3.480 \(\int \sqrt{x} (a+b x)^{3/2} (A+B x) \, dx\)

Optimal. Leaf size=159 \[ -\frac{a^3 (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{5/2}}+\frac{a^2 \sqrt{x} \sqrt{a+b x} (8 A b-3 a B)}{64 b^2}+\frac{a x^{3/2} \sqrt{a+b x} (8 A b-3 a B)}{32 b}+\frac{x^{3/2} (a+b x)^{3/2} (8 A b-3 a B)}{24 b}+\frac{B x^{3/2} (a+b x)^{5/2}}{4 b} \]

[Out]

(a^2*(8*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b^2) + (a*(8*A*b - 3*a*B)*x^(3/2
)*Sqrt[a + b*x])/(32*b) + ((8*A*b - 3*a*B)*x^(3/2)*(a + b*x)^(3/2))/(24*b) + (B*
x^(3/2)*(a + b*x)^(5/2))/(4*b) - (a^3*(8*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/
Sqrt[a + b*x]])/(64*b^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.181922, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^3 (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{5/2}}+\frac{a^2 \sqrt{x} \sqrt{a+b x} (8 A b-3 a B)}{64 b^2}+\frac{a x^{3/2} \sqrt{a+b x} (8 A b-3 a B)}{32 b}+\frac{x^{3/2} (a+b x)^{3/2} (8 A b-3 a B)}{24 b}+\frac{B x^{3/2} (a+b x)^{5/2}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(a^2*(8*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b^2) + (a*(8*A*b - 3*a*B)*x^(3/2
)*Sqrt[a + b*x])/(32*b) + ((8*A*b - 3*a*B)*x^(3/2)*(a + b*x)^(3/2))/(24*b) + (B*
x^(3/2)*(a + b*x)^(5/2))/(4*b) - (a^3*(8*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/
Sqrt[a + b*x]])/(64*b^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.1111, size = 150, normalized size = 0.94 \[ \frac{B x^{\frac{3}{2}} \left (a + b x\right )^{\frac{5}{2}}}{4 b} - \frac{a^{3} \left (8 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{64 b^{\frac{5}{2}}} - \frac{a^{2} \sqrt{x} \sqrt{a + b x} \left (8 A b - 3 B a\right )}{64 b^{2}} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (8 A b - 3 B a\right )}{96 b^{2}} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{5}{2}} \left (8 A b - 3 B a\right )}{24 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)*x**(1/2),x)

[Out]

B*x**(3/2)*(a + b*x)**(5/2)/(4*b) - a**3*(8*A*b - 3*B*a)*atanh(sqrt(a + b*x)/(sq
rt(b)*sqrt(x)))/(64*b**(5/2)) - a**2*sqrt(x)*sqrt(a + b*x)*(8*A*b - 3*B*a)/(64*b
**2) - a*sqrt(x)*(a + b*x)**(3/2)*(8*A*b - 3*B*a)/(96*b**2) + sqrt(x)*(a + b*x)*
*(5/2)*(8*A*b - 3*B*a)/(24*b**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.124982, size = 119, normalized size = 0.75 \[ \frac{3 a^3 (3 a B-8 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (-9 a^3 B+6 a^2 b (4 A+B x)+8 a b^2 x (14 A+9 B x)+16 b^3 x^2 (4 A+3 B x)\right )}{192 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*(a + b*x)^(3/2)*(A + B*x),x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-9*a^3*B + 6*a^2*b*(4*A + B*x) + 16*b^3*x^2*(4*A
+ 3*B*x) + 8*a*b^2*x*(14*A + 9*B*x)) + 3*a^3*(-8*A*b + 3*a*B)*Log[b*Sqrt[x] + S
qrt[b]*Sqrt[a + b*x]])/(192*b^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 218, normalized size = 1.4 \[ -{\frac{1}{384}\sqrt{bx+a}\sqrt{x} \left ( -96\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-128\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-144\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }-224\,Aax\sqrt{x \left ( bx+a \right ) }{b}^{5/2}-12\,B{a}^{2}x\sqrt{x \left ( bx+a \right ) }{b}^{3/2}+24\,A{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-48\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-9\,B{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +18\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)*x^(1/2),x)

[Out]

