3.347 \(\int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=137 \[ -\frac{5}{2} a^{3/2} B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )-\frac{5 \left (a+c x^2\right )^{3/2} (a B-A c x)}{6 x^2}-\frac{5 a c \sqrt{a+c x^2} (A-B x)}{2 x}-\frac{\left (a+c x^2\right )^{5/2} (A-B x)}{3 x^3}+\frac{5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.339142, antiderivative size = 137, 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 \[ -\frac{5}{2} a^{3/2} B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )-\frac{5 \left (a+c x^2\right )^{3/2} (a B-A c x)}{6 x^2}-\frac{5 a c \sqrt{a+c x^2} (A-B x)}{2 x}-\frac{\left (a+c x^2\right )^{5/2} (A-B x)}{3 x^3}+\frac{5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(5/2))/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 39.6784, size = 138, normalized size = 1.01 \[ \frac{5 A a c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2} - \frac{5 B a^{\frac{3}{2}} c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2} - \frac{5 a c \left (24 A - 24 B x\right ) \sqrt{a + c x^{2}}}{48 x} - \frac{5 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (- 12 A c x + 12 B a\right )}{72 x^{2}} - \frac{\left (3 A - 3 B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{9 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(5/2)/x**4,x)

[Out]

5*A*a*c**(3/2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/2 - 5*B*a**(3/2)*c*atanh(sqrt(a
+ c*x**2)/sqrt(a))/2 - 5*a*c*(24*A - 24*B*x)*sqrt(a + c*x**2)/(48*x) - 5*(a + c
*x**2)**(3/2)*(-12*A*c*x + 12*B*a)/(72*x**2) - (3*A - 3*B*x)*(a + c*x**2)**(5/2)
/(9*x**3)

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Mathematica [A]  time = 0.33783, size = 137, normalized size = 1. \[ \frac{1}{2} \left (-5 a^{3/2} B c \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+5 a^{3/2} B c \log (x)+\frac{\sqrt{a+c x^2} \left (-a^2 (2 A+3 B x)+14 a c x^2 (B x-A)+c^2 x^4 (3 A+2 B x)\right )}{3 x^3}+5 a A c^{3/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^4,x]

[Out]

((Sqrt[a + c*x^2]*(14*a*c*x^2*(-A + B*x) + c^2*x^4*(3*A + 2*B*x) - a^2*(2*A + 3*
B*x)))/(3*x^3) + 5*a^(3/2)*B*c*Log[x] - 5*a^(3/2)*B*c*Log[a + Sqrt[a]*Sqrt[a + c
*x^2]] + 5*a*A*c^(3/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/2

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Maple [A]  time = 0.016, size = 207, normalized size = 1.5 \[ -{\frac{A}{3\,a{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{4\,Ac}{3\,{a}^{2}x} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{4\,Ax{c}^{2}}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Ax{c}^{2}}{3\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ax{c}^{2}}{2}\sqrt{c{x}^{2}+a}}+{\frac{5\,aA}{2}{c}^{{\frac{3}{2}}}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) }-{\frac{B}{2\,a{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bc}{2\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Bc}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Bc}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{5\,aBc}{2}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(5/2)/x^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313557, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A a c^{\frac{3}{2}} x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 15 \, B a^{\frac{3}{2}} c x^{3} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{12 \, x^{3}}, \frac{30 \, A a \sqrt{-c} c x^{3} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 15 \, B a^{\frac{3}{2}} c x^{3} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{12 \, x^{3}}, -\frac{30 \, B \sqrt{-a} a c x^{3} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) - 15 \, A a c^{\frac{3}{2}} x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{12 \, x^{3}}, \frac{15 \, A a \sqrt{-c} c x^{3} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 15 \, B \sqrt{-a} a c x^{3} \arctan \left (\frac{a}{\sqrt{c x^{2} + a} \sqrt{-a}}\right ) +{\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{6 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

