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

Optimal. Leaf size=169 \[ -\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{9/2}}+\frac{5 \sqrt{x} \sqrt{a+b x} (4 A b-7 a B)}{4 b^4}-\frac{5 x^{3/2} \sqrt{a+b x} (4 A b-7 a B)}{6 a b^3}+\frac{2 x^{5/2} (4 A b-7 a B)}{3 a b^2 \sqrt{a+b x}}+\frac{2 x^{7/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.192091, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{9/2}}+\frac{5 \sqrt{x} \sqrt{a+b x} (4 A b-7 a B)}{4 b^4}-\frac{5 x^{3/2} \sqrt{a+b x} (4 A b-7 a B)}{6 a b^3}+\frac{2 x^{5/2} (4 A b-7 a B)}{3 a b^2 \sqrt{a+b x}}+\frac{2 x^{7/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.0439, size = 162, normalized size = 0.96 \[ - \frac{5 a \left (4 A b - 7 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{4 b^{\frac{9}{2}}} + \frac{5 \sqrt{x} \sqrt{a + b x} \left (4 A b - 7 B a\right )}{4 b^{4}} + \frac{2 x^{\frac{7}{2}} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{\frac{3}{2}}} + \frac{2 x^{\frac{5}{2}} \left (4 A b - 7 B a\right )}{3 a b^{2} \sqrt{a + b x}} - \frac{5 x^{\frac{3}{2}} \sqrt{a + b x} \left (4 A b - 7 B a\right )}{6 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.187098, size = 115, normalized size = 0.68 \[ \frac{\sqrt{x} \left (-105 a^3 B+20 a^2 b (3 A-7 B x)+a b^2 x (80 A-21 B x)+6 b^3 x^2 (2 A+B x)\right )}{12 b^4 (a+b x)^{3/2}}+\frac{5 a (7 a B-4 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{4 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(-105*a^3*B + a*b^2*x*(80*A - 21*B*x) + 20*a^2*b*(3*A - 7*B*x) + 6*b^3*
x^2*(2*A + B*x)))/(12*b^4*(a + b*x)^(3/2)) + (5*a*(-4*A*b + 7*a*B)*Log[b*Sqrt[x]
+ Sqrt[b]*Sqrt[a + b*x]])/(4*b^(9/2))

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Maple [B]  time = 0.024, size = 362, normalized size = 2.1 \[ -{\frac{1}{24} \left ( -12\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+60\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}a{b}^{3}-24\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-105\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{a}^{2}{b}^{2}+42\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+120\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}{b}^{2}-160\,Aax\sqrt{x \left ( bx+a \right ) }{b}^{5/2}-210\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{3}b+280\,B{a}^{2}x\sqrt{x \left ( bx+a \right ) }{b}^{3/2}+60\,A{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-120\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-105\,B{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +210\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ) \sqrt{x}{b}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(-12*B*x^3*b^(7/2)*(x*(b*x+a))^(1/2)+60*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1
/2)+2*b*x+a)/b^(1/2))*x^2*a*b^3-24*A*x^2*b^(7/2)*(x*(b*x+a))^(1/2)-105*B*ln(1/2*
(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x^2*a^2*b^2+42*B*x^2*a*b^(5/2)*(x
*(b*x+a))^(1/2)+120*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^2-160*A*a*x*(x*(b*x+a))^(1/2)*b^(5/2)-210*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+280*B*a^2*x*(x*(b*x+a))^(1/2)*b^(3/2)+60*A*a^3*ln(
1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-120*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)^(3/2)

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24889, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b +{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (6 \, B b^{3} x^{4} - 3 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 20 \,{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} - 15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} x\right )} \sqrt{b}}{24 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}, \frac{15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b +{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (6 \, B b^{3} x^{4} - 3 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 20 \,{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} - 15 \,{\left (7 \, B a^{3} - 4 \, A a^{2} b\right )} x\right )} \sqrt{-b}}{12 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/24*(15*(7*B*a^3 - 4*A*a^2*b + (7*B*a^2*b - 4*A*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*(6*B*b^3*x^4 - 3*(7
*B*a*b^2 - 4*A*b^3)*x^3 - 20*(7*B*a^2*b - 4*A*a*b^2)*x^2 - 15*(7*B*a^3 - 4*A*a^2
*b)*x)*sqrt(b))/((b^5*x + a*b^4)*sqrt(b*x + a)*sqrt(b)*sqrt(x)), 1/12*(15*(7*B*a
^3 - 4*A*a^2*b + (7*B*a^2*b - 4*A*a*b^2)*x)*sqrt(b*x + a)*sqrt(x)*arctan(sqrt(b*
x + a)*sqrt(-b)/(b*sqrt(x))) + (6*B*b^3*x^4 - 3*(7*B*a*b^2 - 4*A*b^3)*x^3 - 20*(
7*B*a^2*b - 4*A*a*b^2)*x^2 - 15*(7*B*a^3 - 4*A*a^2*b)*x)*sqrt(-b))/((b^5*x + a*b
^4)*sqrt(b*x + a)*sqrt(-b)*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28652, size = 467, normalized size = 2.76 \[ \frac{1}{4} \, \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} B{\left | b \right |}}{b^{6}} - \frac{13 \, B a b^{11}{\left | b \right |} - 4 \, A b^{12}{\left | b \right |}}{b^{17}}\right )} - \frac{5 \,{\left (7 \, B a^{2} \sqrt{b}{\left | b \right |} - 4 \, A a b^{\frac{3}{2}}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{8 \, b^{6}} - \frac{4 \,{\left (12 \, B a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b}{\left | b \right |} + 18 \, B a^{4}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}}{\left | b \right |} - 9 \, A a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{3}{2}}{\left | b \right |} + 10 \, B a^{5} b^{\frac{5}{2}}{\left | b \right |} - 12 \, A a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{5}{2}}{\left | b \right |} - 7 \, A a^{4} b^{\frac{7}{2}}{\left | b \right |}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*(2*(b*x + a)*B*abs(b)/b^6 - (13*B*a*b^
11*abs(b) - 4*A*b^12*abs(b))/b^17) - 5/8*(7*B*a^2*sqrt(b)*abs(b) - 4*A*a*b^(3/2)
*abs(b))*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^6 - 4/3*(12*B
*a^3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*sqrt(b)*abs(b) + 18*B*a
^4*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(3/2)*abs(b) - 9*A*a^2*
(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(3/2)*abs(b) + 10*B*a^5*b^
(5/2)*abs(b) - 12*A*a^3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(5
/2)*abs(b) - 7*A*a^4*b^(7/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b
- a*b))^2 + a*b)^3*b^5)