Optimal. Leaf size=242 \[ -\frac {2 a^2 \sqrt {c+d x^2}}{11 c e (e x)^{11/2}}-\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{231 c^{13/4} e^{13/2} \sqrt {c+d x^2}}-\frac {2 \sqrt {c+d x^2} \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right )}{231 c^3 e^5 (e x)^{3/2}}-\frac {2 a \sqrt {c+d x^2} (22 b c-9 a d)}{77 c^2 e^3 (e x)^{7/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 242, 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, 453, 325, 329, 220} \[ -\frac {2 a^2 \sqrt {c+d x^2}}{11 c e (e x)^{11/2}}-\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{231 c^{13/4} e^{13/2} \sqrt {c+d x^2}}-\frac {2 \sqrt {c+d x^2} \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right )}{231 c^3 e^5 (e x)^{3/2}}-\frac {2 a \sqrt {c+d x^2} (22 b c-9 a d)}{77 c^2 e^3 (e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 325
Rule 329
Rule 453
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \sqrt {c+d x^2}} \, dx &=-\frac {2 a^2 \sqrt {c+d x^2}}{11 c e (e x)^{11/2}}+\frac {2 \int \frac {\frac {1}{2} a (22 b c-9 a d)+\frac {11}{2} b^2 c x^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx}{11 c e^2}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{11 c e (e x)^{11/2}}-\frac {2 a (22 b c-9 a d) \sqrt {c+d x^2}}{77 c^2 e^3 (e x)^{7/2}}+\frac {\left (77 b^2 c^2-5 a d (22 b c-9 a d)\right ) \int \frac {1}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx}{77 c^2 e^4}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{11 c e (e x)^{11/2}}-\frac {2 a (22 b c-9 a d) \sqrt {c+d x^2}}{77 c^2 e^3 (e x)^{7/2}}-\frac {2 \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right ) \sqrt {c+d x^2}}{231 c^3 e^5 (e x)^{3/2}}-\frac {\left (d \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right )\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{231 c^3 e^6}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{11 c e (e x)^{11/2}}-\frac {2 a (22 b c-9 a d) \sqrt {c+d x^2}}{77 c^2 e^3 (e x)^{7/2}}-\frac {2 \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right ) \sqrt {c+d x^2}}{231 c^3 e^5 (e x)^{3/2}}-\frac {\left (2 d \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{231 c^3 e^7}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{11 c e (e x)^{11/2}}-\frac {2 a (22 b c-9 a d) \sqrt {c+d x^2}}{77 c^2 e^3 (e x)^{7/2}}-\frac {2 \left (77 b^2 c^2-5 a d (22 b c-9 a d)\right ) \sqrt {c+d x^2}}{231 c^3 e^5 (e x)^{3/2}}-\frac {d^{3/4} \left (77 b^2 c^2-5 a d (22 b c-9 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 )}{231 c^{13/4} e^{13/2} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 196, normalized size = 0.81 \[ \frac {x^{13/2} \left (-\frac {2 \left (c+d x^2\right ) \left (3 a^2 \left (7 c^2-9 c d x^2+15 d^2 x^4\right )+22 a b c x^2 \left (3 c-5 d x^2\right )+77 b^2 c^2 x^4\right )}{c^3 x^{11/2}}-\frac {2 i d x \sqrt {\frac {c}{d x^2}+1} \left (45 a^2 d^2-110 a b c d+77 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right )\right |-1\right )}{c^3 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{231 (e x)^{13/2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d e^{7} x^{9} + c e^{7} x^{7}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 411, normalized size = 1.70 \[ -\frac {90 a^{2} d^{3} x^{6}-220 a b c \,d^{2} x^{6}+154 b^{2} c^{2} d \,x^{6}+45 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, a^{2} d^{2} x^{5} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-110 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, a b c d \,x^{5} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+77 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, b^{2} c^{2} x^{5} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+36 a^{2} c \,d^{2} x^{4}-88 a b \,c^{2} d \,x^{4}+154 b^{2} c^{3} x^{4}-12 a^{2} c^{2} d \,x^{2}+132 a b \,c^{3} x^{2}+42 a^{2} c^{3}}{231 \sqrt {d \,x^{2}+c}\, \sqrt {e x}\, c^{3} e^{6} x^{5}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {13}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{13/2}\,\sqrt {d\,x^2+c}} \,d x \]
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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