Optimal. Leaf size=259 \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)^3}{4 b^5}+\frac{B e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^5} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.79556, antiderivative size = 259, 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 \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)^3}{4 b^5}+\frac{B e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^5} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 48.1151, size = 282, normalized size = 1.09 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{16 b e} + \frac{\left (d + e x\right )^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (2 A b e - B a e - B b d\right )}{14 b e^{2}} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e - B a e - B b d\right )}{84 b e^{3}} + \frac{\left (d + e x\right )^{4} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e - B a e - B b d\right )}{70 b e^{4}} + \frac{\left (d + e x\right )^{4} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (2 A b e - B a e - B b d\right )}{280 b e^{5} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.267252, size = 320, normalized size = 1.24 \[ \frac{x \sqrt{(a+b x)^2} \left (14 a^3 \left (5 A \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+14 a^2 b x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )+2 a b^2 x^2 \left (7 A \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+3 B x \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 x^3 \left (2 A \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+B x \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )\right )}{280 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.011, size = 428, normalized size = 1.7 \[{\frac{x \left ( 35\,B{e}^{3}{b}^{3}{x}^{7}+40\,{x}^{6}A{b}^{3}{e}^{3}+120\,{x}^{6}B{e}^{3}a{b}^{2}+120\,{x}^{6}B{b}^{3}d{e}^{2}+140\,{x}^{5}Aa{b}^{2}{e}^{3}+140\,{x}^{5}A{b}^{3}d{e}^{2}+140\,{x}^{5}B{e}^{3}{a}^{2}b+420\,{x}^{5}Ba{b}^{2}d{e}^{2}+140\,{x}^{5}B{b}^{3}{d}^{2}e+168\,{x}^{4}A{a}^{2}b{e}^{3}+504\,{x}^{4}Aa{b}^{2}d{e}^{2}+168\,{x}^{4}A{b}^{3}{d}^{2}e+56\,{x}^{4}B{e}^{3}{a}^{3}+504\,{x}^{4}B{a}^{2}bd{e}^{2}+504\,{x}^{4}Ba{b}^{2}{d}^{2}e+56\,{x}^{4}B{b}^{3}{d}^{3}+70\,{x}^{3}A{a}^{3}{e}^{3}+630\,{x}^{3}A{a}^{2}bd{e}^{2}+630\,{x}^{3}Aa{b}^{2}{d}^{2}e+70\,{x}^{3}A{b}^{3}{d}^{3}+210\,{x}^{3}B{a}^{3}d{e}^{2}+630\,{x}^{3}B{a}^{2}b{d}^{2}e+210\,{x}^{3}a{b}^{2}B{d}^{3}+280\,A{a}^{3}d{e}^{2}{x}^{2}+840\,A{a}^{2}b{d}^{2}e{x}^{2}+280\,Aa{b}^{2}{d}^{3}{x}^{2}+280\,B{a}^{3}{d}^{2}e{x}^{2}+280\,B{a}^{2}b{d}^{3}{x}^{2}+420\,xA{a}^{3}{d}^{2}e+420\,xA{d}^{3}{a}^{2}b+140\,xB{a}^{3}{d}^{3}+280\,A{d}^{3}{a}^{3} \right ) }{280\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*(e*x + d)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.267638, size = 439, normalized size = 1.69 \[ \frac{1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac{1}{7} \,{\left (3 \, B b^{3} d e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B b^{3} d^{2} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} +{\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (A a^{3} e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} +{\left (A a^{3} d e^{2} +{\left (B a^{2} b + A a b^{2}\right )} d^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a^{3} d^{2} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*(e*x + d)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (A + B x\right ) \left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.290556, size = 802, normalized size = 3.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*(e*x + d)^3,x, algorithm="giac")
[Out]