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

Optimal. Leaf size=136 \[ a^{5/2} (-B) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{15}{8} a^2 A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+\frac{1}{8} a \sqrt{a+c x^2} (8 a B+15 A c x)-\frac{\left (a+c x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac{1}{12} \left (a+c x^2\right )^{3/2} (4 a B+15 A c x) \]

[Out]

(a*(8*a*B + 15*A*c*x)*Sqrt[a + c*x^2])/8 + ((4*a*B + 15*A*c*x)*(a + c*x^2)^(3/2)
)/12 - ((5*A - B*x)*(a + c*x^2)^(5/2))/(5*x) + (15*a^2*A*Sqrt[c]*ArcTanh[(Sqrt[c
]*x)/Sqrt[a + c*x^2]])/8 - a^(5/2)*B*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]

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Rubi [A]  time = 0.348791, antiderivative size = 136, 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 \[ a^{5/2} (-B) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{15}{8} a^2 A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+\frac{1}{8} a \sqrt{a+c x^2} (8 a B+15 A c x)-\frac{\left (a+c x^2\right )^{5/2} (5 A-B x)}{5 x}+\frac{1}{12} \left (a+c x^2\right )^{3/2} (4 a B+15 A c x) \]

Antiderivative was successfully verified.

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

[Out]

(a*(8*a*B + 15*A*c*x)*Sqrt[a + c*x^2])/8 + ((4*a*B + 15*A*c*x)*(a + c*x^2)^(3/2)
)/12 - ((5*A - B*x)*(a + c*x^2)^(5/2))/(5*x) + (15*a^2*A*Sqrt[c]*ArcTanh[(Sqrt[c
]*x)/Sqrt[a + c*x^2]])/8 - a^(5/2)*B*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]

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Rubi in Sympy [A]  time = 44.241, size = 124, normalized size = 0.91 \[ \frac{15 A a^{2} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8} - B a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )} + \frac{a \sqrt{a + c x^{2}} \left (30 A c x + 16 B a\right )}{16} + \frac{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (30 A c x + 8 B a\right )}{24} - \frac{\left (5 A - B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{5 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*A*a**2*sqrt(c)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/8 - B*a**(5/2)*atanh(sqrt(a
+ c*x**2)/sqrt(a)) + a*sqrt(a + c*x**2)*(30*A*c*x + 16*B*a)/16 + (a + c*x**2)**(
3/2)*(30*A*c*x + 8*B*a)/24 - (5*A - B*x)*(a + c*x**2)**(5/2)/(5*x)

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Mathematica [A]  time = 0.222108, size = 135, normalized size = 0.99 \[ -a^{5/2} B \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+a^{5/2} B \log (x)+\frac{\sqrt{a+c x^2} \left (-8 a^2 (15 A-23 B x)+a c x^2 (135 A+88 B x)+6 c^2 x^4 (5 A+4 B x)\right )}{120 x}+\frac{15}{8} a^2 A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + c*x^2]*(-8*a^2*(15*A - 23*B*x) + 6*c^2*x^4*(5*A + 4*B*x) + a*c*x^2*(13
5*A + 88*B*x)))/(120*x) + a^(5/2)*B*Log[x] - a^(5/2)*B*Log[a + Sqrt[a]*Sqrt[a +
c*x^2]] + (15*a^2*A*Sqrt[c]*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/8

