Optimal. Leaf size=396 \[ -\frac {x \sqrt {c+d x^2} \left (10 a d f (d e-2 c f)-b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right )}{15 d f^2 \sqrt {e+f x^2}}+\frac {\sqrt {e} \sqrt {c+d x^2} \left (10 a d f (d e-2 c f)-b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {e} \sqrt {c+d x^2} \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (-5 a d f-3 b c f+4 b d e)}{15 f^2}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 f} \]
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Rubi [A] time = 0.45, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {528, 531, 418, 492, 411} \[ -\frac {x \sqrt {c+d x^2} \left (10 a d f (d e-2 c f)-b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right )}{15 d f^2 \sqrt {e+f x^2}}+\frac {\sqrt {e} \sqrt {c+d x^2} \left (10 a d f (d e-2 c f)-b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {e} \sqrt {c+d x^2} \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 f^{5/2} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (-5 a d f-3 b c f+4 b d e)}{15 f^2}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 f} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 492
Rule 528
Rule 531
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{\sqrt {e+f x^2}} \, dx &=\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 f}+\frac {\int \frac {\sqrt {c+d x^2} \left (-c (b e-5 a f)+(-4 b d e+3 b c f+5 a d f) x^2\right )}{\sqrt {e+f x^2}} \, dx}{5 f}\\ &=-\frac {(4 b d e-3 b c f-5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 f^2}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 f}+\frac {\int \frac {-c (5 a f (d e-3 c f)-2 b e (2 d e-3 c f))+\left (-10 a d f (d e-2 c f)+b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 f^2}\\ &=-\frac {(4 b d e-3 b c f-5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 f^2}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 f}-\frac {\left (c \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right )\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 f^2}-\frac {\left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 f^2}\\ &=-\frac {\left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d f^2 \sqrt {e+f x^2}}-\frac {(4 b d e-3 b c f-5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 f^2}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 f}-\frac {\sqrt {e} \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\left (e \left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 d f^2}\\ &=-\frac {\left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d f^2 \sqrt {e+f x^2}}-\frac {(4 b d e-3 b c f-5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 f^2}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 f}+\frac {\sqrt {e} \left (10 a d f (d e-2 c f)-b \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \left (5 a f (d e-3 c f)-b \left (4 d e^2-6 c e f\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 f^{5/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}
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Mathematica [C] time = 0.98, size = 279, normalized size = 0.70 \[ \frac {-i e \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (b \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-10 a d f (d e-2 c f)\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+f x \sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (e+f x^2\right ) \left (5 a d f+b \left (6 c f-4 d e+3 d f x^2\right )\right )+i \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} (c f-d e) (5 a f (2 d e-3 c f)+b e (9 c f-8 d e)) F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{15 f^3 \sqrt {\frac {d}{c}} \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b d x^{4} + {\left (b c + a d\right )} x^{2} + a c\right )} \sqrt {d x^{2} + c}}{\sqrt {f x^{2} + e}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\sqrt {f x^{2} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 924, normalized size = 2.33 \[ \frac {\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}\, \left (3 \sqrt {-\frac {d}{c}}\, b \,d^{2} f^{3} x^{7}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} f^{3} x^{5}+9 \sqrt {-\frac {d}{c}}\, b c d \,f^{3} x^{5}-\sqrt {-\frac {d}{c}}\, b \,d^{2} e \,f^{2} x^{5}+5 \sqrt {-\frac {d}{c}}\, a c d \,f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} e \,f^{2} x^{3}+6 \sqrt {-\frac {d}{c}}\, b \,c^{2} f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, b c d e \,f^{2} x^{3}-4 \sqrt {-\frac {d}{c}}\, b \,d^{2} e^{2} f \,x^{3}+15 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a \,c^{2} f^{3} \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+5 \sqrt {-\frac {d}{c}}\, a c d e \,f^{2} x +20 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a c d e \,f^{2} \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-25 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a c d e \,f^{2} \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-10 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a \,d^{2} e^{2} f \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+10 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, a \,d^{2} e^{2} f \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+6 \sqrt {-\frac {d}{c}}\, b \,c^{2} e \,f^{2} x +3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b \,c^{2} e \,f^{2} \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-9 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b \,c^{2} e \,f^{2} \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-4 \sqrt {-\frac {d}{c}}\, b c d \,e^{2} f x -13 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b c d \,e^{2} f \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+17 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b c d \,e^{2} f \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )+8 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b \,d^{2} e^{3} \EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )-8 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, b \,d^{2} e^{3} \EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \sqrt {\frac {c f}{d e}}\right )\right )}{15 \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) \sqrt {-\frac {d}{c}}\, f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\sqrt {f x^{2} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{3/2}}{\sqrt {f\,x^2+e}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}{\sqrt {e + f x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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