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

Optimal. Leaf size=143 \[ -\frac{5 c \sqrt{a+c x^2} (4 a B-3 A c x)}{8 x}-\frac{\left (a+c x^2\right )^{5/2} (A-2 B x)}{4 x^4}-\frac{5 \left (a+c x^2\right )^{3/2} (4 a B+3 A c x)}{24 x^3}-\frac{15}{8} \sqrt{a} A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{5}{2} a B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

[Out]

(-5*c*(4*a*B - 3*A*c*x)*Sqrt[a + c*x^2])/(8*x) - (5*(4*a*B + 3*A*c*x)*(a + c*x^2
)^(3/2))/(24*x^3) - ((A - 2*B*x)*(a + c*x^2)^(5/2))/(4*x^4) + (5*a*B*c^(3/2)*Arc
Tanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/2 - (15*Sqrt[a]*A*c^2*ArcTanh[Sqrt[a + c*x^2]
/Sqrt[a]])/8

_______________________________________________________________________________________

Rubi [A]  time = 0.34395, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{5 c \sqrt{a+c x^2} (4 a B-3 A c x)}{8 x}-\frac{\left (a+c x^2\right )^{5/2} (A-2 B x)}{4 x^4}-\frac{5 \left (a+c x^2\right )^{3/2} (4 a B+3 A c x)}{24 x^3}-\frac{15}{8} \sqrt{a} A c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{5}{2} a B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*c*(4*a*B - 3*A*c*x)*Sqrt[a + c*x^2])/(8*x) - (5*(4*a*B + 3*A*c*x)*(a + c*x^2
)^(3/2))/(24*x^3) - ((A - 2*B*x)*(a + c*x^2)^(5/2))/(4*x^4) + (5*a*B*c^(3/2)*Arc
Tanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/2 - (15*Sqrt[a]*A*c^2*ArcTanh[Sqrt[a + c*x^2]
/Sqrt[a]])/8

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 43.8745, size = 141, normalized size = 0.99 \[ - \frac{15 A \sqrt{a} c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8} + \frac{5 B a c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2} - \frac{5 c \sqrt{a + c x^{2}} \left (- 24 A c x + 32 B a\right )}{64 x} - \frac{5 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (12 A c x + 16 B a\right )}{96 x^{3}} - \frac{\left (2 A - 4 B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{8 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*A*sqrt(a)*c**2*atanh(sqrt(a + c*x**2)/sqrt(a))/8 + 5*B*a*c**(3/2)*atanh(sqrt
(c)*x/sqrt(a + c*x**2))/2 - 5*c*sqrt(a + c*x**2)*(-24*A*c*x + 32*B*a)/(64*x) - 5
*(a + c*x**2)**(3/2)*(12*A*c*x + 16*B*a)/(96*x**3) - (2*A - 4*B*x)*(a + c*x**2)*
*(5/2)/(8*x**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.239291, size = 138, normalized size = 0.97 \[ \frac{1}{24} \left (-\frac{\sqrt{a+c x^2} \left (a^2 (6 A+8 B x)+a c x^2 (27 A+56 B x)-12 c^2 x^4 (2 A+B x)\right )}{x^4}-45 \sqrt{a} A c^2 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+45 \sqrt{a} A c^2 \log (x)+60 a B c^{3/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((Sqrt[a + c*x^2]*(-12*c^2*x^4*(2*A + B*x) + a^2*(6*A + 8*B*x) + a*c*x^2*(27*A
+ 56*B*x)))/x^4) + 45*Sqrt[a]*A*c^2*Log[x] - 45*Sqrt[a]*A*c^2*Log[a + Sqrt[a]*S
qrt[a + c*x^2]] + 60*a*B*c^(3/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/24

