Integrand size = 26, antiderivative size = 182 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}}+\frac {2 (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^2 e^3 (e x)^{7/2}}-\frac {4 b (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^3 e^5 (e x)^{3/2}}+\frac {8 b^{5/2} (10 b c-11 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{77 a^{7/2} e^8 \left (a+b x^2\right )^{3/4}} \]
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Time = 0.10 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {464, 331, 335, 243, 342, 281, 237} \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {8 b^{5/2} (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (10 b c-11 a d) \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{77 a^{7/2} e^8 \left (a+b x^2\right )^{3/4}}-\frac {4 b \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^3 e^5 (e x)^{3/2}}+\frac {2 \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^2 e^3 (e x)^{7/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}} \]
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Rule 237
Rule 243
Rule 281
Rule 331
Rule 335
Rule 342
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}}-\frac {(10 b c-11 a d) \int \frac {1}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx}{11 a e^2} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}}+\frac {2 (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^2 e^3 (e x)^{7/2}}+\frac {(6 b (10 b c-11 a d)) \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx}{77 a^2 e^4} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}}+\frac {2 (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^2 e^3 (e x)^{7/2}}-\frac {4 b (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^3 e^5 (e x)^{3/2}}-\frac {\left (4 b^2 (10 b c-11 a d)\right ) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx}{77 a^3 e^6} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}}+\frac {2 (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^2 e^3 (e x)^{7/2}}-\frac {4 b (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^3 e^5 (e x)^{3/2}}-\frac {\left (8 b^2 (10 b c-11 a d)\right ) \text {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{77 a^3 e^7} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}}+\frac {2 (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^2 e^3 (e x)^{7/2}}-\frac {4 b (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^3 e^5 (e x)^{3/2}}-\frac {\left (8 b^2 (10 b c-11 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {e x}\right )}{77 a^3 e^7 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}}+\frac {2 (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^2 e^3 (e x)^{7/2}}-\frac {4 b (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^3 e^5 (e x)^{3/2}}+\frac {\left (8 b^2 (10 b c-11 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a e^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {e x}}\right )}{77 a^3 e^7 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}}+\frac {2 (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^2 e^3 (e x)^{7/2}}-\frac {4 b (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^3 e^5 (e x)^{3/2}}+\frac {\left (4 b^2 (10 b c-11 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{e x}\right )}{77 a^3 e^7 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}}+\frac {2 (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^2 e^3 (e x)^{7/2}}-\frac {4 b (10 b c-11 a d) \sqrt [4]{a+b x^2}}{77 a^3 e^5 (e x)^{3/2}}+\frac {8 b^{5/2} (10 b c-11 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{77 a^{7/2} e^8 \left (a+b x^2\right )^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.48 \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 \sqrt {e x} \left (7 c \left (a+b x^2\right )+(-10 b c+11 a d) x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{4},-\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{77 a e^7 x^6 \left (a+b x^2\right )^{3/4}} \]
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\[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {13}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x\]
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\[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {13}{2}}} \,d x } \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {13}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {13}{2}}} \,d x } \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx=\int \frac {d\,x^2+c}{{\left (e\,x\right )}^{13/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
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