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

Optimal. Leaf size=262 \[ \frac{A b^5 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{5 a A b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{10 a^2 A b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b}+\frac{a^5 A \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^4 A b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^3 A b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

(5*a^4*A*b*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (5*a^3*A*b^2*x^2*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (10*a^2*A*b^3*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(3*(a + b*x)) + (5*a*A*b^4*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x))
+ (A*b^5*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (B*(a + b*x)^5*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(6*b) + (a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/
(a + b*x)

_______________________________________________________________________________________

Rubi [A]  time = 0.237089, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{A b^5 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{5 a A b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{10 a^2 A b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{B (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b}+\frac{a^5 A \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^4 A b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^3 A b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

(5*a^4*A*b*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (5*a^3*A*b^2*x^2*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (10*a^2*A*b^3*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(3*(a + b*x)) + (5*a*A*b^4*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x))
+ (A*b^5*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (B*(a + b*x)^5*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(6*b) + (a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/
(a + b*x)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.7482, size = 212, normalized size = 0.81 \[ \frac{A a^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + A a^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{A a^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6} + \frac{A a^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3} + \frac{A a \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{20} + \frac{A \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5} + \frac{B \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a + b*x) + A*a**4*sqrt(a**2 + 2*
a*b*x + b**2*x**2) + A*a**3*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/6 + A
*a**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/3 + A*a*(5*a + 5*b*x)*(a**2 + 2*a*b*x
+ b**2*x**2)**(3/2)/20 + A*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/5 + B*(2*a + 2*b*
x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(12*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.125733, size = 122, normalized size = 0.47 \[ \frac{\sqrt{(a+b x)^2} \left (60 a^5 A \log (x)+x \left (60 a^5 B+150 a^4 b (2 A+B x)+100 a^3 b^2 x (3 A+2 B x)+50 a^2 b^3 x^2 (4 A+3 B x)+15 a b^4 x^3 (5 A+4 B x)+2 b^5 x^4 (6 A+5 B x)\right )\right )}{60 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(x*(60*a^5*B + 150*a^4*b*(2*A + B*x) + 100*a^3*b^2*x*(3*A + 2
*B*x) + 50*a^2*b^3*x^2*(4*A + 3*B*x) + 15*a*b^4*x^3*(5*A + 4*B*x) + 2*b^5*x^4*(6
*A + 5*B*x)) + 60*a^5*A*Log[x]))/(60*(a + b*x))

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 139, normalized size = 0.5 \[{\frac{10\,B{b}^{5}{x}^{6}+12\,A{x}^{5}{b}^{5}+60\,B{x}^{5}a{b}^{4}+75\,A{x}^{4}a{b}^{4}+150\,B{x}^{4}{a}^{2}{b}^{3}+200\,A{x}^{3}{a}^{2}{b}^{3}+200\,B{x}^{3}{a}^{3}{b}^{2}+300\,A{x}^{2}{a}^{3}{b}^{2}+150\,B{x}^{2}{a}^{4}b+60\,A{a}^{5}\ln \left ( x \right ) +300\,Ax{a}^{4}b+60\,Bx{a}^{5}}{60\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*((b*x+a)^2)^(5/2)*(10*B*b^5*x^6+12*A*x^5*b^5+60*B*x^5*a*b^4+75*A*x^4*a*b^4+
150*B*x^4*a^2*b^3+200*A*x^3*a^2*b^3+200*B*x^3*a^3*b^2+300*A*x^2*a^3*b^2+150*B*x^
2*a^4*b+60*A*a^5*ln(x)+300*A*x*a^4*b+60*B*x*a^5)/(b*x+a)^5

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.278905, size = 154, normalized size = 0.59 \[ \frac{1}{6} \, B b^{5} x^{6} + A a^{5} \log \left (x\right ) + \frac{1}{5} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*b^5*x^6 + A*a^5*log(x) + 1/5*(5*B*a*b^4 + A*b^5)*x^5 + 5/4*(2*B*a^2*b^3 +
A*a*b^4)*x^4 + 10/3*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 5/2*(B*a^4*b + 2*A*a^3*b^2)*x^
2 + (B*a^5 + 5*A*a^4*b)*x

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273683, size = 257, normalized size = 0.98 \[ \frac{1}{6} \, B b^{5} x^{6}{\rm sign}\left (b x + a\right ) + B a b^{4} x^{5}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, A b^{5} x^{5}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, B a^{2} b^{3} x^{4}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, A a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, B a^{3} b^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, A a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, B a^{4} b x^{2}{\rm sign}\left (b x + a\right ) + 5 \, A a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + B a^{5} x{\rm sign}\left (b x + a\right ) + 5 \, A a^{4} b x{\rm sign}\left (b x + a\right ) + A a^{5}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*b^5*x^6*sign(b*x + a) + B*a*b^4*x^5*sign(b*x + a) + 1/5*A*b^5*x^5*sign(b*x
+ a) + 5/2*B*a^2*b^3*x^4*sign(b*x + a) + 5/4*A*a*b^4*x^4*sign(b*x + a) + 10/3*B
*a^3*b^2*x^3*sign(b*x + a) + 10/3*A*a^2*b^3*x^3*sign(b*x + a) + 5/2*B*a^4*b*x^2*
sign(b*x + a) + 5*A*a^3*b^2*x^2*sign(b*x + a) + B*a^5*x*sign(b*x + a) + 5*A*a^4*
b*x*sign(b*x + a) + A*a^5*ln(abs(x))*sign(b*x + a)