Integrand size = 26, antiderivative size = 141 \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (4 b c-3 a d)}{9 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{3/4}}+\frac {16 (4 b c-3 a d) \sqrt [4]{a+b x^2}}{9 a^3 e^3 (e x)^{5/2}}-\frac {64 (4 b c-3 a d) \left (a+b x^2\right )^{5/4}}{45 a^4 e^3 (e x)^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {464, 279, 270} \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {64 \left (a+b x^2\right )^{5/4} (4 b c-3 a d)}{45 a^4 e^3 (e x)^{5/2}}+\frac {16 \sqrt [4]{a+b x^2} (4 b c-3 a d)}{9 a^3 e^3 (e x)^{5/2}}-\frac {2 (4 b c-3 a d)}{9 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{3/4}}-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{3/4}} \]
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Rule 270
Rule 279
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{3/4}}-\frac {(4 b c-3 a d) \int \frac {1}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx}{3 a e^2} \\ & = -\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (4 b c-3 a d)}{9 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{3/4}}-\frac {(8 (4 b c-3 a d)) \int \frac {1}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx}{9 a^2 e^2} \\ & = -\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (4 b c-3 a d)}{9 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{3/4}}+\frac {16 (4 b c-3 a d) \sqrt [4]{a+b x^2}}{9 a^3 e^3 (e x)^{5/2}}+\frac {(32 (4 b c-3 a d)) \int \frac {\sqrt [4]{a+b x^2}}{(e x)^{7/2}} \, dx}{9 a^3 e^2} \\ & = -\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (4 b c-3 a d)}{9 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{3/4}}+\frac {16 (4 b c-3 a d) \sqrt [4]{a+b x^2}}{9 a^3 e^3 (e x)^{5/2}}-\frac {64 (4 b c-3 a d) \left (a+b x^2\right )^{5/4}}{45 a^4 e^3 (e x)^{5/2}} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.65 \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {2 x \left (5 a^3 c-12 a^2 b c x^2+9 a^3 d x^2+96 a b^2 c x^4-72 a^2 b d x^4+128 b^3 c x^6-96 a b^2 d x^6\right )}{45 a^4 (e x)^{11/2} \left (a+b x^2\right )^{3/4}} \]
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Time = 3.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(-\frac {2 x \left (-96 a \,b^{2} d \,x^{6}+128 b^{3} c \,x^{6}-72 a^{2} b d \,x^{4}+96 a \,b^{2} c \,x^{4}+9 a^{3} d \,x^{2}-12 a^{2} b c \,x^{2}+5 c \,a^{3}\right )}{45 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a^{4} \left (e x \right )^{\frac {11}{2}}}\) | \(86\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (-81 a b d \,x^{4}+113 b^{2} c \,x^{4}+9 a^{2} d \,x^{2}-17 a b c \,x^{2}+5 a^{2} c \right )}{45 a^{4} x^{4} e^{5} \sqrt {e x}}+\frac {2 b^{2} \left (a d -b c \right ) x^{2}}{3 a^{4} e^{5} \sqrt {e x}\, \left (b \,x^{2}+a \right )^{\frac {3}{4}}}\) | \(104\) |
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Time = 0.34 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.74 \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {2 \, {\left (32 \, {\left (4 \, b^{3} c - 3 \, a b^{2} d\right )} x^{6} + 24 \, {\left (4 \, a b^{2} c - 3 \, a^{2} b d\right )} x^{4} + 5 \, a^{3} c - 3 \, {\left (4 \, a^{2} b c - 3 \, a^{3} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {e x}}{45 \, {\left (a^{4} b e^{6} x^{7} + a^{5} e^{6} x^{5}\right )}} \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{7/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {11}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {11}{2}}} \,d x } \]
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Time = 5.85 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2\,c}{9\,a\,b\,e^5}-\frac {16\,x^4\,\left (3\,a\,d-4\,b\,c\right )}{15\,a^3\,e^5}+\frac {x^2\,\left (18\,a^3\,d-24\,a^2\,b\,c\right )}{45\,a^4\,b\,e^5}+\frac {x^6\,\left (256\,b^3\,c-192\,a\,b^2\,d\right )}{45\,a^4\,b\,e^5}\right )}{x^6\,\sqrt {e\,x}+\frac {a\,x^4\,\sqrt {e\,x}}{b}} \]
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