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

Optimal. Leaf size=167 \[ \frac{5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{9/2}}-\frac{5 a \sqrt{x} \sqrt{a+b x} (6 A b-7 a B)}{8 b^4}+\frac{5 x^{3/2} \sqrt{a+b x} (6 A b-7 a B)}{12 b^3}-\frac{x^{5/2} \sqrt{a+b x} (6 A b-7 a B)}{3 a b^2}+\frac{2 x^{7/2} (A b-a B)}{a b \sqrt{a+b x}} \]

[Out]

(2*(A*b - a*B)*x^(7/2))/(a*b*Sqrt[a + b*x]) - (5*a*(6*A*b - 7*a*B)*Sqrt[x]*Sqrt[
a + b*x])/(8*b^4) + (5*(6*A*b - 7*a*B)*x^(3/2)*Sqrt[a + b*x])/(12*b^3) - ((6*A*b
- 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(3*a*b^2) + (5*a^2*(6*A*b - 7*a*B)*ArcTanh[(Sqr
t[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(9/2))

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Rubi [A]  time = 0.189823, antiderivative size = 167, 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 \[ \frac{5 a^2 (6 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{9/2}}-\frac{5 a \sqrt{x} \sqrt{a+b x} (6 A b-7 a B)}{8 b^4}+\frac{5 x^{3/2} \sqrt{a+b x} (6 A b-7 a B)}{12 b^3}-\frac{x^{5/2} \sqrt{a+b x} (6 A b-7 a B)}{3 a b^2}+\frac{2 x^{7/2} (A b-a B)}{a b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*x^(7/2))/(a*b*Sqrt[a + b*x]) - (5*a*(6*A*b - 7*a*B)*Sqrt[x]*Sqrt[
a + b*x])/(8*b^4) + (5*(6*A*b - 7*a*B)*x^(3/2)*Sqrt[a + b*x])/(12*b^3) - ((6*A*b
- 7*a*B)*x^(5/2)*Sqrt[a + b*x])/(3*a*b^2) + (5*a^2*(6*A*b - 7*a*B)*ArcTanh[(Sqr
t[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(9/2))

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Rubi in Sympy [A]  time = 19.209, size = 160, normalized size = 0.96 \[ \frac{5 a^{2} \left (6 A b - 7 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{8 b^{\frac{9}{2}}} - \frac{5 a \sqrt{x} \sqrt{a + b x} \left (6 A b - 7 B a\right )}{8 b^{4}} + \frac{5 x^{\frac{3}{2}} \sqrt{a + b x} \left (6 A b - 7 B a\right )}{12 b^{3}} + \frac{2 x^{\frac{7}{2}} \left (A b - B a\right )}{a b \sqrt{a + b x}} - \frac{x^{\frac{5}{2}} \sqrt{a + b x} \left (6 A b - 7 B a\right )}{3 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**2*(6*A*b - 7*B*a)*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(8*b**(9/2)) - 5*a*s
qrt(x)*sqrt(a + b*x)*(6*A*b - 7*B*a)/(8*b**4) + 5*x**(3/2)*sqrt(a + b*x)*(6*A*b
- 7*B*a)/(12*b**3) + 2*x**(7/2)*(A*b - B*a)/(a*b*sqrt(a + b*x)) - x**(5/2)*sqrt(
a + b*x)*(6*A*b - 7*B*a)/(3*a*b**2)

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Mathematica [A]  time = 0.191447, size = 119, normalized size = 0.71 \[ \frac{\sqrt{x} \left (105 a^3 B+a^2 (35 b B x-90 A b)-2 a b^2 x (15 A+7 B x)+4 b^3 x^2 (3 A+2 B x)\right )}{24 b^4 \sqrt{a+b x}}-\frac{5 a^2 (7 a B-6 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{8 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(105*a^3*B + 4*b^3*x^2*(3*A + 2*B*x) - 2*a*b^2*x*(15*A + 7*B*x) + a^2*(
-90*A*b + 35*b*B*x)))/(24*b^4*Sqrt[a + b*x]) - (5*a^2*(-6*A*b + 7*a*B)*Log[b*Sqr
t[x] + Sqrt[b]*Sqrt[a + b*x]])/(8*b^(9/2))

