Optimal. Leaf size=237 \[ \frac {(d e+c f) x \sqrt {a+b x^2}}{c (b c+a d) \sqrt {c-d x^2}}-\frac {(d e+c f) \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {e \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {541, 538, 438,
437, 435, 432, 430} \begin {gather*} -\frac {\sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} (c f+d e) E\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2} (a d+b c)}+\frac {e \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} F\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 432
Rule 435
Rule 437
Rule 438
Rule 538
Rule 541
Rubi steps
\begin {align*} \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx &=\frac {(d e+c f) x \sqrt {a+b x^2}}{c (b c+a d) \sqrt {c-d x^2}}+\frac {\int \frac {c (b e-a f)-b (d e+c f) x^2}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx}{c (b c+a d)}\\ &=\frac {(d e+c f) x \sqrt {a+b x^2}}{c (b c+a d) \sqrt {c-d x^2}}+\frac {e \int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx}{c}-\frac {(d e+c f) \int \frac {\sqrt {a+b x^2}}{\sqrt {c-d x^2}} \, dx}{c (b c+a d)}\\ &=\frac {(d e+c f) x \sqrt {a+b x^2}}{c (b c+a d) \sqrt {c-d x^2}}+\frac {\left (e \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}} \, dx}{c \sqrt {c-d x^2}}-\frac {\left ((d e+c f) \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {a+b x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{c (b c+a d) \sqrt {c-d x^2}}\\ &=\frac {(d e+c f) x \sqrt {a+b x^2}}{c (b c+a d) \sqrt {c-d x^2}}-\frac {\left ((d e+c f) \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{c (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\left (e \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}}} \, dx}{c \sqrt {a+b x^2} \sqrt {c-d x^2}}\\ &=\frac {(d e+c f) x \sqrt {a+b x^2}}{c (b c+a d) \sqrt {c-d x^2}}-\frac {(d e+c f) \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {e \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 7.38, size = 213, normalized size = 0.90 \begin {gather*} \frac {\sqrt {\frac {b}{a}} d (d e+c f) x \left (a+b x^2\right )-i b c (d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c (b c+a d) f \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c d (b c+a d) \sqrt {a+b x^2} \sqrt {c-d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 333, normalized size = 1.41
method | result | size |
default | \(\frac {\left (\sqrt {\frac {d}{c}}\, b c f \,x^{3}+\sqrt {\frac {d}{c}}\, b d e \,x^{3}+\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e +\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b c e -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a c f -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e +\sqrt {\frac {d}{c}}\, a c f x +\sqrt {\frac {d}{c}}\, a d e x \right ) \sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}{c \sqrt {\frac {d}{c}}\, \left (a d +b c \right ) \left (-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c \right )}\) | \(333\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (-\frac {\left (-b d \,x^{2}-a d \right ) x \left (c f +d e \right )}{d c \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {c}{d}\right ) \left (-b d \,x^{2}-a d \right )}}+\frac {\left (-\frac {f}{d}+\frac {c f +d e}{d c}-\frac {a \left (c f +d e \right )}{c \left (a d +b c \right )}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c}}+\frac {\left (c f +d e \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{c \left (a d +b c \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c \,x^{2} b +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}\) | \(376\) |
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 {e + f x^{2}}{\sqrt {a + b 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 {f\,x^2+e}{\sqrt {b\,x^2+a}\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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