Optimal. Leaf size=48 \[ -\frac{(d+e x)^4}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.0220864, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 37} \[ -\frac{(d+e x)^4}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 37
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\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*}
Mathematica [B] time = 0.0431586, size = 106, normalized size = 2.21 \[ \frac{-a^2 b e^2 (d+4 e x)-a^3 e^3-a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (-\left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )}{4 b^4 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.155, size = 119, normalized size = 2.5 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( 4\,{x}^{3}{b}^{3}{e}^{3}+6\,{x}^{2}a{b}^{2}{e}^{3}+6\,{x}^{2}{b}^{3}d{e}^{2}+4\,x{a}^{2}b{e}^{3}+4\,xa{b}^{2}d{e}^{2}+4\,x{b}^{3}{d}^{2}e+{a}^{3}{e}^{3}+d{e}^{2}{a}^{2}b+a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3} \right ) }{4\,{b}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1404, size = 392, normalized size = 8.17 \begin{align*} -\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 \, a^{2} b^{2} d e^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} - \frac{a^{3} b e^{3}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \, a b d e^{2}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} + \frac{2 \, a^{2} e^{3}}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{3 \, d e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{a e^{3}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{d^{3}}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{3 \, a d^{2} e}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}} + \frac{a^{3} e^{3}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62099, size = 284, normalized size = 5.92 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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