Optimal. Leaf size=430 \[ -\frac {c^{5/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}}+\frac {2 c^{5/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}}-\frac {2 c e^2 \sqrt {e x} \sqrt {c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{195 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 e (e x)^{3/2} \sqrt {c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{585 d^3}-\frac {2 b (e x)^{7/2} \sqrt {c+d x^2} (11 b c-26 a d)}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3} \]
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Rubi [A] time = 0.41, antiderivative size = 430, 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 = {464, 459, 321, 329, 305, 220, 1196} \[ -\frac {c^{5/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}}+\frac {2 c^{5/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}}-\frac {2 c e^2 \sqrt {e x} \sqrt {c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{195 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 e (e x)^{3/2} \sqrt {c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{585 d^3}-\frac {2 b (e x)^{7/2} \sqrt {c+d x^2} (11 b c-26 a d)}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 321
Rule 329
Rule 459
Rule 464
Rule 1196
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx &=\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}+\frac {2 \int \frac {(e x)^{5/2} \left (\frac {13 a^2 d}{2}-\frac {1}{2} b (11 b c-26 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{13 d}\\ &=-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {1}{117} \left (-117 a^2-\frac {7 b c (11 b c-26 a d)}{d^2}\right ) \int \frac {(e x)^{5/2}}{\sqrt {c+d x^2}} \, dx\\ &=\frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {\left (c \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{195 d}\\ &=\frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {\left (2 c \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d}\\ &=\frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {\left (2 c^{3/2} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d^{3/2}}+\frac {\left (2 c^{3/2} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2\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 )}{195 d^{3/2}}\\ &=\frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {2 c \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{195 d^{3/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 c^{5/4} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^{5/2} \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 )}{195 d^{7/4} \sqrt {c+d x^2}}-\frac {c^{5/4} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^{5/2} \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 )}{195 d^{7/4} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 143, normalized size = 0.33 \[ \frac {2 e (e x)^{3/2} \left (\left (c+d x^2\right ) \left (117 a^2 d^2+26 a b d \left (5 d x^2-7 c\right )+b^2 \left (77 c^2-55 c d x^2+45 d^2 x^4\right )\right )-3 c \sqrt {\frac {c}{d x^2}+1} \left (117 a^2 d^2-182 a b c d+77 b^2 c^2\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{585 d^3 \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} e^{2} x^{6} + 2 \, a b e^{2} x^{4} + a^{2} e^{2} x^{2}\right )} \sqrt {e x}}{\sqrt {d x^{2} + c}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{\sqrt {d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 661, normalized size = 1.54 \[ -\frac {\sqrt {e x}\, \left (-90 b^{2} d^{4} x^{8}-260 a b \,d^{4} x^{6}+20 b^{2} c \,d^{3} x^{6}-234 a^{2} d^{4} x^{4}+104 a b c \,d^{3} x^{4}-44 b^{2} c^{2} d^{2} x^{4}-234 a^{2} c \,d^{3} x^{2}+364 a b \,c^{2} d^{2} x^{2}-154 b^{2} c^{3} d \,x^{2}+702 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c^{2} d^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-351 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c^{2} d^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-1092 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{3} d \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+546 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{3} d \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+462 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{4} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-231 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{4} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )\right ) e^{2}}{585 \sqrt {d \,x^{2}+c}\, d^{4} x} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{\sqrt {d x^{2} + c}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2}{\sqrt {d\,x^2+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 49.39, size = 144, normalized size = 0.33 \[ \frac {a^{2} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {11}{4}\right )} + \frac {a b e^{\frac {5}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} \Gamma \left (\frac {15}{4}\right )} + \frac {b^{2} e^{\frac {5}{2}} x^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {19}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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