Optimal. Leaf size=178 \[ -\frac {256 \left (a+b x^2\right )^{7/4} (16 b c-11 a d)}{385 a^5 e^3 (e x)^{7/2}}+\frac {64 \left (a+b x^2\right )^{3/4} (16 b c-11 a d)}{55 a^4 e^3 (e x)^{7/2}}-\frac {24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.09, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {453, 273, 264} \[ -\frac {256 \left (a+b x^2\right )^{7/4} (16 b c-11 a d)}{385 a^5 e^3 (e x)^{7/2}}+\frac {64 \left (a+b x^2\right )^{3/4} (16 b c-11 a d)}{55 a^4 e^3 (e x)^{7/2}}-\frac {24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}} \]
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)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx &=-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac {(16 b c-11 a d) \int \frac {1}{(e x)^{9/2} \left (a+b x^2\right )^{9/4}} \, dx}{11 a e^2}\\ &=-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {(12 (16 b c-11 a d)) \int \frac {1}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx}{55 a^2 e^2}\\ &=-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {(96 (16 b c-11 a d)) \int \frac {1}{(e x)^{9/2} \sqrt [4]{a+b x^2}} \, dx}{55 a^3 e^2}\\ &=-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac {64 (16 b c-11 a d) \left (a+b x^2\right )^{3/4}}{55 a^4 e^3 (e x)^{7/2}}+\frac {(128 (16 b c-11 a d)) \int \frac {\left (a+b x^2\right )^{3/4}}{(e x)^{9/2}} \, dx}{55 a^4 e^2}\\ &=-\frac {2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac {24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac {64 (16 b c-11 a d) \left (a+b x^2\right )^{3/4}}{55 a^4 e^3 (e x)^{7/2}}-\frac {256 (16 b c-11 a d) \left (a+b x^2\right )^{7/4}}{385 a^5 e^3 (e x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.44 \[ \frac {2 x \left (a x^2 \left (-5 a^3+20 a^2 b x^2+160 a b^2 x^4+128 b^3 x^6\right ) (11 a d-16 b c)-35 a^5 c\right )}{385 a^6 (e x)^{13/2} \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 143, normalized size = 0.80 \[ -\frac {2 \, {\left (128 \, {\left (16 \, b^{4} c - 11 \, a b^{3} d\right )} x^{8} + 160 \, {\left (16 \, a b^{3} c - 11 \, a^{2} b^{2} d\right )} x^{6} + 35 \, a^{4} c + 20 \, {\left (16 \, a^{2} b^{2} c - 11 \, a^{3} b d\right )} x^{4} - 5 \, {\left (16 \, a^{3} b c - 11 \, a^{4} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{385 \, {\left (a^{5} b^{2} e^{7} x^{10} + 2 \, a^{6} b e^{7} x^{8} + a^{7} e^{7} x^{6}\right )}} \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 110, normalized size = 0.62 \[ -\frac {2 \left (-1408 a \,b^{3} d \,x^{8}+2048 b^{4} c \,x^{8}-1760 a^{2} b^{2} d \,x^{6}+2560 a \,b^{3} c \,x^{6}-220 a^{3} b d \,x^{4}+320 a^{2} b^{2} c \,x^{4}+55 a^{4} d \,x^{2}-80 a^{3} b c \,x^{2}+35 c \,a^{4}\right ) x}{385 \left (b \,x^{2}+a \right )^{\frac {5}{4}} \left (e x \right )^{\frac {13}{2}} a^{5}} \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 156, normalized size = 0.88 \[ \frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {64\,x^6\,\left (11\,a\,d-16\,b\,c\right )}{77\,a^4\,e^6}-\frac {2\,c}{11\,a\,b^2\,e^6}+\frac {8\,x^4\,\left (11\,a\,d-16\,b\,c\right )}{77\,a^3\,b\,e^6}-\frac {x^2\,\left (110\,a^4\,d-160\,a^3\,b\,c\right )}{385\,a^5\,b^2\,e^6}+\frac {256\,b\,x^8\,\left (11\,a\,d-16\,b\,c\right )}{385\,a^5\,e^6}\right )}{x^9\,\sqrt {e\,x}+\frac {a^2\,x^5\,\sqrt {e\,x}}{b^2}+\frac {2\,a\,x^7\,\sqrt {e\,x}}{b}} \]
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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