Optimal. Leaf size=393 \[ -\frac {(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c^2 d e^3 \sqrt {c+d x^2}}+\frac {\sqrt {e x} \sqrt {c+d x^2} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c^2 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}} \]
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Rubi [A] time = 0.36, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {462, 457, 329, 305, 220, 1196} \[ \frac {\sqrt {e x} \sqrt {c+d x^2} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )}{c^2 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c^2 d e^3 \sqrt {c+d x^2}}+\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 457
Rule 462
Rule 1196
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}+\frac {2 \int \frac {\sqrt {e x} \left (\frac {1}{2} a (2 b c-3 a d)+\frac {1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{c e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}-\frac {\left (2 a b-\frac {3 b^2 c}{d}-\frac {3 a^2 d}{c}\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{2 c e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}-\frac {\left (2 a b-\frac {3 b^2 c}{d}-\frac {3 a^2 d}{c}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{c e^3}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{c^{3/2} d^{3/2} e^2}-\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{c^{3/2} d^{3/2} e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{c^2 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 126, normalized size = 0.32 \[ \frac {x \left (x^2 \sqrt {\frac {d x^2}{c}+1} \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {d x^2}{c}\right )-3 a^2 d \left (2 c+3 d x^2\right )+6 a b c d x^2-3 b^2 c^2 x^2\right )}{3 c^2 d (e x)^{3/2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d^{2} e^{2} x^{6} + 2 \, c d e^{2} x^{4} + c^{2} e^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 594, normalized size = 1.51 \[ \frac {-6 a^{2} d^{3} x^{2}+4 a b c \,d^{2} x^{2}-2 b^{2} c^{2} d \,x^{2}+6 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-4 a^{2} c \,d^{2}}{2 \sqrt {d \,x^{2}+c}\, \sqrt {e x}\, c^{2} d^{2} e} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right )^{2}}{\left (e x\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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