[1/12*(15*A*a*c^(3/2)*x^3*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 15*B
*a^(3/2)*c*x^3*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(2*B*c^2*
x^5 + 3*A*c^2*x^4 + 14*B*a*c*x^3 - 14*A*a*c*x^2 - 3*B*a^2*x - 2*A*a^2)*sqrt(c*x^
2 + a))/x^3, 1/12*(30*A*a*sqrt(-c)*c*x^3*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c)))
+ 15*B*a^(3/2)*c*x^3*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(2*
B*c^2*x^5 + 3*A*c^2*x^4 + 14*B*a*c*x^3 - 14*A*a*c*x^2 - 3*B*a^2*x - 2*A*a^2)*sqr
t(c*x^2 + a))/x^3, -1/12*(30*B*sqrt(-a)*a*c*x^3*arctan(a/(sqrt(c*x^2 + a)*sqrt(-
a))) - 15*A*a*c^(3/2)*x^3*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(2
*B*c^2*x^5 + 3*A*c^2*x^4 + 14*B*a*c*x^3 - 14*A*a*c*x^2 - 3*B*a^2*x - 2*A*a^2)*sq
rt(c*x^2 + a))/x^3, 1/6*(15*A*a*sqrt(-c)*c*x^3*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(
-c))) - 15*B*sqrt(-a)*a*c*x^3*arctan(a/(sqrt(c*x^2 + a)*sqrt(-a))) + (2*B*c^2*x^
5 + 3*A*c^2*x^4 + 14*B*a*c*x^3 - 14*A*a*c*x^2 - 3*B*a^2*x - 2*A*a^2)*sqrt(c*x^2
+ a))/x^3]

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Sympy [A]  time = 24.6617, size = 277, normalized size = 2.02 \[ - \frac{2 A a^{\frac{3}{2}} c}{x \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{A \sqrt{a} c^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} - \frac{2 A \sqrt{a} c^{2} x}{\sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 x^{2}} - \frac{A a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3} + \frac{5 A a c^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2} - \frac{5 B a^{\frac{3}{2}} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2} - \frac{B a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} + \frac{2 B a^{2} \sqrt{c}}{x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{2 B a c^{\frac{3}{2}} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + B c^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(5/2)/x**4,x)

[Out]

-2*A*a**(3/2)*c/(x*sqrt(1 + c*x**2/a)) + A*sqrt(a)*c**2*x*sqrt(1 + c*x**2/a)/2 -
2*A*sqrt(a)*c**2*x/sqrt(1 + c*x**2/a) - A*a**2*sqrt(c)*sqrt(a/(c*x**2) + 1)/(3*
x**2) - A*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/3 + 5*A*a*c**(3/2)*asinh(sqrt(c)*x/sqr
t(a))/2 - 5*B*a**(3/2)*c*asinh(sqrt(a)/(sqrt(c)*x))/2 - B*a**2*sqrt(c)*sqrt(a/(c
*x**2) + 1)/(2*x) + 2*B*a**2*sqrt(c)/(x*sqrt(a/(c*x**2) + 1)) + 2*B*a*c**(3/2)*x
/sqrt(a/(c*x**2) + 1) + B*c**2*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**
2)**(3/2)/(3*c), True))

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GIAC/XCAS [A]  time = 0.282542, size = 323, normalized size = 2.36 \[ \frac{5 \, B a^{2} c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5}{2} \, A a c^{\frac{3}{2}}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{1}{6} \,{\left (14 \, B a c +{\left (2 \, B c^{2} x + 3 \, A c^{2}\right )} x\right )} \sqrt{c x^{2} + a} + \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B a^{2} c + 18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{2} c^{\frac{3}{2}} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{3} c^{\frac{3}{2}} - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{4} c + 14 \, A a^{4} c^{\frac{3}{2}}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^4,x, algorithm="giac")

[Out]

5*B*a^2*c*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - 5/2*A*a*c^(
3/2)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + 1/6*(14*B*a*c + (2*B*c^2*x + 3*A*c^
2)*x)*sqrt(c*x^2 + a) + 1/3*(3*(sqrt(c)*x - sqrt(c*x^2 + a))^5*B*a^2*c + 18*(sqr
t(c)*x - sqrt(c*x^2 + a))^4*A*a^2*c^(3/2) - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^2*A
*a^3*c^(3/2) - 3*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^4*c + 14*A*a^4*c^(3/2))/((sqr
t(c)*x - sqrt(c*x^2 + a))^2 - a)^3