Optimal. Leaf size=234 \[ -\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}-\frac {2 \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (b^2 c^2-7 a d (a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{c} d^{5/4} e^{5/2} \sqrt {c+d x^2}}-\frac {2 \sqrt {e x} \sqrt {c+d x^2} \left (b^2 c^2-7 a d (a d+2 b c)\right )}{21 c d e^3}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e^3} \]
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Rubi [A] time = 0.19, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {462, 459, 279, 329, 220} \[ -\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}-\frac {2 \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (b^2 c^2-7 a d (a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{c} d^{5/4} e^{5/2} \sqrt {c+d x^2}}-\frac {2 \sqrt {e x} \sqrt {c+d x^2} \left (b^2 c^2-7 a d (a d+2 b c)\right )}{21 c d e^3}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e^3} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 329
Rule 459
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}+\frac {2 \int \frac {\left (\frac {3}{2} a (2 b c+a d)+\frac {3}{2} b^2 c x^2\right ) \sqrt {c+d x^2}}{\sqrt {e x}} \, dx}{3 c e^2}\\ &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e^3}-\frac {\left (b^2 c^2-7 a d (2 b c+a d)\right ) \int \frac {\sqrt {c+d x^2}}{\sqrt {e x}} \, dx}{7 c d e^2}\\ &=-\frac {2 \left (b^2 c^2-7 a d (2 b c+a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{21 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e^3}-\frac {\left (2 \left (b^2 c^2-7 a d (2 b c+a d)\right )\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{21 d e^2}\\ &=-\frac {2 \left (b^2 c^2-7 a d (2 b c+a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{21 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e^3}-\frac {\left (4 \left (b^2 c^2-7 a d (2 b c+a d)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 d e^3}\\ &=-\frac {2 \left (b^2 c^2-7 a d (2 b c+a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{21 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 d e^3}-\frac {2 \left (b^2 c^2-7 a d (2 b c+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 )}{21 \sqrt [4]{c} d^{5/4} e^{5/2} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 171, normalized size = 0.73 \[ \frac {x^{5/2} \left (\frac {2 \left (c+d x^2\right ) \left (-7 a^2 d+14 a b d x^2+b^2 x^2 \left (2 c+3 d x^2\right )\right )}{d x^{3/2}}+\frac {4 i x \sqrt {\frac {c}{d x^2}+1} \left (7 a^2 d^2+14 a b c d-b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right )\right |-1\right )}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{21 (e x)^{5/2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{e^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 383, normalized size = 1.64 \[ \frac {\frac {2 b^{2} d^{3} x^{6}}{7}+\frac {4 a b \,d^{3} x^{4}}{3}+\frac {10 b^{2} c \,d^{2} x^{4}}{21}-\frac {2 a^{2} d^{3} x^{2}}{3}+\frac {4 a b c \,d^{2} x^{2}}{3}+\frac {4 b^{2} c^{2} d \,x^{2}}{21}+\frac {2 \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} d^{2} x \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{3}+\frac {4 \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b c d x \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{2} x \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{21}-\frac {2 a^{2} c \,d^{2}}{3}}{\sqrt {d \,x^{2}+c}\, \sqrt {e x}\, d^{2} e^{2} x} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\left (e x\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{{\left (e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 9.91, size = 153, normalized size = 0.65 \[ \frac {a^{2} \sqrt {c} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {a b \sqrt {c} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {b^{2} \sqrt {c} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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