Optimal. Leaf size=273 \[ -\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b x \sqrt {c+d x^2}}{a \sqrt {a+b x^2} (b c-a d)}-\frac {d x \sqrt {a+b x^2}}{a \sqrt {c+d x^2} (b c-a d)}+\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.14, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {414, 21, 422, 418, 492, 411} \[ \frac {b x \sqrt {c+d x^2}}{a \sqrt {a+b x^2} (b c-a d)}-\frac {d x \sqrt {a+b x^2}}{a \sqrt {c+d x^2} (b c-a d)}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 411
Rule 414
Rule 418
Rule 422
Rule 492
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx &=\frac {b x \sqrt {c+d x^2}}{a (b c-a d) \sqrt {a+b x^2}}-\frac {\int \frac {a d+b d x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a (b c-a d)}\\ &=\frac {b x \sqrt {c+d x^2}}{a (b c-a d) \sqrt {a+b x^2}}-\frac {d \int \frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx}{a (b c-a d)}\\ &=\frac {b x \sqrt {c+d x^2}}{a (b c-a d) \sqrt {a+b x^2}}-\frac {d \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{b c-a d}-\frac {(b d) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a (b c-a d)}\\ &=-\frac {d x \sqrt {a+b x^2}}{a (b c-a d) \sqrt {c+d x^2}}+\frac {b x \sqrt {c+d x^2}}{a (b c-a d) \sqrt {a+b x^2}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {(c d) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{a (b c-a d)}\\ &=-\frac {d x \sqrt {a+b x^2}}{a (b c-a d) \sqrt {c+d x^2}}+\frac {b x \sqrt {c+d x^2}}{a (b c-a d) \sqrt {a+b x^2}}+\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 112, normalized size = 0.41 \[ \frac {\frac {a d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} E\left (\sin ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|\frac {b c}{a d}\right )}{\sqrt {-\frac {d}{c}}}-b x \left (c+d x^2\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{b^{2} d x^{6} + {\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c + {\left (2 \, a b c + a^{2} d\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 248, normalized size = 0.91 \[ \frac {\left (-\sqrt {-\frac {b}{a}}\, b d \,x^{3}+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a d \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-\sqrt {-\frac {b}{a}}\, b c x +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, b c \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, b c \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )\right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{\sqrt {-\frac {b}{a}}\, \left (a d -b c \right ) \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right ) a} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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