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

Optimal. Leaf size=132 \[ -a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{5 a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{1}{16} a^2 \sqrt{a+c x^2} (16 A+5 B x)+\frac{1}{24} a \left (a+c x^2\right )^{3/2} (8 A+5 B x)+\frac{1}{30} \left (a+c x^2\right )^{5/2} (6 A+5 B x) \]

[Out]

(a^2*(16*A + 5*B*x)*Sqrt[a + c*x^2])/16 + (a*(8*A + 5*B*x)*(a + c*x^2)^(3/2))/24
+ ((6*A + 5*B*x)*(a + c*x^2)^(5/2))/30 + (5*a^3*B*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(16*Sqrt[c]) - a^(5/2)*A*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]

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Rubi [A]  time = 0.352998, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{5 a^3 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{1}{16} a^2 \sqrt{a+c x^2} (16 A+5 B x)+\frac{1}{24} a \left (a+c x^2\right )^{3/2} (8 A+5 B x)+\frac{1}{30} \left (a+c x^2\right )^{5/2} (6 A+5 B x) \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(16*A + 5*B*x)*Sqrt[a + c*x^2])/16 + (a*(8*A + 5*B*x)*(a + c*x^2)^(3/2))/24
+ ((6*A + 5*B*x)*(a + c*x^2)^(5/2))/30 + (5*a^3*B*ArcTanh[(Sqrt[c]*x)/Sqrt[a +
c*x^2]])/(16*Sqrt[c]) - a^(5/2)*A*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]]

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Rubi in Sympy [A]  time = 47.5866, size = 121, normalized size = 0.92 \[ - A a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )} + \frac{5 B a^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 \sqrt{c}} + \frac{a^{2} \left (48 A + 15 B x\right ) \sqrt{a + c x^{2}}}{48} + \frac{a \left (24 A + 15 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{72} + \frac{\left (6 A + 5 B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**(5/2)*atanh(sqrt(a + c*x**2)/sqrt(a)) + 5*B*a**3*atanh(sqrt(c)*x/sqrt(a +
c*x**2))/(16*sqrt(c)) + a**2*(48*A + 15*B*x)*sqrt(a + c*x**2)/48 + a*(24*A + 15*
B*x)*(a + c*x**2)**(3/2)/72 + (6*A + 5*B*x)*(a + c*x**2)**(5/2)/30

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Mathematica [A]  time = 0.255272, size = 132, normalized size = 1. \[ -a^{5/2} A \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+a^{5/2} A \log (x)+\frac{5 a^3 B \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{16 \sqrt{c}}+\frac{1}{240} \sqrt{a+c x^2} \left (a^2 (368 A+165 B x)+2 a c x^2 (88 A+65 B x)+8 c^2 x^4 (6 A+5 B x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + c*x^2]*(8*c^2*x^4*(6*A + 5*B*x) + 2*a*c*x^2*(88*A + 65*B*x) + a^2*(368
*A + 165*B*x)))/240 + a^(5/2)*A*Log[x] - a^(5/2)*A*Log[a + Sqrt[a]*Sqrt[a + c*x^
2]] + (5*a^3*B*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(16*Sqrt[c])

