3.491 \(\int \sqrt{x} (a+b x)^{5/2} (A+B x) \, dx\)

Optimal. Leaf size=192 \[ -\frac{a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{5/2}}+\frac{a^3 \sqrt{x} \sqrt{a+b x} (10 A b-3 a B)}{128 b^2}+\frac{a^2 x^{3/2} \sqrt{a+b x} (10 A b-3 a B)}{64 b}+\frac{a x^{3/2} (a+b x)^{3/2} (10 A b-3 a B)}{48 b}+\frac{x^{3/2} (a+b x)^{5/2} (10 A b-3 a B)}{40 b}+\frac{B x^{3/2} (a+b x)^{7/2}}{5 b} \]

[Out]

(a^3*(10*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^2) + (a^2*(10*A*b - 3*a*B)*x
^(3/2)*Sqrt[a + b*x])/(64*b) + (a*(10*A*b - 3*a*B)*x^(3/2)*(a + b*x)^(3/2))/(48*
b) + ((10*A*b - 3*a*B)*x^(3/2)*(a + b*x)^(5/2))/(40*b) + (B*x^(3/2)*(a + b*x)^(7
/2))/(5*b) - (a^4*(10*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(12
8*b^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.216913, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{5/2}}+\frac{a^3 \sqrt{x} \sqrt{a+b x} (10 A b-3 a B)}{128 b^2}+\frac{a^2 x^{3/2} \sqrt{a+b x} (10 A b-3 a B)}{64 b}+\frac{a x^{3/2} (a+b x)^{3/2} (10 A b-3 a B)}{48 b}+\frac{x^{3/2} (a+b x)^{5/2} (10 A b-3 a B)}{40 b}+\frac{B x^{3/2} (a+b x)^{7/2}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(a^3*(10*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^2) + (a^2*(10*A*b - 3*a*B)*x
^(3/2)*Sqrt[a + b*x])/(64*b) + (a*(10*A*b - 3*a*B)*x^(3/2)*(a + b*x)^(3/2))/(48*
b) + ((10*A*b - 3*a*B)*x^(3/2)*(a + b*x)^(5/2))/(40*b) + (B*x^(3/2)*(a + b*x)^(7
/2))/(5*b) - (a^4*(10*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(12
8*b^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.5505, size = 182, normalized size = 0.95 \[ \frac{B x^{\frac{3}{2}} \left (a + b x\right )^{\frac{7}{2}}}{5 b} - \frac{a^{4} \left (10 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{128 b^{\frac{5}{2}}} - \frac{a^{3} \sqrt{x} \sqrt{a + b x} \left (10 A b - 3 B a\right )}{128 b^{2}} - \frac{a^{2} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (10 A b - 3 B a\right )}{192 b^{2}} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{5}{2}} \left (10 A b - 3 B a\right )}{240 b^{2}} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{7}{2}} \left (10 A b - 3 B a\right )}{40 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)*x**(1/2),x)

[Out]

B*x**(3/2)*(a + b*x)**(7/2)/(5*b) - a**4*(10*A*b - 3*B*a)*atanh(sqrt(b)*sqrt(x)/
sqrt(a + b*x))/(128*b**(5/2)) - a**3*sqrt(x)*sqrt(a + b*x)*(10*A*b - 3*B*a)/(128
*b**2) - a**2*sqrt(x)*(a + b*x)**(3/2)*(10*A*b - 3*B*a)/(192*b**2) - a*sqrt(x)*(
a + b*x)**(5/2)*(10*A*b - 3*B*a)/(240*b**2) + sqrt(x)*(a + b*x)**(7/2)*(10*A*b -
3*B*a)/(40*b**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.16678, size = 138, normalized size = 0.72 \[ \frac{15 a^4 (3 a B-10 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (-45 a^4 B+30 a^3 b (5 A+B x)+4 a^2 b^2 x (295 A+186 B x)+16 a b^3 x^2 (85 A+63 B x)+96 b^4 x^3 (5 A+4 B x)\right )}{1920 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-45*a^4*B + 30*a^3*b*(5*A + B*x) + 96*b^4*x^3*(5
*A + 4*B*x) + 16*a*b^3*x^2*(85*A + 63*B*x) + 4*a^2*b^2*x*(295*A + 186*B*x)) + 15
*a^4*(-10*A*b + 3*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(1920*b^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 260, normalized size = 1.4 \[ -{\frac{1}{3840}\sqrt{bx+a}\sqrt{x} \left ( -768\,B{x}^{4}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-960\,A{x}^{3}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-2016\,B{x}^{3}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-2720\,A{x}^{2}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-1488\,B{x}^{2}{a}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }-2360\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-60\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+150\,A{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-300\,A{a}^{3}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-45\,B{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +90\,B{a}^{4}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)*x^(1/2),x)

[Out]

-1/3840*(b*x+a)^(1/2)*x^(1/2)/b^(5/2)*(-768*B*x^4*b^(9/2)*(x*(b*x+a))^(1/2)-960*
A*x^3*b^(9/2)*(x*(b*x+a))^(1/2)-2016*B*x^3*a*b^(7/2)*(x*(b*x+a))^(1/2)-2720*A*x^
2*a*b^(7/2)*(x*(b*x+a))^(1/2)-1488*B*x^2*a^2*b^(5/2)*(x*(b*x+a))^(1/2)-2360*A*a^
2*(x*(b*x+a))^(1/2)*x*b^(5/2)-60*B*a^3*(x*(b*x+a))^(1/2)*x*b^(3/2)+150*A*a^4*ln(
1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-300*A*a^3*(x*(b*x+a))^(1/2)
*b^(3/2)-45*B*a^5*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+90*B*a^4
*(x*(b*x+a))^(1/2)*b^(1/2))/(x*(b*x+a))^(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*x + A)*(b*x + a)^(5/2)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.246532, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (384 \, B b^{4} x^{4} - 45 \, B a^{4} + 150 \, A a^{3} b + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} x^{3} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} x^{2} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{3840 \, b^{\frac{5}{2}}}, \frac{{\left (384 \, B b^{4} x^{4} - 45 \, B a^{4} + 150 \, A a^{3} b + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} x^{3} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} x^{2} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} + 15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{1920 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3840*(2*(384*B*b^4*x^4 - 45*B*a^4 + 150*A*a^3*b + 48*(21*B*a*b^3 + 10*A*b^4)*
x^3 + 8*(93*B*a^2*b^2 + 170*A*a*b^3)*x^2 + 10*(3*B*a^3*b + 118*A*a^2*b^2)*x)*sqr
t(b*x + a)*sqrt(b)*sqrt(x) - 15*(3*B*a^5 - 10*A*a^4*b)*log(-2*sqrt(b*x + a)*b*sq
rt(x) + (2*b*x + a)*sqrt(b)))/b^(5/2), 1/1920*((384*B*b^4*x^4 - 45*B*a^4 + 150*A
*a^3*b + 48*(21*B*a*b^3 + 10*A*b^4)*x^3 + 8*(93*B*a^2*b^2 + 170*A*a*b^3)*x^2 + 1
0*(3*B*a^3*b + 118*A*a^2*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) + 15*(3*B*a^5 -
10*A*a^4*b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^2)]

_______________________________________________________________________________________

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*x+A)*x**(1/2),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*x + A)*(b*x + a)^(5/2)*sqrt(x),x, algorithm="giac")

[Out]

Timed out