Optimal. Leaf size=323 \[ -\frac {b \sqrt {c} \sqrt {d} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} \sqrt {a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 b x (b c-3 a d)}{3 a^2 \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)} \]
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Rubi [A] time = 0.26, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {414, 527, 525, 418, 411} \[ \frac {\sqrt {d} \sqrt {a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 b x (b c-3 a d)}{3 a^2 \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2}-\frac {b \sqrt {c} \sqrt {d} \sqrt {a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 411
Rule 414
Rule 418
Rule 525
Rule 527
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx &=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}-\frac {\int \frac {-2 b c+3 a d-3 b d x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx}{3 a (b c-a d)}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\int \frac {a d (b c+3 a d)+2 b d (b c-3 a d) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx}{3 a^2 (b c-a d)^2}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}-\frac {(b d (b c-9 a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a (b c-a d)^3}+\frac {\left (d \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a^2 (b c-a d)^3}\\ &=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {c} \sqrt {d} (b c-9 a 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)^3 \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 = 1.07, size = 337, normalized size = 1.04 \[ \frac {2 i b c \left (a+b x^2\right ) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (x \sqrt {\frac {b}{a}}\right ),\frac {a d}{b c}\right )+i b c \left (a+b x^2\right ) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (3 a^2 d^2+7 a b c d-2 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+x \sqrt {\frac {b}{a}} \left (3 a^4 d^3+6 a^3 b d^3 x^2+a^2 b^2 d \left (8 c^2+8 c d x^2+3 d^2 x^4\right )+a b^3 c \left (-3 c^2+4 c d x^2+7 d^2 x^4\right )-2 b^4 c^2 x^2 \left (c+d x^2\right )\right )}{3 a^2 c \sqrt {\frac {b}{a}} \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (a d-b c)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{b^{3} d^{2} x^{10} + {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{8} + {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{6} + a^{3} c^{2} + {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{4} + {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 964, normalized size = 2.98 \[ -\frac {-3 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} d^{3} x^{5}-7 \sqrt {-\frac {b}{a}}\, a \,b^{3} c \,d^{2} x^{5}+2 \sqrt {-\frac {b}{a}}\, b^{4} c^{2} d \,x^{5}-6 \sqrt {-\frac {b}{a}}\, a^{3} b \,d^{3} x^{3}-8 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c \,d^{2} x^{3}+3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a^{2} b^{2} c \,d^{2} x^{2} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+6 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a^{2} b^{2} c \,d^{2} x^{2} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-4 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{2} d \,x^{3}+7 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a \,b^{3} c^{2} d \,x^{2} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-8 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a \,b^{3} c^{2} d \,x^{2} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+2 \sqrt {-\frac {b}{a}}\, b^{4} c^{3} x^{3}-2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, b^{4} c^{3} x^{2} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, b^{4} c^{3} x^{2} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-3 \sqrt {-\frac {b}{a}}\, a^{4} d^{3} x +3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a^{3} b c \,d^{2} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+6 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a^{3} b c \,d^{2} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-8 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c^{2} d x +7 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a^{2} b^{2} c^{2} d \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-8 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a^{2} b^{2} c^{2} d \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+3 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{3} x -2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a \,b^{3} c^{3} \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a \,b^{3} c^{3} \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )}{3 \sqrt {d \,x^{2}+c}\, \left (a d -b c \right )^{3} \sqrt {-\frac {b}{a}}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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