Optimal. Leaf size=356 \[ \frac {x \sqrt {d x^2+2} \left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right )}{15 d^2 f \sqrt {f x^2+3}}-\frac {\sqrt {2} \sqrt {d x^2+2} \left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{15 d^2 f^{3/2} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}-\frac {\sqrt {2} \sqrt {d x^2+2} (-10 a d f+3 b d+2 b f) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{5 d f^{3/2} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}+\frac {x \sqrt {d x^2+2} \sqrt {f x^2+3} (5 a d f+3 b d-4 b f)}{15 d f}+\frac {b x \left (d x^2+2\right )^{3/2} \sqrt {f x^2+3}}{5 d} \]
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Rubi [A] time = 0.32, antiderivative size = 356, 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} \[ \frac {x \sqrt {d x^2+2} \left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right )}{15 d^2 f \sqrt {f x^2+3}}-\frac {\sqrt {2} \sqrt {d x^2+2} \left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{15 d^2 f^{3/2} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}-\frac {\sqrt {2} \sqrt {d x^2+2} (-10 a d f+3 b d+2 b f) F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{5 d f^{3/2} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}+\frac {x \sqrt {d x^2+2} \sqrt {f x^2+3} (5 a d f+3 b d-4 b f)}{15 d f}+\frac {b x \left (d x^2+2\right )^{3/2} \sqrt {f x^2+3}}{5 d} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 492
Rule 528
Rule 531
Rubi steps
\begin {align*} \int \left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2} \, dx &=\frac {b x \left (2+d x^2\right )^{3/2} \sqrt {3+f x^2}}{5 d}+\frac {\int \frac {\sqrt {2+d x^2} \left (-3 (2 b-5 a d)+(3 b d-4 b f+5 a d f) x^2\right )}{\sqrt {3+f x^2}} \, dx}{5 d}\\ &=\frac {(3 b d-4 b f+5 a d f) x \sqrt {2+d x^2} \sqrt {3+f x^2}}{15 d f}+\frac {b x \left (2+d x^2\right )^{3/2} \sqrt {3+f x^2}}{5 d}+\frac {\int \frac {-6 (3 b d+2 b f-10 a d f)+\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) x^2}{\sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx}{15 d f}\\ &=\frac {(3 b d-4 b f+5 a d f) x \sqrt {2+d x^2} \sqrt {3+f x^2}}{15 d f}+\frac {b x \left (2+d x^2\right )^{3/2} \sqrt {3+f x^2}}{5 d}-\frac {(2 (3 b d+2 b f-10 a d f)) \int \frac {1}{\sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx}{5 d f}+\frac {\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) \int \frac {x^2}{\sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx}{15 d f}\\ &=\frac {\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) x \sqrt {2+d x^2}}{15 d^2 f \sqrt {3+f x^2}}+\frac {(3 b d-4 b f+5 a d f) x \sqrt {2+d x^2} \sqrt {3+f x^2}}{15 d f}+\frac {b x \left (2+d x^2\right )^{3/2} \sqrt {3+f x^2}}{5 d}-\frac {\sqrt {2} (3 b d+2 b f-10 a d f) \sqrt {2+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{5 d f^{3/2} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}-\frac {\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) \int \frac {\sqrt {2+d x^2}}{\left (3+f x^2\right )^{3/2}} \, dx}{5 d^2 f}\\ &=\frac {\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) x \sqrt {2+d x^2}}{15 d^2 f \sqrt {3+f x^2}}+\frac {(3 b d-4 b f+5 a d f) x \sqrt {2+d x^2} \sqrt {3+f x^2}}{15 d f}+\frac {b x \left (2+d x^2\right )^{3/2} \sqrt {3+f x^2}}{5 d}-\frac {\sqrt {2} \left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) \sqrt {2+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{15 d^2 f^{3/2} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}-\frac {\sqrt {2} (3 b d+2 b f-10 a d f) \sqrt {2+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{5 d f^{3/2} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}\\ \end {align*}
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Mathematica [C] time = 0.35, size = 186, normalized size = 0.52 \[ \frac {i \sqrt {3} \left (2 b \left (9 d^2-6 d f+4 f^2\right )-5 a d f (3 d+2 f)\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )+\sqrt {d} f x \sqrt {d x^2+2} \sqrt {f x^2+3} \left (5 a d f+3 b d \left (f x^2+1\right )+2 b f\right )+i \sqrt {3} (3 d-2 f) (5 a d f-6 b d+2 b f) F\left (i \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )}{15 d^{3/2} f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}, x\right ) \]
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 \[ \int {\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 775, normalized size = 2.18 \[ \frac {\sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, \left (3 \sqrt {-f}\, b \,d^{3} f^{2} x^{7}+5 \sqrt {-f}\, a \,d^{3} f^{2} x^{5}+12 \sqrt {-f}\, b \,d^{3} f \,x^{5}+8 \sqrt {-f}\, b \,d^{2} f^{2} x^{5}+15 \sqrt {-f}\, a \,d^{3} f \,x^{3}+10 \sqrt {-f}\, a \,d^{2} f^{2} x^{3}+9 \sqrt {-f}\, b \,d^{3} x^{3}+30 \sqrt {-f}\, b \,d^{2} f \,x^{3}+4 \sqrt {-f}\, b d \,f^{2} x^{3}+30 \sqrt {-f}\, a \,d^{2} f x +15 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, a \,d^{2} f \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )+15 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, a \,d^{2} f \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )+10 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, a d \,f^{2} \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )-10 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, a d \,f^{2} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )+18 \sqrt {-f}\, b \,d^{2} x -18 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, b \,d^{2} \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )+9 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, b \,d^{2} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )+12 \sqrt {-f}\, b d f x +12 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, b d f \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )-18 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, b d f \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )-8 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, b \,f^{2} \EllipticE \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )+8 \sqrt {2}\, \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, b \,f^{2} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-f}\, x}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right )\right )}{15 \left (d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6\right ) \sqrt {-f}\, d^{2} f} \]
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 \[ \int {\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}\,{d x} \]
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 \[ \int \left (b\,x^2+a\right )\,\sqrt {d\,x^2+2}\,\sqrt {f\,x^2+3} \,d x \]
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 \[ \int \left (a + b x^{2}\right ) \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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