Optimal. Leaf size=144 \[ -\frac {4 \sqrt {b} \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} (6 b c-5 a d) E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 a^{5/2} e^4 \sqrt [4]{a+b x^2}}+\frac {2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}-\frac {2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {453, 286, 284, 335, 196} \[ \frac {2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}-\frac {4 \sqrt {b} \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} (6 b c-5 a d) E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 a^{5/2} e^4 \sqrt [4]{a+b x^2}}-\frac {2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 284
Rule 286
Rule 335
Rule 453
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac {2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {(6 b c-5 a d) \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx}{5 a e^2}\\ &=-\frac {2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}+\frac {2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}+\frac {(2 b (6 b c-5 a d)) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a^2 e^4}\\ &=-\frac {2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}+\frac {2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}+\frac {\left (2 (6 b c-5 a d) \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}{5 a^2 e^4 \sqrt [4]{a+b x^2}}\\ &=-\frac {2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}+\frac {2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}-\frac {\left (2 (6 b c-5 a d) \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 )}{5 a^2 e^4 \sqrt [4]{a+b x^2}}\\ &=-\frac {2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}+\frac {2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt {e x} \sqrt [4]{a+b x^2}}-\frac {4 \sqrt {b} (6 b c-5 a d) \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 )}{5 a^{5/2} e^4 \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 78, normalized size = 0.54 \[ \frac {x \left (2 x^2 \sqrt [4]{\frac {b x^2}{a}+1} (6 b c-5 a d) \, _2F_1\left (-\frac {1}{4},\frac {5}{4};\frac {3}{4};-\frac {b x^2}{a}\right )-2 a c\right )}{5 a^2 (e x)^{7/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )} \sqrt {e x}}{b^{2} e^{4} x^{8} + 2 \, a b e^{4} x^{6} + a^{2} e^{4} x^{4}}, x\right ) \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {7}{2}} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {d\,x^2+c}{{\left (e\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 169.39, size = 85, normalized size = 0.59 \[ - \frac {c {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{5 b^{\frac {5}{4}} e^{\frac {7}{2}} x^{5}} + \frac {d \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{4}} e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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