3.4.80 \(\int \frac {A+B x}{x^2 (a+c x^2)^{5/2}} \, dx\) [380]

Optimal. Leaf size=104 \[ \frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}} \]

[Out]

1/3*(B*x+A)/a/x/(c*x^2+a)^(3/2)-B*arctanh((c*x^2+a)^(1/2)/a^(1/2))/a^(5/2)+1/3*(3*B*x+4*A)/a^2/x/(c*x^2+a)^(1/
2)-8/3*A*(c*x^2+a)^(1/2)/a^3/x

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Rubi [A]
time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {837, 821, 272, 65, 214} \begin {gather*} -\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}+\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {A+B x}{x^2 \left (a+c x^2\right )^{5/2}} \, dx &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-4 a A c-3 a B c x}{x^2 \left (a+c x^2\right )^{3/2}} \, dx}{3 a^2 c}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}+\frac {\int \frac {8 a^2 A c^2+3 a^2 B c^2 x}{x^2 \sqrt {a+c x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}+\frac {B \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{a^2}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}+\frac {B \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}+\frac {B \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{a^2 c}\\ &=\frac {A+B x}{3 a x \left (a+c x^2\right )^{3/2}}+\frac {4 A+3 B x}{3 a^2 x \sqrt {a+c x^2}}-\frac {8 A \sqrt {a+c x^2}}{3 a^3 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 94, normalized size = 0.90 \begin {gather*} \frac {-8 A c^2 x^4+3 a c x^2 (-4 A+B x)+a^2 (-3 A+4 B x)}{3 a^3 x \left (a+c x^2\right )^{3/2}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.57, size = 122, normalized size = 1.17

method	result	size
default	\(B \left (\frac {1}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {c \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )+A \left (-\frac {1}{a x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 c \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{a}\right )\)	\(122\)
risch	\(-\frac {A \sqrt {c \,x^{2}+a}}{a^{3} x}-\frac {B \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {5}{2}}}-\frac {5 \sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}\, A}{6 a^{3} \left (x -\frac {\sqrt {-a c}}{c}\right )}+\frac {7 \sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}\, B}{12 a^{2} \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}+\frac {\sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}\, A}{12 a^{2} \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}\, B}{12 a^{2} c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {5 \sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}\, A}{6 a^{3} \left (x +\frac {\sqrt {-a c}}{c}\right )}-\frac {7 \sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}\, B}{12 a^{2} \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}-\frac {\sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}\, A}{12 a^{2} \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}\, B}{12 a^{2} c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}}\)	\(563\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^2/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

B*(1/3/a/(c*x^2+a)^(3/2)+1/a*(1/a/(c*x^2+a)^(1/2)-1/a^(3/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)))+A*(-1/a/x/
(c*x^2+a)^(3/2)-4*c/a*(1/3*x/a/(c*x^2+a)^(3/2)+2/3*x/a^2/(c*x^2+a)^(1/2)))

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Maxima [A]
time = 0.28, size = 100, normalized size = 0.96 \begin {gather*} -\frac {8 \, A c x}{3 \, \sqrt {c x^{2} + a} a^{3}} - \frac {4 \, A c x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{a^{\frac {5}{2}}} + \frac {B}{\sqrt {c x^{2} + a} a^{2}} + \frac {B}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^2/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

-8/3*A*c*x/(sqrt(c*x^2 + a)*a^3) - 4/3*A*c*x/((c*x^2 + a)^(3/2)*a^2) - B*arcsinh(a/(sqrt(a*c)*abs(x)))/a^(5/2)
+ B/(sqrt(c*x^2 + a)*a^2) + 1/3*B/((c*x^2 + a)^(3/2)*a) - A/((c*x^2 + a)^(3/2)*a*x)

