3.346 \(\int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=141 \[ -\frac{5}{2} a^{3/2} A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{15}{8} a^2 B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-\frac{\left (a+c x^2\right )^{5/2} (2 A-B x)}{4 x^2}-\frac{5 \left (a+c x^2\right )^{3/2} (3 a B-2 A c x)}{12 x}+\frac{5}{8} a c \sqrt{a+c x^2} (4 A+3 B x) \]

[Out]

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

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Rubi [A]  time = 0.34794, antiderivative size = 141, 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 \[ -\frac{5}{2} a^{3/2} A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{15}{8} a^2 B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-\frac{\left (a+c x^2\right )^{5/2} (2 A-B x)}{4 x^2}-\frac{5 \left (a+c x^2\right )^{3/2} (3 a B-2 A c x)}{12 x}+\frac{5}{8} a c \sqrt{a+c x^2} (4 A+3 B x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.6019, size = 136, normalized size = 0.96 \[ - \frac{5 A a^{\frac{3}{2}} c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2} + \frac{15 B a^{2} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8} + \frac{5 a c \left (32 A + 24 B x\right ) \sqrt{a + c x^{2}}}{64} - \frac{5 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (- 8 A c x + 12 B a\right )}{48 x} - \frac{\left (4 A - 2 B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{8 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*A*a**(3/2)*c*atanh(sqrt(a + c*x**2)/sqrt(a))/2 + 15*B*a**2*sqrt(c)*atanh(sqrt
(c)*x/sqrt(a + c*x**2))/8 + 5*a*c*(32*A + 24*B*x)*sqrt(a + c*x**2)/64 - 5*(a + c
*x**2)**(3/2)*(-8*A*c*x + 12*B*a)/(48*x) - (4*A - 2*B*x)*(a + c*x**2)**(5/2)/(8*
x**2)

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Mathematica [A]  time = 0.25849, size = 135, normalized size = 0.96 \[ \frac{1}{24} \left (-60 a^{3/2} A c \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+60 a^{3/2} A c \log (x)+\frac{\sqrt{a+c x^2} \left (-12 a^2 (A+2 B x)+a c x^2 (56 A+27 B x)+2 c^2 x^4 (4 A+3 B x)\right )}{x^2}+45 a^2 B \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 181, normalized size = 1.3 \[ -{\frac{A}{2\,a{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Ac}{2\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Ac}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Ac}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{5\,aAc}{2}\sqrt{c{x}^{2}+a}}-{\frac{B}{ax} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bcx}{a} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Bcx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{15\,aBcx}{8}\sqrt{c{x}^{2}+a}}+{\frac{15\,{a}^{2}B}{8}\sqrt{c}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*A/a/x^2*(c*x^2+a)^(7/2)+1/2*A/a*c*(c*x^2+a)^(5/2)+5/6*A*c*(c*x^2+a)^(3/2)-5
/2*A*a^(3/2)*c*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)+5/2*A*a*c*(c*x^2+a)^(1/2)-B
/a/x*(c*x^2+a)^(7/2)+B/a*c*x*(c*x^2+a)^(5/2)+5/4*B*c*x*(c*x^2+a)^(3/2)+15/8*B*a*
c*x*(c*x^2+a)^(1/2)+15/8*B*a^2*c^(1/2)*ln(c^(1/2)*x+(c*x^2+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((c*x^2 + a)^(5/2)*(B*x + A)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308334, size = 1, normalized size = 0.01 \[ \left [\frac{45 \, B a^{2} \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 60 \, A a^{\frac{3}{2}} c x^{2} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (6 \, B c^{2} x^{5} + 8 \, A c^{2} x^{4} + 27 \, B a c x^{3} + 56 \, A a c x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{48 \, x^{2}}, \frac{45 \, B a^{2} \sqrt{-c} x^{2} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 30 \, A a^{\frac{3}{2}} c x^{2} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) +{\left (6 \, B c^{2} x^{5} + 8 \, A c^{2} x^{4} + 27 \, B a c x^{3} + 56 \, A a c x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{24 \, x^{2}}, -\frac{120 \, A \sqrt{-a} a c x^{2} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - 45 \, B a^{2} \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (6 \, B c^{2} x^{5} + 8 \, A c^{2} x^{4} + 27 \, B a c x^{3} + 56 \, A a c x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{48 \, x^{2}}, \frac{45 \, B a^{2} \sqrt{-c} x^{2} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 60 \, A \sqrt{-a} a c x^{2} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) +{\left (6 \, B c^{2} x^{5} + 8 \, A c^{2} x^{4} + 27 \, B a c x^{3} + 56 \, A a c x^{2} - 24 \, B a^{2} x - 12 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{24 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 33.2153, size = 279, normalized size = 1.98 \[ - \frac{5 A a^{\frac{3}{2}} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2} - \frac{A a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} + \frac{2 A a^{2} \sqrt{c}}{x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{2 A a c^{\frac{3}{2}} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + A c^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) - \frac{B a^{\frac{5}{2}}}{x \sqrt{1 + \frac{c x^{2}}{a}}} + B a^{\frac{3}{2}} c x \sqrt{1 + \frac{c x^{2}}{a}} - \frac{7 B a^{\frac{3}{2}} c x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 B \sqrt{a} c^{2} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{15 B a^{2} \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8} + \frac{B c^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282663, size = 296, normalized size = 2.1 \[ \frac{5 \, A a^{2} c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{15}{8} \, B a^{2} \sqrt{c}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{1}{24} \,{\left (56 \, A a c +{\left (27 \, B a c + 2 \,{\left (3 \, B c^{2} x + 4 \, A c^{2}\right )} x\right )} x\right )} \sqrt{c x^{2} + a} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{2} c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{3} \sqrt{c} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{3} c - 2 \, B a^{4} \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*A*a^2*c*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - 15/8*B*a^2*
sqrt(c)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + 1/24*(56*A*a*c + (27*B*a*c + 2*(
3*B*c^2*x + 4*A*c^2)*x)*x)*sqrt(c*x^2 + a) + ((sqrt(c)*x - sqrt(c*x^2 + a))^3*A*
a^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a^3*sqrt(c) + (sqrt(c)*x - sqrt(c*x^
2 + a))*A*a^3*c - 2*B*a^4*sqrt(c))/((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^2