Optimal. Leaf size=41 \[ -\frac{(d+e x)^3}{3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0236631, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {767} \[ -\frac{(d+e x)^3}{3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 767
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac{(d+e x)^3}{3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0332619, size = 60, normalized size = 1.46 \[ \frac{-a^2 e^2-a b e (d+3 e x)+b^2 \left (-\left (d^2+3 d e x+3 e^2 x^2\right )\right )}{3 b^3 \left ((a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 69, normalized size = 1.7 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{2} \left ( 3\,{x}^{2}{b}^{2}{e}^{2}+3\,xab{e}^{2}+3\,x{b}^{2}de+{a}^{2}{e}^{2}+abde+{b}^{2}{d}^{2} \right ) }{3\,{b}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.01561, size = 440, normalized size = 10.73 \begin{align*} -\frac{e^{2} x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b} - \frac{2 \, a^{2} e^{2}}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{3}} - \frac{a^{3} b^{2} e^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \, a^{2} b e^{2}}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{a e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{b d^{2} + 2 \, a d e}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{{\left (2 \, b d e + a e^{2}\right )} a^{2} b^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} - \frac{a d^{2}}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{a^{3} e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b^{2}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \,{\left (2 \, b d e + a e^{2}\right )} a b}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{2 \, b d e + a e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{{\left (b d^{2} + 2 \, a d e\right )} a}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.5933, size = 170, normalized size = 4.15 \begin{align*} -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right ) \left (d + e x\right )^{2}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}{\left (e x + d\right )}^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]