3.349 \(\int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=140 \[ -\frac{c^2 \sqrt{a+c x^2} (8 A-15 B x)}{8 x}-\frac{\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}-\frac{c \left (a+c x^2\right )^{3/2} (8 A+15 B x)}{24 x^3}+A c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-\frac{15}{8} \sqrt{a} B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right ) \]

[Out]

-(c^2*(8*A - 15*B*x)*Sqrt[a + c*x^2])/(8*x) - (c*(8*A + 15*B*x)*(a + c*x^2)^(3/2
))/(24*x^3) - ((4*A + 5*B*x)*(a + c*x^2)^(5/2))/(20*x^5) + A*c^(5/2)*ArcTanh[(Sq
rt[c]*x)/Sqrt[a + c*x^2]] - (15*Sqrt[a]*B*c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/
8

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Rubi [A]  time = 0.360809, antiderivative size = 140, 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 \[ -\frac{c^2 \sqrt{a+c x^2} (8 A-15 B x)}{8 x}-\frac{\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}-\frac{c \left (a+c x^2\right )^{3/2} (8 A+15 B x)}{24 x^3}+A c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-\frac{15}{8} \sqrt{a} B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(5/2))/x^6,x]

[Out]

-(c^2*(8*A - 15*B*x)*Sqrt[a + c*x^2])/(8*x) - (c*(8*A + 15*B*x)*(a + c*x^2)^(3/2
))/(24*x^3) - ((4*A + 5*B*x)*(a + c*x^2)^(5/2))/(20*x^5) + A*c^(5/2)*ArcTanh[(Sq
rt[c]*x)/Sqrt[a + c*x^2]] - (15*Sqrt[a]*B*c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/
8

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Rubi in Sympy [A]  time = 48.2306, size = 129, normalized size = 0.92 \[ A c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )} - \frac{15 B \sqrt{a} c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8} - \frac{c^{2} \left (32 A - 60 B x\right ) \sqrt{a + c x^{2}}}{32 x} - \frac{c \left (16 A + 30 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{48 x^{3}} - \frac{\left (4 A + 5 B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{20 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(5/2)/x**6,x)

[Out]

A*c**(5/2)*atanh(sqrt(c)*x/sqrt(a + c*x**2)) - 15*B*sqrt(a)*c**2*atanh(sqrt(a +
c*x**2)/sqrt(a))/8 - c**2*(32*A - 60*B*x)*sqrt(a + c*x**2)/(32*x) - c*(16*A + 30
*B*x)*(a + c*x**2)**(3/2)/(48*x**3) - (4*A + 5*B*x)*(a + c*x**2)**(5/2)/(20*x**5
)

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Mathematica [A]  time = 0.242919, size = 140, normalized size = 1. \[ -\frac{\sqrt{a+c x^2} \left (6 a^2 (4 A+5 B x)+a c x^2 (88 A+135 B x)+8 c^2 x^4 (23 A-15 B x)\right )}{120 x^5}+A c^{5/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )-\frac{15}{8} \sqrt{a} B c^2 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+\frac{15}{8} \sqrt{a} B c^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^6,x]

[Out]

-(Sqrt[a + c*x^2]*(8*c^2*x^4*(23*A - 15*B*x) + 6*a^2*(4*A + 5*B*x) + a*c*x^2*(88
*A + 135*B*x)))/(120*x^5) + (15*Sqrt[a]*B*c^2*Log[x])/8 - (15*Sqrt[a]*B*c^2*Log[
a + Sqrt[a]*Sqrt[a + c*x^2]])/8 + A*c^(5/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]]

