Optimal. Leaf size=69 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (b d-a e)}{6 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0235983, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {640, 609} \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (b d-a e)}{6 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 640
Rule 609
Rubi steps
\begin{align*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}+\frac{\left (2 b^2 d-2 a b e\right ) \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx}{2 b^2}\\ &=\frac{(b d-a e) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}\\ \end{align*}
Mathematica [A] time = 0.0453663, size = 121, normalized size = 1.75 \[ \frac{x \sqrt{(a+b x)^2} \left (35 a^3 b^2 x^2 (4 d+3 e x)+21 a^2 b^3 x^3 (5 d+4 e x)+35 a^4 b x (3 d+2 e x)+21 a^5 (2 d+e x)+7 a b^4 x^4 (6 d+5 e x)+b^5 x^5 (7 d+6 e x)\right )}{42 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.16, size = 138, normalized size = 2. \begin{align*}{\frac{x \left ( 6\,e{b}^{5}{x}^{6}+35\,{x}^{5}ea{b}^{4}+7\,{x}^{5}d{b}^{5}+84\,{a}^{2}{b}^{3}e{x}^{4}+42\,a{b}^{4}d{x}^{4}+105\,{x}^{3}e{a}^{3}{b}^{2}+105\,{x}^{3}d{a}^{2}{b}^{3}+70\,{x}^{2}e{a}^{4}b+140\,{x}^{2}d{a}^{3}{b}^{2}+21\,xe{a}^{5}+105\,xd{a}^{4}b+42\,d{a}^{5} \right ) }{42\, \left ( bx+a \right ) ^{5}} \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 [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 = 1.51008, size = 247, normalized size = 3.58 \begin{align*} \frac{1}{7} \, b^{5} e x^{7} + a^{5} d x + \frac{1}{6} \,{\left (b^{5} d + 5 \, a b^{4} e\right )} x^{6} +{\left (a b^{4} d + 2 \, a^{2} b^{3} e\right )} x^{5} + \frac{5}{2} \,{\left (a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{4} + \frac{5}{3} \,{\left (2 \, a^{3} b^{2} d + a^{4} b e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d + a^{5} e\right )} x^{2} \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 \left (d + e x\right ) \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 [B] time = 1.24066, size = 269, normalized size = 3.9 \begin{align*} \frac{1}{7} \, b^{5} x^{7} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, b^{5} d x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{6} \, a b^{4} x^{6} e \mathrm{sgn}\left (b x + a\right ) + a b^{4} d x^{5} \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{2} b^{3} x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b^{3} d x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{3} b^{2} x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{3} b^{2} d x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, a^{4} b x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{4} b d x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, a^{5} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{5} d x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]