3.527 \(\int \frac{x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=133 \[ \frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}-\frac{\sqrt{x} \sqrt{a+b x} (2 A b-5 a B)}{a b^3}+\frac{2 x^{3/2} (2 A b-5 a B)}{3 a b^2 \sqrt{a+b x}}+\frac{2 x^{5/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

[Out]

(2*(A*b - a*B)*x^(5/2))/(3*a*b*(a + b*x)^(3/2)) + (2*(2*A*b - 5*a*B)*x^(3/2))/(3
*a*b^2*Sqrt[a + b*x]) - ((2*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(a*b^3) + ((2*A*
b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(7/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.152166, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}-\frac{\sqrt{x} \sqrt{a+b x} (2 A b-5 a B)}{a b^3}+\frac{2 x^{3/2} (2 A b-5 a B)}{3 a b^2 \sqrt{a+b x}}+\frac{2 x^{5/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^(3/2)*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

(2*(A*b - a*B)*x^(5/2))/(3*a*b*(a + b*x)^(3/2)) + (2*(2*A*b - 5*a*B)*x^(3/2))/(3
*a*b^2*Sqrt[a + b*x]) - ((2*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(a*b^3) + ((2*A*
b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(7/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.819, size = 126, normalized size = 0.95 \[ \frac{2 \left (A b - \frac{5 B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{7}{2}}} + \frac{2 x^{\frac{5}{2}} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{\frac{3}{2}}} + \frac{4 x^{\frac{3}{2}} \left (A b - \frac{5 B a}{2}\right )}{3 a b^{2} \sqrt{a + b x}} - \frac{2 \sqrt{x} \sqrt{a + b x} \left (A b - \frac{5 B a}{2}\right )}{a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(B*x+A)/(b*x+a)**(5/2),x)

[Out]

2*(A*b - 5*B*a/2)*atanh(sqrt(a + b*x)/(sqrt(b)*sqrt(x)))/b**(7/2) + 2*x**(5/2)*(
A*b - B*a)/(3*a*b*(a + b*x)**(3/2)) + 4*x**(3/2)*(A*b - 5*B*a/2)/(3*a*b**2*sqrt(
a + b*x)) - 2*sqrt(x)*sqrt(a + b*x)*(A*b - 5*B*a/2)/(a*b**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.145789, size = 93, normalized size = 0.7 \[ \frac{\sqrt{x} \left (15 a^2 B+a (20 b B x-6 A b)+b^2 x (3 B x-8 A)\right )}{3 b^3 (a+b x)^{3/2}}+\frac{(2 A b-5 a B) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(3/2)*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

(Sqrt[x]*(15*a^2*B + b^2*x*(-8*A + 3*B*x) + a*(-6*A*b + 20*b*B*x)))/(3*b^3*(a +
b*x)^(3/2)) + ((2*A*b - 5*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/b^(7/2)

_______________________________________________________________________________________

Maple [B]  time = 0.023, size = 315, normalized size = 2.4 \[{\frac{1}{6} \left ( 6\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{b}^{3}-15\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}a{b}^{2}+6\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+12\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) xa{b}^{2}-16\,A\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-30\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}b+40\,Ba\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+6\,A{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-12\,A\sqrt{x \left ( bx+a \right ) }a{b}^{3/2}-15\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +30\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ) \sqrt{x}{b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(B*x+A)/(b*x+a)^(5/2),x)

[Out]

1/6*(6*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x^2*b^3-15*B*ln(1
/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x^2*a*b^2+6*B*x^2*b^(5/2)*(x*(
b*x+a))^(1/2)+12*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a*b^2
-16*A*(x*(b*x+a))^(1/2)*x*b^(5/2)-30*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x
+a)/b^(1/2))*x*a^2*b+40*B*a*(x*(b*x+a))^(1/2)*x*b^(3/2)+6*A*a^2*ln(1/2*(2*(x*(b*
x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-12*A*(x*(b*x+a))^(1/2)*a*b^(3/2)-15*B*a^
3*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+30*B*a^2*(x*(b*x+a))^(1/
2)*b^(1/2))/b^(7/2)*x^(1/2)/(x*(b*x+a))^(1/2)/(b*x+a)^(3/2)

