Optimal. Leaf size=237 \[ -\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c+d x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 237, 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 = {412, 525, 418, 411} \[ -\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c+d x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 412
Rule 418
Rule 525
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {-2 c-d x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx}{3 a}\\ &=\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}-\frac {(c d) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a (b c-a d)}+\frac {(2 b c-a d) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a (b c-a d)}\\ &=\frac {x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2}}+\frac {(2 b c-a d) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \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 )}{3 a^2 (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 [C] time = 0.47, size = 243, normalized size = 1.03 \[ \frac {-2 i c \left (a+b x^2\right ) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (a d-b c) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (x \sqrt {\frac {b}{a}}\right ),\frac {a d}{b c}\right )+x \sqrt {\frac {b}{a}} \left (c+d x^2\right ) \left (2 a^2 d+a b \left (d x^2-3 c\right )-2 b^2 c x^2\right )+i c \left (a+b x^2\right ) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (a d-2 b c) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}}, x\right ) \]
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 \[ \int \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 617, normalized size = 2.60 \[ \frac {\sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{5}-2 \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{5}+2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{3}-2 \sqrt {-\frac {b}{a}}\, a b c d \,x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c d \,x^{2} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b c d \,x^{2} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-2 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{3}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c^{2} x^{2} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b^{2} c^{2} x^{2} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+2 \sqrt {-\frac {b}{a}}\, a^{2} c d x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} c d \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a^{2} c d \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-3 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b \,c^{2} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a b \,c^{2} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )}{3 \sqrt {d \,x^{2}+c}\, \sqrt {-\frac {b}{a}}\, \left (a d -b c \right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}} \]
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 \[ \int \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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