Optimal. Leaf size=141 \[ \frac {64 \left (a+b x^2\right )^{3/4} (12 b c-7 a d)}{105 a^4 e^3 (e x)^{3/2}}-\frac {16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac {2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.07, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {453, 273, 264} \begin {gather*} \frac {64 \left (a+b x^2\right )^{3/4} (12 b c-7 a d)}{105 a^4 e^3 (e x)^{3/2}}-\frac {16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac {2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rule 453
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{9/4}} \, dx &=-\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {(12 b c-7 a d) \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right )^{9/4}} \, dx}{7 a e^2}\\ &=-\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac {(8 (12 b c-7 a d)) \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx}{35 a^2 e^2}\\ &=-\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac {16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac {(32 (12 b c-7 a d)) \int \frac {1}{(e x)^{5/2} \sqrt [4]{a+b x^2}} \, dx}{35 a^3 e^2}\\ &=-\frac {2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac {16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}+\frac {64 (12 b c-7 a d) \left (a+b x^2\right )^{3/4}}{105 a^4 e^3 (e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 94, normalized size = 0.67 \begin {gather*} \frac {\sqrt {e x} \left (-10 a^3 \left (3 c+7 d x^2\right )+40 a^2 b x^2 \left (3 c-14 d x^2\right )+64 a b^2 x^4 \left (15 c-7 d x^2\right )+768 b^3 c x^6\right )}{105 a^4 e^5 x^4 \left (a+b x^2\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 38.39, size = 130, normalized size = 0.92 \begin {gather*} -\frac {2 \left (a+b x^2\right )^{3/4} \left (15 a^3 c e^6+35 a^3 d e^6 x^2-60 a^2 b c e^6 x^2+280 a^2 b d e^6 x^4-480 a b^2 c e^6 x^4+224 a b^2 d e^6 x^6-384 b^3 c e^6 x^6\right )}{105 a^4 e^3 (e x)^{7/2} \left (a e^2+b e^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 119, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (32 \, {\left (12 \, b^{3} c - 7 \, a b^{2} d\right )} x^{6} + 40 \, {\left (12 \, a b^{2} c - 7 \, a^{2} b d\right )} x^{4} - 15 \, a^{3} c + 5 \, {\left (12 \, a^{2} b c - 7 \, a^{3} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{105 \, {\left (a^{4} b^{2} e^{5} x^{8} + 2 \, a^{5} b e^{5} x^{6} + a^{6} e^{5} x^{4}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {9}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 86, normalized size = 0.61 \begin {gather*} -\frac {2 \left (224 a \,b^{2} d \,x^{6}-384 b^{3} c \,x^{6}+280 a^{2} b d \,x^{4}-480 a \,b^{2} c \,x^{4}+35 a^{3} d \,x^{2}-60 a^{2} b c \,x^{2}+15 c \,a^{3}\right ) x}{105 \left (b \,x^{2}+a \right )^{\frac {5}{4}} \left (e x \right )^{\frac {9}{2}} a^{4}} \end {gather*}
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 \begin {gather*} \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {9}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 144, normalized size = 1.02 \begin {gather*} -\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2\,c}{7\,a\,b^2\,e^4}+\frac {16\,x^4\,\left (7\,a\,d-12\,b\,c\right )}{21\,a^3\,b\,e^4}+\frac {x^2\,\left (70\,a^3\,d-120\,a^2\,b\,c\right )}{105\,a^4\,b^2\,e^4}-\frac {x^6\,\left (768\,b^3\,c-448\,a\,b^2\,d\right )}{105\,a^4\,b^2\,e^4}\right )}{x^7\,\sqrt {e\,x}+\frac {a^2\,x^3\,\sqrt {e\,x}}{b^2}+\frac {2\,a\,x^5\,\sqrt {e\,x}}{b}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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