3.1145 \(\int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx\)

Optimal. Leaf size=136 \[ \frac{5 a^6 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{8 b}+\frac{5}{16} a^4 c^2 x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2} \]

[Out]

(5*a^4*c^2*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/16 + (5*a^2*c*x*(a + b*x)^(3/2)*(a
*c - b*c*x)^(3/2))/24 + (x*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2))/6 + (5*a^6*c^(5/
2)*ArcTan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a*c - b*c*x]])/(8*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.151592, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{5 a^6 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{8 b}+\frac{5}{16} a^4 c^2 x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2),x]

[Out]

(5*a^4*c^2*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/16 + (5*a^2*c*x*(a + b*x)^(3/2)*(a
*c - b*c*x)^(3/2))/24 + (x*(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2))/6 + (5*a^6*c^(5/
2)*ArcTan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a*c - b*c*x]])/(8*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.4125, size = 126, normalized size = 0.93 \[ - \frac{5 a^{6} c^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a c - b c x}}{\sqrt{c} \sqrt{a + b x}} \right )}}{8 b} + \frac{5 a^{4} c^{2} x \sqrt{a + b x} \sqrt{a c - b c x}}{16} + \frac{5 a^{2} c x \left (a + b x\right )^{\frac{3}{2}} \left (a c - b c x\right )^{\frac{3}{2}}}{24} + \frac{x \left (a + b x\right )^{\frac{5}{2}} \left (a c - b c x\right )^{\frac{5}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(-b*c*x+a*c)**(5/2),x)

[Out]

-5*a**6*c**(5/2)*atan(sqrt(a*c - b*c*x)/(sqrt(c)*sqrt(a + b*x)))/(8*b) + 5*a**4*
c**2*x*sqrt(a + b*x)*sqrt(a*c - b*c*x)/16 + 5*a**2*c*x*(a + b*x)**(3/2)*(a*c - b
*c*x)**(3/2)/24 + x*(a + b*x)**(5/2)*(a*c - b*c*x)**(5/2)/6

_______________________________________________________________________________________

Mathematica [A]  time = 0.177654, size = 105, normalized size = 0.77 \[ \frac{(c (a-b x))^{5/2} \left (15 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right )+b x \sqrt{a-b x} \sqrt{a+b x} \left (33 a^4-26 a^2 b^2 x^2+8 b^4 x^4\right )\right )}{48 b (a-b x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2)*(a*c - b*c*x)^(5/2),x]

[Out]

((c*(a - b*x))^(5/2)*(b*x*Sqrt[a - b*x]*Sqrt[a + b*x]*(33*a^4 - 26*a^2*b^2*x^2 +
8*b^4*x^4) + 15*a^6*ArcTan[(b*x)/(Sqrt[a - b*x]*Sqrt[a + b*x])]))/(48*b*(a - b*
x)^(5/2))

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 243, normalized size = 1.8 \[ -{\frac{1}{6\,bc} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( -bcx+ac \right ) ^{{\frac{7}{2}}}}-{\frac{a}{6\,bc} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( -bcx+ac \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}}{8\,bc}\sqrt{bx+a} \left ( -bcx+ac \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{3}}{24\,b} \left ( -bcx+ac \right ) ^{{\frac{5}{2}}}\sqrt{bx+a}}+{\frac{5\,{a}^{4}c}{48\,b} \left ( -bcx+ac \right ) ^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{5\,{a}^{5}{c}^{2}}{16\,b}\sqrt{bx+a}\sqrt{-bcx+ac}}+{\frac{5\,{a}^{6}{c}^{3}}{16}\sqrt{ \left ( bx+a \right ) \left ( -bcx+ac \right ) }\arctan \left ({x\sqrt{{b}^{2}c}{\frac{1}{\sqrt{-{b}^{2}c{x}^{2}+{a}^{2}c}}}} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{-bcx+ac}}}{\frac{1}{\sqrt{{b}^{2}c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(-b*c*x+a*c)^(5/2),x)

[Out]

-1/6/b/c*(b*x+a)^(5/2)*(-b*c*x+a*c)^(7/2)-1/6*a/b/c*(b*x+a)^(3/2)*(-b*c*x+a*c)^(
7/2)-1/8*a^2/b/c*(b*x+a)^(1/2)*(-b*c*x+a*c)^(7/2)+1/24*a^3/b*(-b*c*x+a*c)^(5/2)*
(b*x+a)^(1/2)+5/48*a^4*c/b*(-b*c*x+a*c)^(3/2)*(b*x+a)^(1/2)+5/16*a^5*c^2/b*(-b*c
*x+a*c)^(1/2)*(b*x+a)^(1/2)+5/16*a^6*c^3*((b*x+a)*(-b*c*x+a*c))^(1/2)/(-b*c*x+a*
c)^(1/2)/(b*x+a)^(1/2)/(b^2*c)^(1/2)*arctan((b^2*c)^(1/2)*x/(-b^2*c*x^2+a^2*c)^(
1/2))

_______________________________________________________________________________________

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*c*x + a*c)^(5/2)*(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.240066, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{6} \sqrt{-c} c^{2} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right ) + 2 \,{\left (8 \, b^{5} c^{2} x^{5} - 26 \, a^{2} b^{3} c^{2} x^{3} + 33 \, a^{4} b c^{2} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{96 \, b}, \frac{15 \, a^{6} c^{\frac{5}{2}} \arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right ) +{\left (8 \, b^{5} c^{2} x^{5} - 26 \, a^{2} b^{3} c^{2} x^{3} + 33 \, a^{4} b c^{2} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{48 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*c*x + a*c)^(5/2)*(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[1/96*(15*a^6*sqrt(-c)*c^2*log(2*b^2*c*x^2 + 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*
b*sqrt(-c)*x - a^2*c) + 2*(8*b^5*c^2*x^5 - 26*a^2*b^3*c^2*x^3 + 33*a^4*b*c^2*x)*
sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b, 1/48*(15*a^6*c^(5/2)*arctan(b*sqrt(c)*x/(sq
rt(-b*c*x + a*c)*sqrt(b*x + a))) + (8*b^5*c^2*x^5 - 26*a^2*b^3*c^2*x^3 + 33*a^4*
b*c^2*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)*(-b*c*x+a*c)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*c*x + a*c)^(5/2)*(b*x + a)^(5/2),x, algorithm="giac")

[Out]

Timed out