Optimal. Leaf size=210 \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^3}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^5 (a+b x)}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.102529, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ -\frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x)}+\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^3}-\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x)}{e^5 (a+b x)}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{d+e x} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{(a+b x) \left (a b+b^2 x\right )^3}{d+e x} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^4}{d+e x} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (-\frac{b (b d-a e)^3}{e^4}+\frac{b (b d-a e)^2 (a+b x)}{e^3}-\frac{b (b d-a e) (a+b x)^2}{e^2}+\frac{b (a+b x)^3}{e}+\frac{(-b d+a e)^4}{e^4 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{b (b d-a e)^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac{(b d-a e)^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^3}-\frac{(b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2}+\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e}+\frac{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0681799, size = 133, normalized size = 0.63 \[ \frac{\sqrt{(a+b x)^2} \left (b e x \left (36 a^2 b e^2 (e x-2 d)+48 a^3 e^3+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (d+e x)\right )}{12 e^5 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 225, normalized size = 1.1 \begin{align*}{\frac{3\,{x}^{4}{b}^{4}{e}^{4}+16\,{x}^{3}a{b}^{3}{e}^{4}-4\,{x}^{3}{b}^{4}d{e}^{3}+36\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-24\,{x}^{2}a{b}^{3}d{e}^{3}+6\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){a}^{4}{e}^{4}-48\,\ln \left ( ex+d \right ){a}^{3}bd{e}^{3}+72\,\ln \left ( ex+d \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-48\,\ln \left ( ex+d \right ) a{b}^{3}{d}^{3}e+12\,\ln \left ( ex+d \right ){b}^{4}{d}^{4}+48\,x{a}^{3}b{e}^{4}-72\,x{a}^{2}{b}^{2}d{e}^{3}+48\,xa{b}^{3}{d}^{2}{e}^{2}-12\,x{b}^{4}{d}^{3}e}{12\, \left ( bx+a \right ) ^{3}{e}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54742, size = 369, normalized size = 1.76 \begin{align*} \frac{3 \, b^{4} e^{4} x^{4} - 4 \,{\left (b^{4} d e^{3} - 4 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} - 12 \,{\left (b^{4} d^{3} e - 4 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \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 (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16008, size = 359, normalized size = 1.71 \begin{align*}{\left (b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, b^{4} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) - 4 \, b^{4} d x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, b^{4} d^{2} x^{2} e \mathrm{sgn}\left (b x + a\right ) - 12 \, b^{4} d^{3} x \mathrm{sgn}\left (b x + a\right ) + 16 \, a b^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 24 \, a b^{3} d x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 48 \, a b^{3} d^{2} x e \mathrm{sgn}\left (b x + a\right ) + 36 \, a^{2} b^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 72 \, a^{2} b^{2} d x e^{2} \mathrm{sgn}\left (b x + a\right ) + 48 \, a^{3} b x e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]