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Maple [B]  time = 0.027, size = 288, normalized size = 1.7 \[{\frac{1}{48} \left ( 16\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+24\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-28\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+90\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}{b}^{2}-60\,Aax\sqrt{x \left ( bx+a \right ) }{b}^{5/2}-105\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{3}b+70\,B{a}^{2}x\sqrt{x \left ( bx+a \right ) }{b}^{3/2}+90\,A{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-180\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-105\,B{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +210\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ) \sqrt{x}{b}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(16*B*x^3*b^(7/2)*(x*(b*x+a))^(1/2)+24*A*x^2*b^(7/2)*(x*(b*x+a))^(1/2)-28*B
*x^2*a*b^(5/2)*(x*(b*x+a))^(1/2)+90*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+
a)/b^(1/2))*x*a^2*b^2-60*A*a*x*(x*(b*x+a))^(1/2)*b^(5/2)-105*B*ln(1/2*(2*(x*(b*x
+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a^3*b+70*B*a^2*x*(x*(b*x+a))^(1/2)*b^(3/2
)+90*A*a^3*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-180*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))/b^(9/2)*x^(1/2)/(x*(b*x+a))^(1/2)/(b
*x+a)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246775, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} \sqrt{b x + a} \sqrt{x} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (8 \, B b^{3} x^{4} - 2 \,{\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} + 5 \,{\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{2} + 15 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} \sqrt{b}}{48 \, \sqrt{b x + a} b^{\frac{9}{2}} \sqrt{x}}, -\frac{15 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (8 \, B b^{3} x^{4} - 2 \,{\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{3} + 5 \,{\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{2} + 15 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x\right )} \sqrt{-b}}{24 \, \sqrt{b x + a} \sqrt{-b} b^{4} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(15*(7*B*a^3 - 6*A*a^2*b)*sqrt(b*x + a)*sqrt(x)*log(2*sqrt(b*x + a)*b*sqr
t(x) + (2*b*x + a)*sqrt(b)) - 2*(8*B*b^3*x^4 - 2*(7*B*a*b^2 - 6*A*b^3)*x^3 + 5*(
7*B*a^2*b - 6*A*a*b^2)*x^2 + 15*(7*B*a^3 - 6*A*a^2*b)*x)*sqrt(b))/(sqrt(b*x + a)
*b^(9/2)*sqrt(x)), -1/24*(15*(7*B*a^3 - 6*A*a^2*b)*sqrt(b*x + a)*sqrt(x)*arctan(
sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (8*B*b^3*x^4 - 2*(7*B*a*b^2 - 6*A*b^3)*x^3
+ 5*(7*B*a^2*b - 6*A*a*b^2)*x^2 + 15*(7*B*a^3 - 6*A*a^2*b)*x)*sqrt(-b))/(sqrt(b
*x + a)*sqrt(-b)*b^4*sqrt(x))]

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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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.263173, size = 293, normalized size = 1.75 \[ \frac{1}{24} \, \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )} B{\left | b \right |}}{b^{6}} - \frac{19 \, B a b^{17}{\left | b \right |} - 6 \, A b^{18}{\left | b \right |}}{b^{23}}\right )} + \frac{3 \,{\left (29 \, B a^{2} b^{17}{\left | b \right |} - 18 \, A a b^{18}{\left | b \right |}\right )}}{b^{23}}\right )} + \frac{5 \,{\left (7 \, B a^{3} \sqrt{b}{\left | b \right |} - 6 \, A a^{2} b^{\frac{3}{2}}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{16 \, b^{6}} + \frac{4 \,{\left (B a^{4} \sqrt{b}{\left | b \right |} - A a^{3} b^{\frac{3}{2}}{\left | b \right |}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*B*abs(b)/b^
6 - (19*B*a*b^17*abs(b) - 6*A*b^18*abs(b))/b^23) + 3*(29*B*a^2*b^17*abs(b) - 18*
A*a*b^18*abs(b))/b^23) + 5/16*(7*B*a^3*sqrt(b)*abs(b) - 6*A*a^2*b^(3/2)*abs(b))*
ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^6 + 4*(B*a^4*sqrt(b)*a
bs(b) - A*a^3*b^(3/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b)
)^2 + a*b)*b^5)