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

Optimal. Leaf size=159 \[ -\frac{5 a^3 (8 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{9/2}}+\frac{5 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-7 a B)}{64 b^4}-\frac{5 a x^{3/2} \sqrt{a+b x} (8 A b-7 a B)}{96 b^3}+\frac{x^{5/2} \sqrt{a+b x} (8 A b-7 a B)}{24 b^2}+\frac{B x^{7/2} \sqrt{a+b x}}{4 b} \]

[Out]

(5*a^2*(8*A*b - 7*a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b^4) - (5*a*(8*A*b - 7*a*B)*x^
(3/2)*Sqrt[a + b*x])/(96*b^3) + ((8*A*b - 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(24*b^2)
+ (B*x^(7/2)*Sqrt[a + b*x])/(4*b) - (5*a^3*(8*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*Sqr
t[x])/Sqrt[a + b*x]])/(64*b^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.186749, 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 \[ -\frac{5 a^3 (8 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{9/2}}+\frac{5 a^2 \sqrt{x} \sqrt{a+b x} (8 A b-7 a B)}{64 b^4}-\frac{5 a x^{3/2} \sqrt{a+b x} (8 A b-7 a B)}{96 b^3}+\frac{x^{5/2} \sqrt{a+b x} (8 A b-7 a B)}{24 b^2}+\frac{B x^{7/2} \sqrt{a+b x}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

(5*a^2*(8*A*b - 7*a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b^4) - (5*a*(8*A*b - 7*a*B)*x^
(3/2)*Sqrt[a + b*x])/(96*b^3) + ((8*A*b - 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(24*b^2)
+ (B*x^(7/2)*Sqrt[a + b*x])/(4*b) - (5*a^3*(8*A*b - 7*a*B)*ArcTanh[(Sqrt[b]*Sqr
t[x])/Sqrt[a + b*x]])/(64*b^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.0467, size = 155, normalized size = 0.97 \[ \frac{B x^{\frac{7}{2}} \sqrt{a + b x}}{4 b} - \frac{5 a^{3} \left (8 A b - 7 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{64 b^{\frac{9}{2}}} + \frac{5 a^{2} \sqrt{x} \sqrt{a + b x} \left (8 A b - 7 B a\right )}{64 b^{4}} - \frac{5 a x^{\frac{3}{2}} \sqrt{a + b x} \left (8 A b - 7 B a\right )}{96 b^{3}} + \frac{x^{\frac{5}{2}} \sqrt{a + b x} \left (8 A b - 7 B a\right )}{24 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**(7/2)*sqrt(a + b*x)/(4*b) - 5*a**3*(8*A*b - 7*B*a)*atanh(sqrt(a + b*x)/(sqr
t(b)*sqrt(x)))/(64*b**(9/2)) + 5*a**2*sqrt(x)*sqrt(a + b*x)*(8*A*b - 7*B*a)/(64*
b**4) - 5*a*x**(3/2)*sqrt(a + b*x)*(8*A*b - 7*B*a)/(96*b**3) + x**(5/2)*sqrt(a +
b*x)*(8*A*b - 7*B*a)/(24*b**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.139809, size = 120, normalized size = 0.75 \[ \frac{15 a^3 (7 a B-8 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (-105 a^3 B+10 a^2 b (12 A+7 B x)-8 a b^2 x (10 A+7 B x)+16 b^3 x^2 (4 A+3 B x)\right )}{192 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-105*a^3*B + 16*b^3*x^2*(4*A + 3*B*x) - 8*a*b^2*
x*(10*A + 7*B*x) + 10*a^2*b*(12*A + 7*B*x)) + 15*a^3*(-8*A*b + 7*a*B)*Log[b*Sqrt
[x] + Sqrt[b]*Sqrt[a + b*x]])/(192*b^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 218, normalized size = 1.4 \[ -{\frac{1}{384}\sqrt{x}\sqrt{bx+a} \left ( -96\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-128\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+112\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+160\,Aax\sqrt{x \left ( bx+a \right ) }{b}^{5/2}-140\,B{a}^{2}x\sqrt{x \left ( bx+a \right ) }{b}^{3/2}+120\,A{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-240\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-105\,B{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +210\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*x^(1/2)*(b*x+a)^(1/2)/b^(9/2)*(-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)+112*B*x^2*a*b^(5/2)*(x*(b*x+a))^(1/2)+160*A*a*x*(x
*(b*x+a))^(1/2)*b^(5/2)-140*B*a^2*x*(x*(b*x+a))^(1/2)*b^(3/2)+120*A*a^3*ln(1/2*(
2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-240*A*a^2*(x*(b*x+a))^(1/2)*b^(3
/2)-105*B*a^4*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+210*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)*x^(5/2)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238199, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B b^{3} x^{3} - 105 \, B a^{3} + 120 \, A a^{2} b - 8 \,{\left (7 \, B a b^{2} - 8 \, A b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{2} b - 8 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 15 \,{\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{384 \, b^{\frac{9}{2}}}, \frac{{\left (48 \, B b^{3} x^{3} - 105 \, B a^{3} + 120 \, A a^{2} b - 8 \,{\left (7 \, B a b^{2} - 8 \, A b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{2} b - 8 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} + 15 \,{\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{192 \, \sqrt{-b} b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(2*(48*B*b^3*x^3 - 105*B*a^3 + 120*A*a^2*b - 8*(7*B*a*b^2 - 8*A*b^3)*x^2
+ 10*(7*B*a^2*b - 8*A*a*b^2)*x)*sqrt(b*x + a)*sqrt(b)*sqrt(x) - 15*(7*B*a^4 - 8*
A*a^3*b)*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)))/b^(9/2), 1/192*(
(48*B*b^3*x^3 - 105*B*a^3 + 120*A*a^2*b - 8*(7*B*a*b^2 - 8*A*b^3)*x^2 + 10*(7*B*
a^2*b - 8*A*a*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) + 15*(7*B*a^4 - 8*A*a^3*b)*
arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^4)]

_______________________________________________________________________________________

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

[Out]

Timed out