Integrand size = 19, antiderivative size = 79 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 \left (c d^2+a e^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt {a+c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {737, 651} \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]
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Rule 651
Rule 737
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {\left (2 \left (c d^2+a e^2\right )\right ) \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c} \\ & = -\frac {(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 \left (c d^2+a e^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt {a+c x^2}} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {-2 a^3 e^3+2 c^3 d^3 x^3-3 a^2 c e \left (d^2+e^2 x^2\right )+3 a c^2 d x \left (d^2+e^2 x^2\right )}{3 a^2 c^2 \left (a+c x^2\right )^{3/2}} \]
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Time = 2.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.05
method | result | size |
gosper | \(-\frac {-3 a \,c^{2} d \,e^{2} x^{3}-2 c^{3} d^{3} x^{3}+3 a^{2} c \,e^{3} x^{2}-3 d^{3} c^{2} a x +2 a^{3} e^{3}+3 d^{2} e \,a^{2} c}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c^{2}}\) | \(83\) |
trager | \(-\frac {-3 a \,c^{2} d \,e^{2} x^{3}-2 c^{3} d^{3} x^{3}+3 a^{2} c \,e^{3} x^{2}-3 d^{3} c^{2} a x +2 a^{3} e^{3}+3 d^{2} e \,a^{2} c}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c^{2}}\) | \(83\) |
default | \(d^{3} \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )+e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )+3 d \,e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{2 c}\right )-\frac {d^{2} e}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(151\) |
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Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {{\left (3 \, a^{2} c e^{3} x^{2} - 3 \, a c^{2} d^{3} x + 3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} - {\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x^{3}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} \]
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\[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.68 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, d^{3} x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d^{3} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {d e^{2} x}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {d e^{2} x}{\sqrt {c x^{2} + a} a c} - \frac {d^{2} e}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {2 \, a e^{3}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {{\left (\frac {3 \, d^{3}}{a} - {\left (\frac {3 \, e^{3}}{c} - \frac {{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x}{a^{2} c^{2}}\right )} x\right )} x - \frac {3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3}}{a^{2} c^{2}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \]
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Time = 9.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {2\,a^3\,e^3+3\,a^2\,c\,d^2\,e+3\,a^2\,c\,e^3\,x^2-3\,a\,c^2\,d^3\,x-3\,a\,c^2\,d\,e^2\,x^3-2\,c^3\,d^3\,x^3}{3\,a^2\,c^2\,{\left (c\,x^2+a\right )}^{3/2}} \]
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