Optimal. Leaf size=387 \[ \frac {2 \sqrt {e x} \sqrt {c+d x^2} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right )}{5 c^2 \sqrt {d} e^4 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt {c+d x^2}}-\frac {2 \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt {c+d x^2}}-\frac {2 a^2 \sqrt {c+d x^2}}{5 c e (e x)^{5/2}}-\frac {2 a \sqrt {c+d x^2} (10 b c-3 a d)}{5 c^2 e^3 \sqrt {e x}} \]
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Rubi [A] time = 0.35, antiderivative size = 387, 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, 453, 329, 305, 220, 1196} \[ \frac {2 \sqrt {e x} \sqrt {c+d x^2} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right )}{5 c^2 \sqrt {d} e^4 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt {c+d x^2}}-\frac {2 \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt {c+d x^2}}-\frac {2 a^2 \sqrt {c+d x^2}}{5 c e (e x)^{5/2}}-\frac {2 a \sqrt {c+d x^2} (10 b c-3 a d)}{5 c^2 e^3 \sqrt {e x}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 453
Rule 462
Rule 1196
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \sqrt {c+d x^2}} \, dx &=-\frac {2 a^2 \sqrt {c+d x^2}}{5 c e (e x)^{5/2}}+\frac {2 \int \frac {\frac {1}{2} a (10 b c-3 a d)+\frac {5}{2} b^2 c x^2}{(e x)^{3/2} \sqrt {c+d x^2}} \, dx}{5 c e^2}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{5 c e (e x)^{5/2}}-\frac {2 a (10 b c-3 a d) \sqrt {c+d x^2}}{5 c^2 e^3 \sqrt {e x}}+\frac {\left (5 b^2 c^2+10 a b c d-3 a^2 d^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{5 c^2 e^4}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{5 c e (e x)^{5/2}}-\frac {2 a (10 b c-3 a d) \sqrt {c+d x^2}}{5 c^2 e^3 \sqrt {e x}}+\frac {\left (2 \left (5 b^2 c^2+10 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 c^2 e^5}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{5 c e (e x)^{5/2}}-\frac {2 a (10 b c-3 a d) \sqrt {c+d x^2}}{5 c^2 e^3 \sqrt {e x}}+\frac {\left (2 \left (5 b^2 c^2+10 a b c d-3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 c^{3/2} \sqrt {d} e^4}-\frac {\left (2 \left (5 b^2 c^2+10 a b c d-3 a^2 d^2\right )\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 )}{5 c^{3/2} \sqrt {d} e^4}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{5 c e (e x)^{5/2}}-\frac {2 a (10 b c-3 a d) \sqrt {c+d x^2}}{5 c^2 e^3 \sqrt {e x}}+\frac {2 \left (5 b^2 c^2+10 a b c d-3 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{5 c^2 \sqrt {d} e^4 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \left (5 b^2 c^2+10 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 )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt {c+d x^2}}+\frac {\left (5 b^2 c^2+10 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 )}{5 c^{7/4} d^{3/4} e^{7/2} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 116, normalized size = 0.30 \[ \frac {x \left (2 x^4 \sqrt {\frac {c}{d x^2}+1} \left (-3 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )-2 a \left (c+d x^2\right ) \left (a \left (c-3 d x^2\right )+10 b c x^2\right )\right )}{5 c^2 (e x)^{7/2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d e^{4} x^{6} + c e^{4} x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 626, normalized size = 1.62 \[ -\frac {-6 a^{2} d^{3} x^{4}+20 a b c \,d^{2} x^{4}+6 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{2} x^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{2} x^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-20 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d \,x^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+10 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d \,x^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-10 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} x^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+5 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} x^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-4 a^{2} c \,d^{2} x^{2}+20 a b \,c^{2} d \,x^{2}+2 a^{2} c^{2} d}{5 \sqrt {d \,x^{2}+c}\, \sqrt {e x}\, c^{2} d \,e^{3} x^{2}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{7/2}\,\sqrt {d\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 40.79, size = 155, normalized size = 0.40 \[ \frac {a^{2} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {a b \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {b^{2} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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