^2\,e\,f^2+3\,a\,c^2\,f^3+10\,b\,c\,d\,e^2\,f+6\,a\,c\,d\,e\,f^2-35\,b\,d^2\,e^3+15\,a\,d^2\,e^2\,f\right )}{8\,e^{5/2}\,f^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 26.16, size = 865, normalized size = 2.97 \[ \frac {b d^{3} x^{3}}{3 f^{3}} + x \left (\frac {a d^{3}}{f^{3}} + \frac {3 b c d^{2}}{f^{3}} - \frac {3 b d^{3} e}{f^{4}}\right ) - \frac {\sqrt {- \frac {1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right ) \log {\left (- \frac {e^{3} f^{4} \sqrt {- \frac {1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right )}{3 a c^{3} f^{4} + 3 a c^{2} d e f^{3} + 9 a c d^{2} e^{2} f^{2} - 15 a d^{3} e^{3} f + b c^{3} e f^{3} + 9 b c^{2} d e^{2} f^{2} - 45 b c d^{2} e^{3} f + 35 b d^{3} e^{4}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right ) \log {\left (\frac {e^{3} f^{4} \sqrt {- \frac {1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right )}{3 a c^{3} f^{4} + 3 a c^{2} d e f^{3} + 9 a c d^{2} e^{2} f^{2} - 15 a d^{3} e^{3} f + b c^{3} e f^{3} + 9 b c^{2} d e^{2} f^{2} - 45 b c d^{2} e^{3} f + 35 b d^{3} e^{4}} + x \right )}}{16} + \frac {x^{3} \left (3 a c^{3} f^{5} + 3 a c^{2} d e f^{4} - 15 a c d^{2} e^{2} f^{3} + 9 a d^{3} e^{3} f^{2} + b c^{3} e f^{4} - 15 b c^{2} d e^{2} f^{3} + 27 b c d^{2} e^{3} f^{2} - 13 b d^{3} e^{4} f\right ) + x \left (5 a c^{3} e f^{4} - 3 a c^{2} d e^{2} f^{3} - 9 a c d^{2} e^{3} f^{2} + 7 a d^{3} e^{4} f - b c^{3} e^{2} f^{3} - 9 b c^{2} d e^{3} f^{2} + 21 b c d^{2} e^{4} f - 11 b d^{3} e^{5}\right )}{8 e^{4} f^{4} + 16 e^{3} f^{5} x^{2} + 8 e^{2} f^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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