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 236, normalized size = 1.7 \[ -{\frac{A}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,Ac}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,A{c}^{2}}{8\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,A{c}^{2}}{8\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,A{c}^{2}}{8}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{15\,A{c}^{2}}{8}\sqrt{c{x}^{2}+a}}-{\frac{B}{3\,a{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{4\,Bc}{3\,{a}^{2}x} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{4\,B{c}^{2}x}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,B{c}^{2}x}{3\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{c}^{2}x}{2}\sqrt{c{x}^{2}+a}}+{\frac{5\,Ba}{2}{c}^{{\frac{3}{2}}}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*A/a/x^4*(c*x^2+a)^(7/2)-3/8*A/a^2*c/x^2*(c*x^2+a)^(7/2)+3/8*A/a^2*c^2*(c*x^
2+a)^(5/2)+5/8*A/a*c^2*(c*x^2+a)^(3/2)-15/8*A*a^(1/2)*c^2*ln((2*a+2*a^(1/2)*(c*x
^2+a)^(1/2))/x)+15/8*A*c^2*(c*x^2+a)^(1/2)-1/3*B/a/x^3*(c*x^2+a)^(7/2)-4/3*B/a^2
*c/x*(c*x^2+a)^(7/2)+4/3*B/a^2*c^2*x*(c*x^2+a)^(5/2)+5/3*B/a*c^2*x*(c*x^2+a)^(3/
2)+5/2*B*c^2*x*(c*x^2+a)^(1/2)+5/2*B*a*c^(3/2)*ln(c^(1/2)*x+(c*x^2+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((c*x^2 + a)^(5/2)*(B*x + A)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.313428, size = 1, normalized size = 0.01 \[ \left [\frac{60 \, B a c^{\frac{3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 45 \, A \sqrt{a} c^{2} x^{4} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (12 \, B c^{2} x^{5} + 24 \, A c^{2} x^{4} - 56 \, B a c x^{3} - 27 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{48 \, x^{4}}, \frac{120 \, B a \sqrt{-c} c x^{4} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 45 \, A \sqrt{a} c^{2} x^{4} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (12 \, B c^{2} x^{5} + 24 \, A c^{2} x^{4} - 56 \, B a c x^{3} - 27 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{48 \, x^{4}}, -\frac{45 \, A \sqrt{-a} c^{2} x^{4} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - 30 \, B a c^{\frac{3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) -{\left (12 \, B c^{2} x^{5} + 24 \, A c^{2} x^{4} - 56 \, B a c x^{3} - 27 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{24 \, x^{4}}, \frac{60 \, B a \sqrt{-c} c x^{4} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 45 \, A \sqrt{-a} c^{2} x^{4} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) +{\left (12 \, B c^{2} x^{5} + 24 \, A c^{2} x^{4} - 56 \, B a c x^{3} - 27 \, A a c x^{2} - 8 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{24 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(60*B*a*c^(3/2)*x^4*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 45*A
*sqrt(a)*c^2*x^4*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(12*B*c
^2*x^5 + 24*A*c^2*x^4 - 56*B*a*c*x^3 - 27*A*a*c*x^2 - 8*B*a^2*x - 6*A*a^2)*sqrt(
c*x^2 + a))/x^4, 1/48*(120*B*a*sqrt(-c)*c*x^4*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-
c))) + 45*A*sqrt(a)*c^2*x^4*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2)
+ 2*(12*B*c^2*x^5 + 24*A*c^2*x^4 - 56*B*a*c*x^3 - 27*A*a*c*x^2 - 8*B*a^2*x - 6*A
*a^2)*sqrt(c*x^2 + a))/x^4, -1/24*(45*A*sqrt(-a)*c^2*x^4*arctan(a/(sqrt(c*x^2 +
a)*sqrt(-a))) - 30*B*a*c^(3/2)*x^4*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x -
a) - (12*B*c^2*x^5 + 24*A*c^2*x^4 - 56*B*a*c*x^3 - 27*A*a*c*x^2 - 8*B*a^2*x - 6*
A*a^2)*sqrt(c*x^2 + a))/x^4, 1/24*(60*B*a*sqrt(-c)*c*x^4*arctan(c*x/(sqrt(c*x^2
+ a)*sqrt(-c))) - 45*A*sqrt(-a)*c^2*x^4*arctan(a/(sqrt(c*x^2 + a)*sqrt(-a))) + (
12*B*c^2*x^5 + 24*A*c^2*x^4 - 56*B*a*c*x^3 - 27*A*a*c*x^2 - 8*B*a^2*x - 6*A*a^2)
*sqrt(c*x^2 + a))/x^4]

