Integrand size = 26, antiderivative size = 104 \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 c \sqrt [4]{a+b x^2}}{9 a e (e x)^{9/2}}+\frac {2 (8 b c-9 a d) \sqrt [4]{a+b x^2}}{9 a^2 e^3 (e x)^{5/2}}-\frac {8 (8 b c-9 a d) \left (a+b x^2\right )^{5/4}}{45 a^3 e^3 (e x)^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {464, 279, 270} \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {8 \left (a+b x^2\right )^{5/4} (8 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2}}+\frac {2 \sqrt [4]{a+b x^2} (8 b c-9 a d)}{9 a^2 e^3 (e x)^{5/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{9 a e (e x)^{9/2}} \]
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Rule 270
Rule 279
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \sqrt [4]{a+b x^2}}{9 a e (e x)^{9/2}}-\frac {(8 b c-9 a d) \int \frac {1}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx}{9 a e^2} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{9 a e (e x)^{9/2}}+\frac {2 (8 b c-9 a d) \sqrt [4]{a+b x^2}}{9 a^2 e^3 (e x)^{5/2}}+\frac {(4 (8 b c-9 a d)) \int \frac {\sqrt [4]{a+b x^2}}{(e x)^{7/2}} \, dx}{9 a^2 e^2} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{9 a e (e x)^{9/2}}+\frac {2 (8 b c-9 a d) \sqrt [4]{a+b x^2}}{9 a^2 e^3 (e x)^{5/2}}-\frac {8 (8 b c-9 a d) \left (a+b x^2\right )^{5/4}}{45 a^3 e^3 (e x)^{5/2}} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 x \sqrt [4]{a+b x^2} \left (5 a^2 c-8 a b c x^2+9 a^2 d x^2+32 b^2 c x^4-36 a b d x^4\right )}{45 a^3 (e x)^{11/2}} \]
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Time = 3.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (-36 a b d \,x^{4}+32 b^{2} c \,x^{4}+9 a^{2} d \,x^{2}-8 a b c \,x^{2}+5 a^{2} c \right )}{45 a^{3} \left (e x \right )^{\frac {11}{2}}}\) | \(62\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (-36 a b d \,x^{4}+32 b^{2} c \,x^{4}+9 a^{2} d \,x^{2}-8 a b c \,x^{2}+5 a^{2} c \right )}{45 e^{5} \sqrt {e x}\, a^{3} x^{4}}\) | \(67\) |
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Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.63 \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 \, {\left (4 \, {\left (8 \, b^{2} c - 9 \, a b d\right )} x^{4} + 5 \, a^{2} c - {\left (8 \, a b c - 9 \, a^{2} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {e x}}{45 \, a^{3} e^{6} x^{5}} \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {11}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {11}{2}}} \,d x } \]
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Time = 5.60 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.72 \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2\,c}{9\,a\,e^5}+\frac {x^2\,\left (18\,a^2\,d-16\,a\,b\,c\right )}{45\,a^3\,e^5}+\frac {x^4\,\left (64\,b^2\,c-72\,a\,b\,d\right )}{45\,a^3\,e^5}\right )}{x^4\,\sqrt {e\,x}} \]
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