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Maple [B]  time = 0.017, size = 257, normalized size = 1.8 \[ -{\frac{A}{5\,a{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Ac}{15\,{a}^{2}{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,A{c}^{2}}{15\,{a}^{3}x} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,A{c}^{3}x}{15\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,A{c}^{3}x}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{A{c}^{3}x}{a}\sqrt{c{x}^{2}+a}}+A{c}^{{\frac{5}{2}}}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) -{\frac{B}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,Bc}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,B{c}^{2}}{8\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,B{c}^{2}}{8\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,B{c}^{2}}{8}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{15\,B{c}^{2}}{8}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(5/2)/x^6,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313919, size = 1, normalized size = 0.01 \[ \left [\frac{120 \, A c^{\frac{5}{2}} x^{5} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 225 \, B \sqrt{a} c^{2} x^{5} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x^{5}}, \frac{240 \, A \sqrt{-c} c^{2} x^{5} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 225 \, B \sqrt{a} c^{2} x^{5} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, x^{5}}, -\frac{225 \, B \sqrt{-a} c^{2} x^{5} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - 60 \, A c^{\frac{5}{2}} x^{5} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) -{\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x^{5}}, \frac{120 \, A \sqrt{-c} c^{2} x^{5} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 225 \, B \sqrt{-a} c^{2} x^{5} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) +{\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 32.2566, size = 294, normalized size = 2.1 \[ - \frac{A \sqrt{a} c^{2}}{x \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{11 A a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 x^{2}} - \frac{8 A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15} + A c^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )} - \frac{A c^{3} x}{\sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{15 B \sqrt{a} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8} - \frac{B a^{3}}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B a^{2} \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{x} + \frac{7 B a c^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B c^{\frac{5}{2}} x}{\sqrt{\frac{a}{c x^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(5/2)/x**6,x)

[Out]

-A*sqrt(a)*c**2/(x*sqrt(1 + c*x**2/a)) - A*a**2*sqrt(c)*sqrt(a/(c*x**2) + 1)/(5*
x**4) - 11*A*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/(15*x**2) - 8*A*c**(5/2)*sqrt(a/(c*
x**2) + 1)/15 + A*c**(5/2)*asinh(sqrt(c)*x/sqrt(a)) - A*c**3*x/(sqrt(a)*sqrt(1 +
c*x**2/a)) - 15*B*sqrt(a)*c**2*asinh(sqrt(a)/(sqrt(c)*x))/8 - B*a**3/(4*sqrt(c)
*x**5*sqrt(a/(c*x**2) + 1)) - 3*B*a**2*sqrt(c)/(8*x**3*sqrt(a/(c*x**2) + 1)) - B
*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/x + 7*B*a*c**(3/2)/(8*x*sqrt(a/(c*x**2) + 1)) +
B*c**(5/2)*x/sqrt(a/(c*x**2) + 1)

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GIAC/XCAS [A]  time = 0.286366, size = 447, normalized size = 3.19 \[ \frac{15 \, B a c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a}} - A c^{\frac{5}{2}}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \sqrt{c x^{2} + a} B c^{2} + \frac{135 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B a c^{2} + 360 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} A a c^{\frac{5}{2}} - 150 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a^{2} c^{2} - 720 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} A a^{2} c^{\frac{5}{2}} + 1120 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{3} c^{\frac{5}{2}} + 150 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{4} c^{2} - 560 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{4} c^{\frac{5}{2}} - 135 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{5} c^{2} + 184 \, A a^{5} c^{\frac{5}{2}}}{60 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^6,x, algorithm="giac")

[Out]

15/4*B*a*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - A*c^(5/2
)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + sqrt(c*x^2 + a)*B*c^2 + 1/60*(135*(sqr
t(c)*x - sqrt(c*x^2 + a))^9*B*a*c^2 + 360*(sqrt(c)*x - sqrt(c*x^2 + a))^8*A*a*c^
(5/2) - 150*(sqrt(c)*x - sqrt(c*x^2 + a))^7*B*a^2*c^2 - 720*(sqrt(c)*x - sqrt(c*
x^2 + a))^6*A*a^2*c^(5/2) + 1120*(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*a^3*c^(5/2) +
150*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B*a^4*c^2 - 560*(sqrt(c)*x - sqrt(c*x^2 + a
))^2*A*a^4*c^(5/2) - 135*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^5*c^2 + 184*A*a^5*c^(
5/2))/((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^5