-1/384*(b*x+a)^(1/2)*x^(1/2)/b^(5/2)*(-96*B*x^3*b^(7/2)*(x*(b*x+a))^(1/2)-128*A*
x^2*b^(7/2)*(x*(b*x+a))^(1/2)-144*B*x^2*a*b^(5/2)*(x*(b*x+a))^(1/2)-224*A*a*x*(x
*(b*x+a))^(1/2)*b^(5/2)-12*B*a^2*x*(x*(b*x+a))^(1/2)*b^(3/2)+24*A*a^3*ln(1/2*(2*
(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-48*A*a^2*(x*(b*x+a))^(1/2)*b^(3/2)
-9*B*a^4*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+18*B*a^3*(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)^(3/2)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.233407, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B b^{3} x^{3} - 9 \, B a^{3} + 24 \, A a^{2} b + 8 \,{\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{384 \, b^{\frac{5}{2}}}, \frac{{\left (48 \, B b^{3} x^{3} - 9 \, B a^{3} + 24 \, A a^{2} b + 8 \,{\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} + 3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{192 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)*sqrt(x),x, algorithm="fricas")

[Out]

[1/384*(2*(48*B*b^3*x^3 - 9*B*a^3 + 24*A*a^2*b + 8*(9*B*a*b^2 + 8*A*b^3)*x^2 + 2
*(3*B*a^2*b + 56*A*a*b^2)*x)*sqrt(b*x + a)*sqrt(b)*sqrt(x) - 3*(3*B*a^4 - 8*A*a^
3*b)*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)))/b^(5/2), 1/192*((48*
B*b^3*x^3 - 9*B*a^3 + 24*A*a^2*b + 8*(9*B*a*b^2 + 8*A*b^3)*x^2 + 2*(3*B*a^2*b +
56*A*a*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) + 3*(3*B*a^4 - 8*A*a^3*b)*arctan(s
qrt(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^2)]

_______________________________________________________________________________________

Sympy [A]  time = 80.4719, size = 298, normalized size = 1.87 \[ \frac{A a^{\frac{5}{2}} \sqrt{x}}{8 b \sqrt{1 + \frac{b x}{a}}} + \frac{17 A a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 \sqrt{1 + \frac{b x}{a}}} + \frac{11 A \sqrt{a} b x^{\frac{5}{2}}}{12 \sqrt{1 + \frac{b x}{a}}} - \frac{A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{A b^{2} x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} - \frac{3 B a^{\frac{7}{2}} \sqrt{x}}{64 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{B a^{\frac{5}{2}} x^{\frac{3}{2}}}{64 b \sqrt{1 + \frac{b x}{a}}} + \frac{13 B a^{\frac{3}{2}} x^{\frac{5}{2}}}{32 \sqrt{1 + \frac{b x}{a}}} + \frac{5 B \sqrt{a} b x^{\frac{7}{2}}}{8 \sqrt{1 + \frac{b x}{a}}} + \frac{3 B a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{5}{2}}} + \frac{B b^{2} x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)*(B*x+A)*x**(1/2),x)

[Out]

A*a**(5/2)*sqrt(x)/(8*b*sqrt(1 + b*x/a)) + 17*A*a**(3/2)*x**(3/2)/(24*sqrt(1 + b
*x/a)) + 11*A*sqrt(a)*b*x**(5/2)/(12*sqrt(1 + b*x/a)) - A*a**3*asinh(sqrt(b)*sqr
t(x)/sqrt(a))/(8*b**(3/2)) + A*b**2*x**(7/2)/(3*sqrt(a)*sqrt(1 + b*x/a)) - 3*B*a
**(7/2)*sqrt(x)/(64*b**2*sqrt(1 + b*x/a)) - B*a**(5/2)*x**(3/2)/(64*b*sqrt(1 + b
*x/a)) + 13*B*a**(3/2)*x**(5/2)/(32*sqrt(1 + b*x/a)) + 5*B*sqrt(a)*b*x**(7/2)/(8
*sqrt(1 + b*x/a)) + 3*B*a**4*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(64*b**(5/2)) + B*b*
*2*x**(9/2)/(4*sqrt(a)*sqrt(1 + b*x/a))

_______________________________________________________________________________________

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)^(3/2)*sqrt(x),x, algorithm="giac")

[Out]

Timed out