_______________________________________________________________________________________

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^(3/2)/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.242317, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, B a^{2} - 2 \, A a b +{\left (5 \, B a b - 2 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (3 \, B b^{2} x^{3} + 4 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} x^{2} + 3 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} x\right )} \sqrt{b}}{6 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}, -\frac{3 \,{\left (5 \, B a^{2} - 2 \, A a b +{\left (5 \, B a b - 2 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (3 \, B b^{2} x^{3} + 4 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} x^{2} + 3 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} x\right )} \sqrt{-b}}{3 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(3/2)/(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(3*(5*B*a^2 - 2*A*a*b + (5*B*a*b - 2*A*b^2)*x)*sqrt(b*x + a)*sqrt(x)*log(2
*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) - 2*(3*B*b^2*x^3 + 4*(5*B*a*b -
2*A*b^2)*x^2 + 3*(5*B*a^2 - 2*A*a*b)*x)*sqrt(b))/((b^4*x + a*b^3)*sqrt(b*x + a)*
sqrt(b)*sqrt(x)), -1/3*(3*(5*B*a^2 - 2*A*a*b + (5*B*a*b - 2*A*b^2)*x)*sqrt(b*x +
a)*sqrt(x)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (3*B*b^2*x^3 + 4*(5*B*a
*b - 2*A*b^2)*x^2 + 3*(5*B*a^2 - 2*A*a*b)*x)*sqrt(-b))/((b^4*x + a*b^3)*sqrt(b*x
+ a)*sqrt(-b)*sqrt(x))]

_______________________________________________________________________________________

Sympy [A]  time = 140.069, size = 729, normalized size = 5.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(B*x+A)/(b*x+a)**(5/2),x)

[Out]

A*(6*a**(39/2)*b**11*x**(27/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3
*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)
*sqrt(1 + b*x/a)) + 6*a**(37/2)*b**12*x**(29/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sq
rt(x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b*
*(29/2)*x**(29/2)*sqrt(1 + b*x/a)) - 6*a**19*b**(23/2)*x**14/(3*a**(39/2)*b**(27
/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a))
- 8*a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*
a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a))) + B*(-15*a**(81/2)*b**22*x**(51/
2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(79/2)*b**(51/2)*x**(51/
2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 + b*x/a)) - 15*a**(7
9/2)*b**23*x**(53/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(79/2)
*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 +
b*x/a)) + 15*a**40*b**(45/2)*x**26/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x
/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 + b*x/a)) + 20*a**39*b**(47/2)*x**2
7/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(5
3/2)*sqrt(1 + b*x/a)) + 3*a**38*b**(49/2)*x**28/(3*a**(79/2)*b**(51/2)*x**(51/2)
*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 + b*x/a)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273975, size = 417, normalized size = 3.14 \[ \frac{\sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a} B{\left | b \right |}}{b^{5}} + \frac{{\left (5 \, B a \sqrt{b}{\left | b \right |} - 2 \, A b^{\frac{3}{2}}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{2 \, b^{5}} + \frac{4 \,{\left (9 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b}{\left | b \right |} + 12 \, B a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}}{\left | b \right |} - 6 \, A a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{3}{2}}{\left | b \right |} + 7 \, B a^{4} b^{\frac{5}{2}}{\left | b \right |} - 6 \, A a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{5}{2}}{\left | b \right |} - 4 \, A a^{3} b^{\frac{7}{2}}{\left | b \right |}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(3/2)/(b*x + a)^(5/2),x, algorithm="giac")

[Out]

sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*B*abs(b)/b^5 + 1/2*(5*B*a*sqrt(b)*abs(b) -
2*A*b^(3/2)*abs(b))*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^5
+ 4/3*(9*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*sqrt(b)*abs(
b) + 12*B*a^3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(3/2)*abs(b)
- 6*A*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(3/2)*abs(b) + 7*
B*a^4*b^(5/2)*abs(b) - 6*A*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))
^2*b^(5/2)*abs(b) - 4*A*a^3*b^(7/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x
+ a)*b - a*b))^2 + a*b)^3*b^4)