Optimal. Leaf size=141 \[ \frac {64 \left (a+b x^2\right )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac {16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac {2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \]
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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 )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac {16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac {2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \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)^{15/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}-\frac {(12 b c-13 a d) \int \frac {1}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx}{13 a e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac {2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}+\frac {(8 (12 b c-13 a d)) \int \frac {\sqrt [4]{a+b x^2}}{(e x)^{11/2}} \, dx}{13 a^2 e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac {2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}-\frac {16 (12 b c-13 a d) \left (a+b x^2\right )^{5/4}}{65 a^3 e^3 (e x)^{9/2}}-\frac {(32 (12 b c-13 a d)) \int \frac {\left (a+b x^2\right )^{5/4}}{(e x)^{11/2}} \, dx}{65 a^3 e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac {2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}-\frac {16 (12 b c-13 a d) \left (a+b x^2\right )^{5/4}}{65 a^3 e^3 (e x)^{9/2}}+\frac {64 (12 b c-13 a d) \left (a+b x^2\right )^{9/4}}{585 a^4 e^3 (e x)^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 94, normalized size = 0.67 \begin {gather*} -\frac {2 \sqrt {e x} \sqrt [4]{a+b x^2} \left (5 a^3 \left (9 c+13 d x^2\right )-4 a^2 b x^2 \left (15 c+26 d x^2\right )+32 a b^2 x^4 \left (3 c+13 d x^2\right )-384 b^3 c x^6\right )}{585 a^4 e^8 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.23, size = 114, normalized size = 0.81 \begin {gather*} -\frac {2 \sqrt [4]{a+b x^2} \left (45 a^3 c e^6+65 a^3 d e^6 x^2-60 a^2 b c e^6 x^2-104 a^2 b d e^6 x^4+96 a b^2 c e^6 x^4+416 a b^2 d e^6 x^6-384 b^3 c e^6 x^6\right )}{585 a^4 e^7 (e x)^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 90, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (32 \, {\left (12 \, b^{3} c - 13 \, a b^{2} d\right )} x^{6} - 8 \, {\left (12 \, a b^{2} c - 13 \, a^{2} b d\right )} x^{4} - 45 \, a^{3} c + 5 \, {\left (12 \, a^{2} b c - 13 \, a^{3} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {e x}}{585 \, a^{4} e^{8} x^{7}} \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 {3}{4}} \left (e x\right )^{\frac {15}{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 (b \,x^{2}+a \right )^{\frac {1}{4}} \left (416 a \,b^{2} d \,x^{6}-384 b^{3} c \,x^{6}-104 a^{2} b d \,x^{4}+96 a \,b^{2} c \,x^{4}+65 a^{3} d \,x^{2}-60 a^{2} b c \,x^{2}+45 c \,a^{3}\right ) x}{585 \left (e x \right )^{\frac {15}{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 {3}{4}} \left (e x\right )^{\frac {15}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 100, normalized size = 0.71 \begin {gather*} -\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2\,c}{13\,a\,e^7}+\frac {x^2\,\left (130\,a^3\,d-120\,a^2\,b\,c\right )}{585\,a^4\,e^7}-\frac {x^6\,\left (768\,b^3\,c-832\,a\,b^2\,d\right )}{585\,a^4\,e^7}-\frac {16\,b\,x^4\,\left (13\,a\,d-12\,b\,c\right )}{585\,a^3\,e^7}\right )}{x^6\,\sqrt {e\,x}} \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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