Optimal. Leaf size=271 \[ \frac {(2 b c-a d) f x \sqrt {c+d x^2}}{c d^2 \sqrt {e+f x^2}}-\frac {(b c-a d) x \sqrt {e+f x^2}}{c d \sqrt {c+d x^2}}-\frac {(2 b c-a d) \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c d^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c d \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {540, 545, 429,
506, 422} \begin {gather*} -\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} (2 b c-a d) E\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c d^2 \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {f x \sqrt {c+d x^2} (2 b c-a d)}{c d^2 \sqrt {e+f x^2}}-\frac {x \sqrt {e+f x^2} (b c-a d)}{c d \sqrt {c+d x^2}}+\frac {b e^{3/2} \sqrt {c+d x^2} F\left (\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 506
Rule 540
Rule 545
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{3/2}} \, dx &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{c d \sqrt {c+d x^2}}-\frac {\int \frac {-b c e-(2 b c-a d) f x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{c d}\\ &=-\frac {(b c-a d) x \sqrt {e+f x^2}}{c d \sqrt {c+d x^2}}+\frac {(b e) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{d}+\frac {((2 b c-a d) f) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{c d}\\ &=\frac {(2 b c-a d) f x \sqrt {c+d x^2}}{c d^2 \sqrt {e+f x^2}}-\frac {(b c-a d) x \sqrt {e+f x^2}}{c d \sqrt {c+d x^2}}+\frac {b e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c d \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {((2 b c-a d) e f) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{c d^2}\\ &=\frac {(2 b c-a d) f x \sqrt {c+d x^2}}{c d^2 \sqrt {e+f x^2}}-\frac {(b c-a d) x \sqrt {e+f x^2}}{c d \sqrt {c+d x^2}}-\frac {(2 b c-a d) \sqrt {e} \sqrt {f} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c d^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c d \sqrt {f} \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] Result contains complex when optimal does not.
time = 2.68, size = 192, normalized size = 0.71 \begin {gather*} \frac {-i (2 b c-a d) e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-(b c-a d) \left (\sqrt {\frac {d}{c}} x \left (e+f x^2\right )-i e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )\right )}{c^2 \left (\frac {d}{c}\right )^{3/2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 328, normalized size = 1.21
method | result | size |
default | \(\frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \left (\sqrt {-\frac {d}{c}}\, a d f \,x^{3}-\sqrt {-\frac {d}{c}}\, b c f \,x^{3}+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a d e -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c e -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a d e +2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c e +\sqrt {-\frac {d}{c}}\, a d e x -\sqrt {-\frac {d}{c}}\, b c e x \right )}{d \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) c \sqrt {-\frac {d}{c}}}\) | \(328\) |
elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (d f \,x^{2}+d e \right ) \left (a d -b c \right ) x}{d^{2} c \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}+\frac {\left (\frac {a d f -b c f +b d e}{d^{2}}-\frac {\left (a d -b c \right ) \left (c f -d e \right )}{d^{2} c}-\frac {e \left (a d -b c \right )}{d c}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (\frac {b f}{d}-\frac {\left (a d -b c \right ) f}{d c}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(379\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b x^{2}\right ) \sqrt {e + f x^{2}}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (b\,x^2+a\right )\,\sqrt {f\,x^2+e}}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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