Optimal. Leaf size=141 \[ -\frac {64 \left (a+b x^2\right )^{7/4} (12 b c-11 a d)}{231 a^4 e^3 (e x)^{7/2}}+\frac {16 \left (a+b x^2\right )^{3/4} (12 b c-11 a d)}{33 a^3 e^3 (e x)^{7/2}}-\frac {2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.06, 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 )^{7/4} (12 b c-11 a d)}{231 a^4 e^3 (e x)^{7/2}}+\frac {16 \left (a+b x^2\right )^{3/4} (12 b c-11 a d)}{33 a^3 e^3 (e x)^{7/2}}-\frac {2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}} \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)^{13/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}-\frac {(12 b c-11 a d) \int \frac {1}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx}{11 a e^2}\\ &=-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}-\frac {2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {(8 (12 b c-11 a d)) \int \frac {1}{(e x)^{9/2} \sqrt [4]{a+b x^2}} \, dx}{11 a^2 e^2}\\ &=-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}-\frac {2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac {16 (12 b c-11 a d) \left (a+b x^2\right )^{3/4}}{33 a^3 e^3 (e x)^{7/2}}+\frac {(32 (12 b c-11 a d)) \int \frac {\left (a+b x^2\right )^{3/4}}{(e x)^{9/2}} \, dx}{33 a^3 e^2}\\ &=-\frac {2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}-\frac {2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac {16 (12 b c-11 a d) \left (a+b x^2\right )^{3/4}}{33 a^3 e^3 (e x)^{7/2}}-\frac {64 (12 b c-11 a d) \left (a+b x^2\right )^{7/4}}{231 a^4 e^3 (e x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 68, normalized size = 0.48 \begin {gather*} \frac {2 x \left (-21 a^3 c-x^2 \left (-3 a^2+8 a b x^2+32 b^2 x^4\right ) (12 b c-11 a d)\right )}{231 a^4 (e x)^{13/2} \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.97, size = 130, normalized size = 0.92 \begin {gather*} \frac {2 \left (a+b x^2\right )^{3/4} \left (-21 a^3 c e^6-33 a^3 d e^6 x^2+36 a^2 b c e^6 x^2+88 a^2 b d e^6 x^4-96 a b^2 c e^6 x^4+352 a b^2 d e^6 x^6-384 b^3 c e^6 x^6\right )}{231 a^4 e^5 (e x)^{11/2} \left (a e^2+b e^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 105, normalized size = 0.74 \begin {gather*} -\frac {2 \, {\left (32 \, {\left (12 \, b^{3} c - 11 \, a b^{2} d\right )} x^{6} + 8 \, {\left (12 \, a b^{2} c - 11 \, a^{2} b d\right )} x^{4} + 21 \, a^{3} c - 3 \, {\left (12 \, a^{2} b c - 11 \, a^{3} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{231 \, {\left (a^{4} b e^{7} x^{8} + a^{5} e^{7} x^{6}\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 {5}{4}} \left (e x\right )^{\frac {13}{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 (-352 a \,b^{2} d \,x^{6}+384 b^{3} c \,x^{6}-88 a^{2} b d \,x^{4}+96 a \,b^{2} c \,x^{4}+33 a^{3} d \,x^{2}-36 a^{2} b c \,x^{2}+21 c \,a^{3}\right ) x}{231 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (e x \right )^{\frac {13}{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 {5}{4}} \left (e x\right )^{\frac {13}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 125, normalized size = 0.89 \begin {gather*} -\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2\,c}{11\,a\,b\,e^6}-\frac {16\,x^4\,\left (11\,a\,d-12\,b\,c\right )}{231\,a^3\,e^6}+\frac {x^2\,\left (66\,a^3\,d-72\,a^2\,b\,c\right )}{231\,a^4\,b\,e^6}+\frac {x^6\,\left (768\,b^3\,c-704\,a\,b^2\,d\right )}{231\,a^4\,b\,e^6}\right )}{x^7\,\sqrt {e\,x}+\frac {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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