Optimal. Leaf size=141 \[ -\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac {2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}-\frac {16 (12 b c-13 a d) \left (a+b x^2\right )^{5/4}}{65 a^3 e^3 (e x)^{9/2}}+\frac {64 (12 b c-13 a d) \left (a+b x^2\right )^{9/4}}{585 a^4 e^3 (e x)^{9/2}} \]
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Rubi [A]
time = 0.04, 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 = {464, 279, 270}
\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 270
Rule 279
Rule 464
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 = 0.46, size = 91, normalized size = 0.65 \begin {gather*} -\frac {2 x \sqrt [4]{a+b x^2} \left (45 a^3 c-60 a^2 b c x^2+65 a^3 d x^2+96 a b^2 c x^4-104 a^2 b d x^4-384 b^3 c x^6+416 a b^2 d x^6\right )}{585 a^4 (e x)^{15/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 86, normalized size = 0.61
method | result | size |
gosper | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (416 a \,b^{2} d \,x^{6}-384 b^{3} c \,x^{6}-104 a^{2} b d \,x^{4}+96 a \,b^{2} c \,x^{4}+65 a^{3} d \,x^{2}-60 a^{2} b c \,x^{2}+45 c \,a^{3}\right )}{585 a^{4} \left (e x \right )^{\frac {15}{2}}}\) | \(86\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (416 a \,b^{2} d \,x^{6}-384 b^{3} c \,x^{6}-104 a^{2} b d \,x^{4}+96 a \,b^{2} c \,x^{4}+65 a^{3} d \,x^{2}-60 a^{2} b c \,x^{2}+45 c \,a^{3}\right )}{585 e^{7} \sqrt {e x}\, a^{4} x^{6}}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 128, normalized size = 0.91 \begin {gather*} -\frac {2}{585} \, {\left (\frac {13 \, {\left (\frac {45 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} b^{2}}{\sqrt {x}} - \frac {18 \, {\left (b x^{2} + a\right )}^{\frac {5}{4}} b}{x^{\frac {5}{2}}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {9}{4}}}{x^{\frac {9}{2}}}\right )} d}{a^{3}} - \frac {3 \, {\left (\frac {195 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} b^{3}}{\sqrt {x}} - \frac {117 \, {\left (b x^{2} + a\right )}^{\frac {5}{4}} b^{2}}{x^{\frac {5}{2}}} + \frac {65 \, {\left (b x^{2} + a\right )}^{\frac {9}{4}} b}{x^{\frac {9}{2}}} - \frac {15 \, {\left (b x^{2} + a\right )}^{\frac {13}{4}}}{x^{\frac {13}{2}}}\right )} c}{a^{4}}\right )} e^{\left (-\frac {15}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.82, size = 84, normalized size = 0.60 \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}} e^{\left (-\frac {15}{2}\right )}}{585 \, a^{4} x^{\frac {13}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.67, 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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