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Maple [A]  time = 0.013, size = 158, normalized size = 1.2 \[ -{\frac{A}{ax} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Acx}{a} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Acx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{15\,aAcx}{8}\sqrt{c{x}^{2}+a}}+{\frac{15\,A{a}^{2}}{8}\sqrt{c}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) }+{\frac{B}{5} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ba}{3} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-B{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) +B\sqrt{c{x}^{2}+a}{a}^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.314231, size = 1, normalized size = 0.01 \[ \left [\frac{225 \, A a^{2} \sqrt{c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 120 \, B a^{\frac{5}{2}} x \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x}, \frac{225 \, A a^{2} \sqrt{-c} x \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 60 \, B a^{\frac{5}{2}} x \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) +{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x}, -\frac{240 \, B \sqrt{-a} a^{2} x \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - 225 \, A a^{2} \sqrt{c} x \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x}, \frac{225 \, A a^{2} \sqrt{-c} x \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 120 \, B \sqrt{-a} a^{2} x \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) +{\left (24 \, B c^{2} x^{5} + 30 \, A c^{2} x^{4} + 88 \, B a c x^{3} + 135 \, A a c x^{2} + 184 \, B a^{2} x - 120 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/240*(225*A*a^2*sqrt(c)*x*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 12
0*B*a^(5/2)*x*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(24*B*c^2*
x^5 + 30*A*c^2*x^4 + 88*B*a*c*x^3 + 135*A*a*c*x^2 + 184*B*a^2*x - 120*A*a^2)*sqr
t(c*x^2 + a))/x, 1/120*(225*A*a^2*sqrt(-c)*x*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c
))) + 60*B*a^(5/2)*x*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + (24*B
*c^2*x^5 + 30*A*c^2*x^4 + 88*B*a*c*x^3 + 135*A*a*c*x^2 + 184*B*a^2*x - 120*A*a^2
)*sqrt(c*x^2 + a))/x, -1/240*(240*B*sqrt(-a)*a^2*x*arctan(a/(sqrt(c*x^2 + a)*sqr
t(-a))) - 225*A*a^2*sqrt(c)*x*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) -
2*(24*B*c^2*x^5 + 30*A*c^2*x^4 + 88*B*a*c*x^3 + 135*A*a*c*x^2 + 184*B*a^2*x - 12
0*A*a^2)*sqrt(c*x^2 + a))/x, 1/120*(225*A*a^2*sqrt(-c)*x*arctan(c*x/(sqrt(c*x^2
+ a)*sqrt(-c))) - 120*B*sqrt(-a)*a^2*x*arctan(a/(sqrt(c*x^2 + a)*sqrt(-a))) + (2
4*B*c^2*x^5 + 30*A*c^2*x^4 + 88*B*a*c*x^3 + 135*A*a*c*x^2 + 184*B*a^2*x - 120*A*
a^2)*sqrt(c*x^2 + a))/x]

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Sympy [A]  time = 29.7623, size = 318, normalized size = 2.34 \[ - \frac{A a^{\frac{5}{2}}}{x \sqrt{1 + \frac{c x^{2}}{a}}} + A a^{\frac{3}{2}} c x \sqrt{1 + \frac{c x^{2}}{a}} - \frac{7 A a^{\frac{3}{2}} c x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A \sqrt{a} c^{2} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{15 A a^{2} \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8} + \frac{A c^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} - B a^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )} + \frac{B a^{3}}{\sqrt{c} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B a^{2} \sqrt{c} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + 2 B a c \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + B c^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**(5/2)/(x*sqrt(1 + c*x**2/a)) + A*a**(3/2)*c*x*sqrt(1 + c*x**2/a) - 7*A*a**
(3/2)*c*x/(8*sqrt(1 + c*x**2/a)) + 3*A*sqrt(a)*c**2*x**3/(8*sqrt(1 + c*x**2/a))
+ 15*A*a**2*sqrt(c)*asinh(sqrt(c)*x/sqrt(a))/8 + A*c**3*x**5/(4*sqrt(a)*sqrt(1 +
c*x**2/a)) - B*a**(5/2)*asinh(sqrt(a)/(sqrt(c)*x)) + B*a**3/(sqrt(c)*x*sqrt(a/(
c*x**2) + 1)) + B*a**2*sqrt(c)*x/sqrt(a/(c*x**2) + 1) + 2*B*a*c*Piecewise((sqrt(
a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + B*c**2*Piecewise((-2*
a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a +
c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.280555, size = 203, normalized size = 1.49 \[ \frac{2 \, B a^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{15}{8} \, A a^{2} \sqrt{c}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{2 \, A a^{3} \sqrt{c}}{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a} + \frac{1}{120} \,{\left (184 \, B a^{2} +{\left (135 \, A a c + 2 \,{\left (44 \, B a c + 3 \,{\left (4 \, B c^{2} x + 5 \, A c^{2}\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*a^3*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - 15/8*A*a^2*sq
rt(c)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + 2*A*a^3*sqrt(c)/((sqrt(c)*x - sqrt
(c*x^2 + a))^2 - a) + 1/120*(184*B*a^2 + (135*A*a*c + 2*(44*B*a*c + 3*(4*B*c^2*x
+ 5*A*c^2)*x)*x)*x)*sqrt(c*x^2 + a)