_______________________________________________________________________________________

Sympy [A]  time = 34.6218, size = 299, normalized size = 2.09 \[ - \frac{15 A \sqrt{a} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8} - \frac{A a^{3}}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 A a^{2} \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{A a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{x} + \frac{7 A a c^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A c^{\frac{5}{2}} x}{\sqrt{\frac{a}{c x^{2}} + 1}} - \frac{2 B a^{\frac{3}{2}} c}{x \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{B \sqrt{a} c^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} - \frac{2 B \sqrt{a} c^{2} x}{\sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 x^{2}} - \frac{B a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3} + \frac{5 B a c^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*A*sqrt(a)*c**2*asinh(sqrt(a)/(sqrt(c)*x))/8 - A*a**3/(4*sqrt(c)*x**5*sqrt(a/
(c*x**2) + 1)) - 3*A*a**2*sqrt(c)/(8*x**3*sqrt(a/(c*x**2) + 1)) - A*a*c**(3/2)*s
qrt(a/(c*x**2) + 1)/x + 7*A*a*c**(3/2)/(8*x*sqrt(a/(c*x**2) + 1)) + A*c**(5/2)*x
/sqrt(a/(c*x**2) + 1) - 2*B*a**(3/2)*c/(x*sqrt(1 + c*x**2/a)) + B*sqrt(a)*c**2*x
*sqrt(1 + c*x**2/a)/2 - 2*B*sqrt(a)*c**2*x/sqrt(1 + c*x**2/a) - B*a**2*sqrt(c)*s
qrt(a/(c*x**2) + 1)/(3*x**2) - B*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/3 + 5*B*a*c**(3
/2)*asinh(sqrt(c)*x/sqrt(a))/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.285397, size = 427, normalized size = 2.99 \[ \frac{15 \, A a c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a}} - \frac{5}{2} \, B a c^{\frac{3}{2}}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{1}{2} \,{\left (B c^{2} x + 2 \, A c^{2}\right )} \sqrt{c x^{2} + a} + \frac{27 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A a c^{2} + 72 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a^{2} c^{\frac{3}{2}} - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a^{2} c^{2} - 168 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{3} c^{\frac{3}{2}} - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{3} c^{2} + 152 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{4} c^{\frac{3}{2}} + 27 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{4} c^{2} - 56 \, B a^{5} c^{\frac{3}{2}}}{12 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15/4*A*a*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - 5/2*B*a*
c^(3/2)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + 1/2*(B*c^2*x + 2*A*c^2)*sqrt(c*x
^2 + a) + 1/12*(27*(sqrt(c)*x - sqrt(c*x^2 + a))^7*A*a*c^2 + 72*(sqrt(c)*x - sqr
t(c*x^2 + a))^6*B*a^2*c^(3/2) - 3*(sqrt(c)*x - sqrt(c*x^2 + a))^5*A*a^2*c^2 - 16
8*(sqrt(c)*x - sqrt(c*x^2 + a))^4*B*a^3*c^(3/2) - 3*(sqrt(c)*x - sqrt(c*x^2 + a)
)^3*A*a^3*c^2 + 152*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a^4*c^(3/2) + 27*(sqrt(c)*
x - sqrt(c*x^2 + a))*A*a^4*c^2 - 56*B*a^5*c^(3/2))/((sqrt(c)*x - sqrt(c*x^2 + a)
)^2 - a)^4