Optimal. Leaf size=442 \[ -\frac {(e x)^{3/2} \left (7 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {\sqrt {e x} \sqrt {c+d x^2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {(e x)^{3/2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d e^3 \sqrt {c+d x^2}} \]
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Rubi [A] time = 0.42, antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {462, 457, 290, 329, 305, 220, 1196} \[ -\frac {(e x)^{3/2} \left (7 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\sqrt {e x} \sqrt {c+d x^2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {(e x)^{3/2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d e^3 \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 290
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 )^{5/2}} \, dx &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {e x} \left (\frac {1}{2} a (2 b c-7 a d)+\frac {1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx}{c e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \int \frac {\sqrt {e x}}{\left (c+d x^2\right )^{3/2}} \, dx}{2 c^2 d e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{4 c^3 d e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^3 d e^3}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^{5/2} d^{3/2} e^2}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\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 )}{2 c^{5/2} d^{3/2} e^2}\\ &=-\frac {2 a^2}{c e \sqrt {e x} \left (c+d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) (e x)^{3/2}}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) (e x)^{3/2}}{2 c^3 d e^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{2 c^3 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (b^2 c^2+a d (2 b c-7 a d)\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 )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2+a d (2 b c-7 a d)\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 )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 161, normalized size = 0.36 \[ \frac {x \left (-x^2 \left (c+d x^2\right ) \sqrt {\frac {d x^2}{c}+1} \left (-7 a^2 d^2+2 a b c d+b^2 c^2\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {d x^2}{c}\right )+a^2 (-d) \left (12 c^2+35 c d x^2+21 d^2 x^4\right )+2 a b c d x^2 \left (5 c+3 d x^2\right )+b^2 c^2 x^2 \left (c+3 d x^2\right )\right )}{6 c^3 d (e x)^{3/2} \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d^{3} e^{2} x^{8} + 3 \, c d^{2} e^{2} x^{6} + 3 \, c^{2} d e^{2} x^{4} + c^{3} e^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1187, normalized size = 2.69 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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