Optimal. Leaf size=242 \[ \frac {b x}{a (b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} (b c+a d) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {2 b \sqrt {c} \sqrt {d} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {425, 539, 429,
422} \begin {gather*} -\frac {2 b \sqrt {c} \sqrt {d} \sqrt {a+b x^2} F\left (\text {ArcTan}\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)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} (a d+b c) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b x}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 425
Rule 429
Rule 539
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx &=\frac {b x}{a (b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}-\frac {\int \frac {a d-b d x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx}{a (b c-a d)}\\ &=\frac {b x}{a (b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}-\frac {(2 b d) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{(b c-a d)^2}+\frac {(d (b c+a d)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{a (b c-a d)^2}\\ &=\frac {b x}{a (b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} (b c+a 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} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {2 b \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)^2 \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 [C] Result contains complex when optimal does not.
time = 4.51, size = 224, normalized size = 0.93 \begin {gather*} \frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} x \left (a^2 d^2+a b d^2 x^2+b^2 c \left (c+d x^2\right )\right )+i b c (b c+a d) \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 b c (-b c+a d) \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 )\right )}{b c (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 354, normalized size = 1.46
method | result | size |
default | \(\frac {\left (\sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2}+\sqrt {-\frac {b}{a}}\, a^{2} d^{2} x +\sqrt {-\frac {b}{a}}\, b^{2} c^{2} x \right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{c \sqrt {-\frac {b}{a}}\, a \left (a d -b c \right )^{2} \left (b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c \right )}\) | \(354\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {2 b d \left (-\frac {\left (a d +b c \right ) x^{3}}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b d}\right )}{\sqrt {\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{b d}+\frac {a c}{b d}\right ) b d}}+\frac {\left (\frac {1}{a c}-\frac {a^{2} d^{2}+b^{2} c^{2}}{a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}+\frac {b \left (a d +b c \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(464\) |
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 {1}{\left (a + b x^{2}\right )^{\frac {3}{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 {1}{{\left (b\,x^2+a\right )}^{3/2}\,{\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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