Optimal. Leaf size=198 \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{6 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e) (-3 a B e+2 A b e+b B d)}{5 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)^2}{4 b^4}+\frac{B e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.235755, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (-3 a B e+A b e+2 b B d)}{6 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e) (-3 a B e+2 A b e+b B d)}{5 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)^2}{4 b^4}+\frac{B e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^2 \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{(A b-a B) (b d-a e)^2 \left (a b+b^2 x\right )^3}{b^3}+\frac{(b d-a e) (b B d+2 A b e-3 a B e) \left (a b+b^2 x\right )^4}{b^4}+\frac{e (2 b B d+A b e-3 a B e) \left (a b+b^2 x\right )^5}{b^5}+\frac{B e^2 \left (a b+b^2 x\right )^6}{b^6}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{(A b-a B) (b d-a e)^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^4}+\frac{(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 b^4}+\frac{e (2 b B d+A b e-3 a B e) (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^4}+\frac{B e^2 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^4}\\ \end{align*}
Mathematica [A] time = 0.108587, size = 233, normalized size = 1.18 \[ \frac{x \sqrt{(a+b x)^2} \left (21 a^2 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+35 a^3 \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+21 a b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )\right )}{420 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.006, size = 304, normalized size = 1.5 \begin{align*}{\frac{x \left ( 60\,{b}^{3}B{e}^{2}{x}^{6}+70\,{x}^{5}A{b}^{3}{e}^{2}+210\,{x}^{5}B{e}^{2}{b}^{2}a+140\,{x}^{5}B{b}^{3}de+252\,{x}^{4}Aa{b}^{2}{e}^{2}+168\,{x}^{4}A{b}^{3}de+252\,{x}^{4}B{e}^{2}{a}^{2}b+504\,{x}^{4}Ba{b}^{2}de+84\,{x}^{4}B{b}^{3}{d}^{2}+315\,{x}^{3}A{a}^{2}b{e}^{2}+630\,{x}^{3}Aa{b}^{2}de+105\,{x}^{3}A{d}^{2}{b}^{3}+105\,{x}^{3}B{e}^{2}{a}^{3}+630\,{x}^{3}B{a}^{2}bde+315\,{x}^{3}Ba{b}^{2}{d}^{2}+140\,{x}^{2}A{a}^{3}{e}^{2}+840\,{x}^{2}A{a}^{2}bde+420\,{x}^{2}A{d}^{2}{b}^{2}a+280\,{x}^{2}B{a}^{3}de+420\,{x}^{2}B{a}^{2}b{d}^{2}+420\,xA{a}^{3}de+630\,xA{d}^{2}{a}^{2}b+210\,xB{a}^{3}{d}^{2}+420\,A{d}^{2}{a}^{3} \right ) }{420\, \left ( bx+a \right ) ^{3}} \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 = 1.50392, size = 512, normalized size = 2.59 \begin{align*} \frac{1}{7} \, B b^{3} e^{2} x^{7} + A a^{3} d^{2} x + \frac{1}{6} \,{\left (2 \, B b^{3} d e +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{2} + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A a^{3} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a^{3} d e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2}\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 (A + B x\right ) \left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.1225, size = 582, normalized size = 2.94 \begin{align*} \frac{1}{7} \, B b^{3} x^{7} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, B b^{3} d x^{6} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{5} \, B b^{3} d^{2} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a b^{2} x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, A b^{3} x^{6} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{6}{5} \, B a b^{2} d x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{2}{5} \, A b^{3} d x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, B a b^{2} d^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, A b^{3} d^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{5} \, B a^{2} b x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{5} \, A a b^{2} x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, B a^{2} b d x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, A a b^{2} d x^{4} e \mathrm{sgn}\left (b x + a\right ) + B a^{2} b d^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + A a b^{2} d^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, B a^{3} x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, A a^{2} b x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, B a^{3} d x^{3} e \mathrm{sgn}\left (b x + a\right ) + 2 \, A a^{2} b d x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a^{3} d^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, A a^{2} b d^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, A a^{3} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + A a^{3} d x^{2} e \mathrm{sgn}\left (b x + a\right ) + A a^{3} d^{2} x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]