Optimal. Leaf size=144 \[ -\frac {4 b^{3/2} (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (6 b c-7 a d) F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{21 a^{5/2} e^6 \left (a+b x^2\right )^{3/4}}+\frac {2 \sqrt [4]{a+b x^2} (6 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {453, 325, 329, 237, 335, 275, 231} \[ -\frac {4 b^{3/2} (e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (6 b c-7 a d) F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{21 a^{5/2} e^6 \left (a+b x^2\right )^{3/4}}+\frac {2 \sqrt [4]{a+b x^2} (6 b c-7 a d)}{21 a^2 e^3 (e x)^{3/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 231
Rule 237
Rule 275
Rule 325
Rule 329
Rule 335
Rule 453
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}-\frac {(6 b c-7 a d) \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right )^{3/4}} \, dx}{7 a e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}+\frac {(2 b (6 b c-7 a d)) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx}{21 a^2 e^4}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}+\frac {(4 b (6 b c-7 a d)) \operatorname {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{21 a^2 e^5}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}+\frac {\left (4 b (6 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \operatorname {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 )}{21 a^2 e^5 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}-\frac {\left (4 b (6 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \operatorname {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 )}{21 a^2 e^5 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}-\frac {\left (2 b (6 b c-7 a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{e x}\right )}{21 a^2 e^5 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{7 a e (e x)^{7/2}}+\frac {2 (6 b c-7 a d) \sqrt [4]{a+b x^2}}{21 a^2 e^3 (e x)^{3/2}}-\frac {4 b^{3/2} (6 b c-7 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 )}{21 a^{5/2} e^6 \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 88, normalized size = 0.61 \[ -\frac {2 \sqrt {e x} \left (x^2 \left (\frac {b x^2}{a}+1\right )^{3/4} (7 a d-6 b c) \, _2F_1\left (-\frac {3}{4},\frac {3}{4};\frac {1}{4};-\frac {b x^2}{a}\right )+3 c \left (a+b x^2\right )\right )}{21 a e^5 x^4 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x^{2} + c\right )} \sqrt {e x}}{b e^{5} x^{7} + a e^{5} x^{5}}, x\right ) \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {d\,x^2+c}{{\left (e\,x\right )}^{9/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 170.51, size = 85, normalized size = 0.59 \[ - \frac {c {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{5 b^{\frac {3}{4}} e^{\frac {9}{2}} x^{5}} + \frac {d \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} e^{\frac {9}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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