Optimal. Leaf size=192 \[ \frac {7 \sqrt {a} e^4 \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} (6 b c-11 a d) E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 b^{7/2} \sqrt [4]{a+b x^2}}+\frac {7 e^3 (e x)^{3/2} (6 b c-11 a d)}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {e (e x)^{7/2} (6 b c-11 a d)}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{11/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.11, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {457, 285, 284, 335, 196} \[ \frac {7 e^3 (e x)^{3/2} (6 b c-11 a d)}{30 b^3 \sqrt [4]{a+b x^2}}+\frac {7 \sqrt {a} e^4 \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} (6 b c-11 a d) E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 b^{7/2} \sqrt [4]{a+b x^2}}-\frac {e (e x)^{7/2} (6 b c-11 a d)}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{11/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 284
Rule 285
Rule 335
Rule 457
Rubi steps
\begin {align*} \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx &=\frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {\left (2 \left (-3 b c+\frac {11 a d}{2}\right )\right ) \int \frac {(e x)^{9/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a b}\\ &=\frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left (7 (6 b c-11 a d) e^2\right ) \int \frac {(e x)^{5/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{30 b^2}\\ &=\frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {7 (6 b c-11 a d) e^3 (e x)^{3/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}-\frac {\left (7 a (6 b c-11 a d) e^4\right ) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{20 b^3}\\ &=\frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {7 (6 b c-11 a d) e^3 (e x)^{3/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}-\frac {\left (7 a (6 b c-11 a d) e^4 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x}\right ) \int \frac {1}{\left (1+\frac {a}{b x^2}\right )^{5/4} x^2} \, dx}{20 b^4 \sqrt [4]{a+b x^2}}\\ &=\frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {7 (6 b c-11 a d) e^3 (e x)^{3/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {\left (7 a (6 b c-11 a d) e^4 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{20 b^4 \sqrt [4]{a+b x^2}}\\ &=\frac {2 (b c-a d) (e x)^{11/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {7 (6 b c-11 a d) e^3 (e x)^{3/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac {(6 b c-11 a d) e (e x)^{7/2}}{15 a b^2 \sqrt [4]{a+b x^2}}+\frac {7 \sqrt {a} (6 b c-11 a d) e^4 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 b^{7/2} \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 116, normalized size = 0.60 \[ \frac {e^3 (e x)^{3/2} \left (-77 a^2 d+7 \left (a+b x^2\right ) \sqrt [4]{\frac {b x^2}{a}+1} (11 a d-6 b c) \, _2F_1\left (\frac {3}{4},\frac {9}{4};\frac {7}{4};-\frac {b x^2}{a}\right )+a b \left (42 c-22 d x^2\right )+4 b^2 x^2 \left (3 c+d x^2\right )\right )}{12 b^3 \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d e^{4} x^{6} + c e^{4} x^{4}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{\frac {9}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {9}{4}}}\, dx \]
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 \[ \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x\right )}^{9/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{9/4}} \,d x \]
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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