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Maple [A]  time = 0.011, size = 138, normalized size = 1.1 \[{\frac{Bx}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,aBx}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}Bx}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,B{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{A}{5} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{aA}{3} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-A{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) +A\sqrt{c{x}^{2}+a}{a}^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*x*(c*x^2+a)^(5/2)+5/24*B*a*x*(c*x^2+a)^(3/2)+5/16*B*a^2*x*(c*x^2+a)^(1/2)+
5/16*B*a^3/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/5*A*(c*x^2+a)^(5/2)+1/3*A*a*(
c*x^2+a)^(3/2)-A*a^(5/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)+A*(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.317056, size = 1, normalized size = 0.01 \[ \left [\frac{75 \, B a^{3} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 240 \, A a^{\frac{5}{2}} \sqrt{c} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (40 \, B c^{2} x^{5} + 48 \, A c^{2} x^{4} + 130 \, B a c x^{3} + 176 \, A a c x^{2} + 165 \, B a^{2} x + 368 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{480 \, \sqrt{c}}, \frac{75 \, B a^{3} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) + 120 \, A a^{\frac{5}{2}} \sqrt{-c} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) +{\left (40 \, B c^{2} x^{5} + 48 \, A c^{2} x^{4} + 130 \, B a c x^{3} + 176 \, A a c x^{2} + 165 \, B a^{2} x + 368 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{240 \, \sqrt{-c}}, -\frac{480 \, A \sqrt{-a} a^{2} \sqrt{c} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - 75 \, B a^{3} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) - 2 \,{\left (40 \, B c^{2} x^{5} + 48 \, A c^{2} x^{4} + 130 \, B a c x^{3} + 176 \, A a c x^{2} + 165 \, B a^{2} x + 368 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{480 \, \sqrt{c}}, \frac{75 \, B a^{3} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 240 \, A \sqrt{-a} a^{2} \sqrt{-c} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) +{\left (40 \, B c^{2} x^{5} + 48 \, A c^{2} x^{4} + 130 \, B a c x^{3} + 176 \, A a c x^{2} + 165 \, B a^{2} x + 368 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{240 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/480*(75*B*a^3*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) + 240*A*a^(
5/2)*sqrt(c)*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(40*B*c^2*x
^5 + 48*A*c^2*x^4 + 130*B*a*c*x^3 + 176*A*a*c*x^2 + 165*B*a^2*x + 368*A*a^2)*sqr
t(c*x^2 + a)*sqrt(c))/sqrt(c), 1/240*(75*B*a^3*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)
) + 120*A*a^(5/2)*sqrt(-c)*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) +
(40*B*c^2*x^5 + 48*A*c^2*x^4 + 130*B*a*c*x^3 + 176*A*a*c*x^2 + 165*B*a^2*x + 36
8*A*a^2)*sqrt(c*x^2 + a)*sqrt(-c))/sqrt(-c), -1/480*(480*A*sqrt(-a)*a^2*sqrt(c)*
arctan(a/(sqrt(c*x^2 + a)*sqrt(-a))) - 75*B*a^3*log(-2*sqrt(c*x^2 + a)*c*x - (2*
c*x^2 + a)*sqrt(c)) - 2*(40*B*c^2*x^5 + 48*A*c^2*x^4 + 130*B*a*c*x^3 + 176*A*a*c
*x^2 + 165*B*a^2*x + 368*A*a^2)*sqrt(c*x^2 + a)*sqrt(c))/sqrt(c), 1/240*(75*B*a^
3*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - 240*A*sqrt(-a)*a^2*sqrt(-c)*arctan(a/(sqr
t(c*x^2 + a)*sqrt(-a))) + (40*B*c^2*x^5 + 48*A*c^2*x^4 + 130*B*a*c*x^3 + 176*A*a
*c*x^2 + 165*B*a^2*x + 368*A*a^2)*sqrt(c*x^2 + a)*sqrt(-c))/sqrt(-c)]

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Sympy [A]  time = 42.5093, size = 323, normalized size = 2.45 \[ - A a^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )} + \frac{A a^{3}}{\sqrt{c} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A a^{2} \sqrt{c} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + 2 A a c \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + A c^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \frac{B a^{\frac{5}{2}} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 B a^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 B a^{\frac{3}{2}} c x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 B \sqrt{a} c^{2} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 \sqrt{c}} + \frac{B c^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**(5/2)*asinh(sqrt(a)/(sqrt(c)*x)) + A*a**3/(sqrt(c)*x*sqrt(a/(c*x**2) + 1))
+ A*a**2*sqrt(c)*x/sqrt(a/(c*x**2) + 1) + 2*A*a*c*Piecewise((sqrt(a)*x**2/2, Eq
(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + A*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, N
e(c, 0)), (sqrt(a)*x**4/4, True)) + B*a**(5/2)*x*sqrt(1 + c*x**2/a)/2 + 3*B*a**(
5/2)*x/(16*sqrt(1 + c*x**2/a)) + 35*B*a**(3/2)*c*x**3/(48*sqrt(1 + c*x**2/a)) +
17*B*sqrt(a)*c**2*x**5/(24*sqrt(1 + c*x**2/a)) + 5*B*a**3*asinh(sqrt(c)*x/sqrt(a
))/(16*sqrt(c)) + B*c**3*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.276999, size = 169, normalized size = 1.28 \[ \frac{2 \, A a^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \, B a^{3}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, \sqrt{c}} + \frac{1}{240} \,{\left (368 \, A a^{2} +{\left (165 \, B a^{2} + 2 \,{\left (88 \, A a c +{\left (65 \, B a c + 4 \,{\left (5 \, B c^{2} x + 6 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a^3*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - 5/16*B*a^3*ln
(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/sqrt(c) + 1/240*(368*A*a^2 + (165*B*a^2 + 2*
(88*A*a*c + (65*B*a*c + 4*(5*B*c^2*x + 6*A*c^2)*x)*x)*x)*x)*sqrt(c*x^2 + a)