Integrand size = 28, antiderivative size = 48 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {(d+e x)^4}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 37} \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {(d+e x)^4}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
[In]
[Out]
Rule 37
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^3}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(d+e x)^4}{4 (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(106\) vs. \(2(48)=96\).
Time = 1.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.21 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-a^3 e^3-a^2 b e^2 (d+4 e x)-a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )-b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{4 b^4 (a+b x)^3 \sqrt {(a+b x)^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(35)=70\).
Time = 2.51 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.35
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {e^{3} x^{3}}{b}-\frac {3 e^{2} \left (a e +b d \right ) x^{2}}{2 b^{2}}-\frac {e \left (a^{2} e^{2}+a b d e +b^{2} d^{2}\right ) x}{b^{3}}-\frac {a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}}{4 b^{4}}\right )}{\left (b x +a \right )^{5}}\) | \(113\) |
gosper | \(-\frac {\left (b x +a \right ) \left (4 e^{3} x^{3} b^{3}+6 x^{2} a \,b^{2} e^{3}+6 x^{2} b^{3} d \,e^{2}+4 a^{2} b \,e^{3} x +4 x a \,b^{2} d \,e^{2}+4 b^{3} d^{2} e x +a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right )}{4 b^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(119\) |
default | \(-\frac {\left (b x +a \right ) \left (4 e^{3} x^{3} b^{3}+6 x^{2} a \,b^{2} e^{3}+6 x^{2} b^{3} d \,e^{2}+4 a^{2} b \,e^{3} x +4 x a \,b^{2} d \,e^{2}+4 b^{3} d^{2} e x +a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right )}{4 b^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(119\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (35) = 70\).
Time = 0.26 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.98 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \, {\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]
[In]
[Out]
\[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (35) = 70\).
Time = 0.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 4.96 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {e^{3} x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {d^{2} e}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, a^{2} e^{3}}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} - \frac {3 \, d e^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {a e^{3}}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, a d e^{2}}{b^{6} {\left (x + \frac {a}{b}\right )}^{3}} + \frac {2 \, a^{2} e^{3}}{3 \, b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {d^{3}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {3 \, a d^{2} e}{4 \, b^{6} {\left (x + \frac {a}{b}\right )}^{4}} - \frac {3 \, a^{2} d e^{2}}{4 \, b^{7} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {a^{3} e^{3}}{4 \, b^{8} {\left (x + \frac {a}{b}\right )}^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (35) = 70\).
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.48 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {4 \, b^{3} e^{3} x^{3} + 6 \, b^{3} d e^{2} x^{2} + 6 \, a b^{2} e^{3} x^{2} + 4 \, b^{3} d^{2} e x + 4 \, a b^{2} d e^{2} x + 4 \, a^{2} b e^{3} x + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3}}{4 \, {\left (b x + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x + a\right )} \]
[In]
[Out]
Time = 10.09 (sec) , antiderivative size = 254, normalized size of antiderivative = 5.29 \[ \int \frac {(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\left (\frac {2\,a\,e^3-3\,b\,d\,e^2}{2\,b^4}+\frac {a\,e^3}{2\,b^4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{{\left (a+b\,x\right )}^3}-\frac {\left (\frac {d^3}{4\,b}-\frac {a\,\left (\frac {3\,d^2\,e}{4\,b}+\frac {a\,\left (\frac {a\,e^3}{4\,b^2}-\frac {3\,d\,e^2}{4\,b}\right )}{b}\right )}{b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{{\left (a+b\,x\right )}^5}-\frac {\left (\frac {a^2\,e^3-3\,a\,b\,d\,e^2+3\,b^2\,d^2\,e}{3\,b^4}+\frac {a\,\left (\frac {a\,e^3}{3\,b^3}+\frac {e^2\,\left (a\,e-3\,b\,d\right )}{3\,b^3}\right )}{b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{{\left (a+b\,x\right )}^4}-\frac {e^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{b^4\,{\left (a+b\,x\right )}^2} \]
[In]
[Out]