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Fricas [A]
time = 3.51, size = 264, normalized size = 2.54 \begin {gather*} \left [\frac {3 \, {\left (B c^{2} x^{5} + 2 \, B a c x^{3} + B a^{2} x\right )} \sqrt {a} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (8 \, A c^{2} x^{4} - 3 \, B a c x^{3} + 12 \, A a c x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )}}, \frac {3 \, {\left (B c^{2} x^{5} + 2 \, B a c x^{3} + B a^{2} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (8 \, A c^{2} x^{4} - 3 \, B a c x^{3} + 12 \, A a c x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*(B*c^2*x^5 + 2*B*a*c*x^3 + B*a^2*x)*sqrt(a)*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(8
*A*c^2*x^4 - 3*B*a*c*x^3 + 12*A*a*c*x^2 - 4*B*a^2*x + 3*A*a^2)*sqrt(c*x^2 + a))/(a^3*c^2*x^5 + 2*a^4*c*x^3 + a
^5*x), 1/3*(3*(B*c^2*x^5 + 2*B*a*c*x^3 + B*a^2*x)*sqrt(-a)*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (8*A*c^2*x^4 - 3
*B*a*c*x^3 + 12*A*a*c*x^2 - 4*B*a^2*x + 3*A*a^2)*sqrt(c*x^2 + a))/(a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (88) = 176\).
time = 9.57, size = 910, normalized size = 8.75 \begin {gather*} A \left (- \frac {3 a^{2} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} + 6 a^{4} c^{5} x^{2} + 3 a^{3} c^{6} x^{4}} - \frac {12 a c^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} + 6 a^{4} c^{5} x^{2} + 3 a^{3} c^{6} x^{4}} - \frac {8 c^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{3 a^{5} c^{4} + 6 a^{4} c^{5} x^{2} + 3 a^{3} c^{6} x^{4}}\right ) + B \left (\frac {8 a^{7} \sqrt {1 + \frac {c x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {3 a^{7} \log {\left (\frac {c x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} - \frac {6 a^{7} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {14 a^{6} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {9 a^{6} c x^{2} \log {\left (\frac {c x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} - \frac {18 a^{6} c x^{2} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {6 a^{5} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {9 a^{5} c^{2} x^{4} \log {\left (\frac {c x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} - \frac {18 a^{5} c^{2} x^{4} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} + \frac {3 a^{4} c^{3} x^{6} \log {\left (\frac {c x^{2}}{a} \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}} - \frac {6 a^{4} c^{3} x^{6} \log {\left (\sqrt {1 + \frac {c x^{2}}{a}} + 1 \right )}}{6 a^{\frac {19}{2}} + 18 a^{\frac {17}{2}} c x^{2} + 18 a^{\frac {15}{2}} c^{2} x^{4} + 6 a^{\frac {13}{2}} c^{3} x^{6}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(-3*a**2*c**(9/2)*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4 + 6*a**4*c**5*x**2 + 3*a**3*c**6*x**4) - 12*a*c**(11/2)*
x**2*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4 + 6*a**4*c**5*x**2 + 3*a**3*c**6*x**4) - 8*c**(13/2)*x**4*sqrt(a/(c*x**
2) + 1)/(3*a**5*c**4 + 6*a**4*c**5*x**2 + 3*a**3*c**6*x**4)) + B*(8*a**7*sqrt(1 + c*x**2/a)/(6*a**(19/2) + 18*
a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) + 3*a**7*log(c*x**2/a)/(6*a**(19/2) + 18*a*
*(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) - 6*a**7*log(sqrt(1 + c*x**2/a) + 1)/(6*a**(1
9/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) + 14*a**6*c*x**2*sqrt(1 + c*x**2/
a)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) + 9*a**6*c*x**2*log(c*
x**2/a)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) - 18*a**6*c*x**2*
log(sqrt(1 + c*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**
6) + 6*a**5*c**2*x**4*sqrt(1 + c*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(1
3/2)*c**3*x**6) + 9*a**5*c**2*x**4*log(c*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 +
6*a**(13/2)*c**3*x**6) - 18*a**5*c**2*x**4*log(sqrt(1 + c*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 1
8*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) + 3*a**4*c**3*x**6*log(c*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*c*
x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) - 6*a**4*c**3*x**6*log(sqrt(1 + c*x**2/a) + 1)/(6*a**(1
9/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6))

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Giac [A]
time = 0.66, size = 119, normalized size = 1.14 \begin {gather*} -\frac {{\left ({\left (\frac {5 \, A c^{2} x}{a^{3}} - \frac {3 \, B c}{a^{2}}\right )} x + \frac {6 \, A c}{a^{2}}\right )} x - \frac {4 \, B}{a}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} + \frac {2 \, B \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, A \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(((5*A*c^2*x/a^3 - 3*B*c/a^2)*x + 6*A*c/a^2)*x - 4*B/a)/(c*x^2 + a)^(3/2) + 2*B*arctan(-(sqrt(c)*x - sqrt
(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) + 2*A*sqrt(c)/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)*a^2)

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Mupad [B]
time = 1.69, size = 96, normalized size = 0.92 \begin {gather*} \frac {\frac {B}{3\,a}+\frac {B\,\left (c\,x^2+a\right )}{a^2}}{{\left (c\,x^2+a\right )}^{3/2}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {A\,a^2-8\,A\,{\left (c\,x^2+a\right )}^2+4\,A\,a\,\left (c\,x^2+a\right )}{3\,a^3\,x\,{\left (c\,x^2+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B/(3*a) + (B*(a + c*x^2))/a^2)/(a + c*x^2)^(3/2) - (B*atanh((a + c*x^2)^(1/2)/a^(1/2)))/a^(5/2) + (A*a^2 - 8*
A*(a + c*x^2)^2 + 4*A*a*(a + c*x^2))/(3*a^3*x*(a + c*x^2)^(3/2))

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