Optimal. Leaf size=421 \[ -\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}-\frac {2 \sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {4 \sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {4 \sqrt {e x} \sqrt {c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 (e x)^{3/2} \sqrt {c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 c d e^3}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3} \]
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Rubi [A] time = 0.40, antiderivative size = 421, 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, 459, 279, 329, 305, 220, 1196} \[ -\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}-\frac {4 \sqrt {e x} \sqrt {c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {4 \sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {2 (e x)^{3/2} \sqrt {c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 c d e^3}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 305
Rule 329
Rule 459
Rule 462
Rule 1196
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{3/2}} \, dx &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 \int \sqrt {e x} \left (\frac {1}{2} a (2 b c+5 a d)+\frac {1}{2} b^2 c x^2\right ) \sqrt {c+d x^2} \, dx}{c e^2}\\ &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}-\frac {\left (b^2 c^2-3 a d (2 b c+5 a d)\right ) \int \sqrt {e x} \sqrt {c+d x^2} \, dx}{3 c d e^2}\\ &=-\frac {2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}-\frac {\left (2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right )\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{15 d e^2}\\ &=-\frac {2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}-\frac {\left (4 \left (b^2 c^2-3 a d (2 b c+5 a d)\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 d e^3}\\ &=-\frac {2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}-\frac {\left (4 \sqrt {c} \left (b^2 c^2-3 a d (2 b c+5 a d)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 d^{3/2} e^2}+\frac {\left (4 \sqrt {c} \left (b^2 c^2-3 a d (2 b c+5 a d)\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 )}{15 d^{3/2} e^2}\\ &=-\frac {2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c d e^3}-\frac {4 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}+\frac {4 \sqrt [4]{c} \left (b^2 c^2-3 a d (2 b c+5 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 )}{15 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {2 \sqrt [4]{c} \left (b^2 c^2-3 a d (2 b c+5 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 )}{15 d^{7/4} e^{3/2} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 129, normalized size = 0.31 \[ \frac {x \left (12 x^2 \sqrt {\frac {c}{d x^2}+1} \left (15 a^2 d^2+6 a b c d-b^2 c^2\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )+2 \left (c+d x^2\right ) \left (-45 a^2 d+18 a b d x^2+b^2 x^2 \left (2 c+5 d x^2\right )\right )\right )}{45 d (e x)^{3/2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{e^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 624, normalized size = 1.48 \[ \frac {\frac {2 b^{2} d^{3} x^{6}}{9}+\frac {4 a b \,d^{3} x^{4}}{5}+\frac {14 b^{2} c \,d^{2} x^{4}}{45}-2 a^{2} d^{3} x^{2}+\frac {4 a b c \,d^{2} x^{2}}{5}+\frac {4 b^{2} c^{2} d \,x^{2}}{45}+4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-2 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c \,d^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+\frac {8 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{2} d \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{15}+\frac {2 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{3} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{15}-2 a^{2} c \,d^{2}}{\sqrt {d \,x^{2}+c}\, \sqrt {e x}\, d^{2} e} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\left (e x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{{\left (e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.65, size = 153, normalized size = 0.36 \[ \frac {a^{2} \sqrt {c} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {a b \sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {b^{2} \sqrt {c} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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