3.333 \(\int \frac{(A+B x) (a+c x^2)^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=111 \[ -\frac{\left (a+c x^2\right )^{3/2} (A-B x)}{2 x^2}-\frac{3 \sqrt{a+c x^2} (a B-A c x)}{2 x}-\frac{3}{2} \sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{3}{2} a B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.078875, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {813, 844, 217, 206, 266, 63, 208} \[ -\frac{\left (a+c x^2\right )^{3/2} (A-B x)}{2 x^2}-\frac{3 \sqrt{a+c x^2} (a B-A c x)}{2 x}-\frac{3}{2} \sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{3}{2} a B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 813

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

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 266

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 63

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 208

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

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{x^3} \, dx &=-\frac{(A-B x) \left (a+c x^2\right )^{3/2}}{2 x^2}-\frac{3}{8} \int \frac{(-4 a B-4 A c x) \sqrt{a+c x^2}}{x^2} \, dx\\ &=-\frac{3 (a B-A c x) \sqrt{a+c x^2}}{2 x}-\frac{(A-B x) \left (a+c x^2\right )^{3/2}}{2 x^2}+\frac{3}{16} \int \frac{8 a A c+8 a B c x}{x \sqrt{a+c x^2}} \, dx\\ &=-\frac{3 (a B-A c x) \sqrt{a+c x^2}}{2 x}-\frac{(A-B x) \left (a+c x^2\right )^{3/2}}{2 x^2}+\frac{1}{2} (3 a A c) \int \frac{1}{x \sqrt{a+c x^2}} \, dx+\frac{1}{2} (3 a B c) \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=-\frac{3 (a B-A c x) \sqrt{a+c x^2}}{2 x}-\frac{(A-B x) \left (a+c x^2\right )^{3/2}}{2 x^2}+\frac{1}{4} (3 a A c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )+\frac{1}{2} (3 a B c) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=-\frac{3 (a B-A c x) \sqrt{a+c x^2}}{2 x}-\frac{(A-B x) \left (a+c x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} a B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+\frac{1}{2} (3 a A) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )\\ &=-\frac{3 (a B-A c x) \sqrt{a+c x^2}}{2 x}-\frac{(A-B x) \left (a+c x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} a B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-\frac{3}{2} \sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0371102, size = 90, normalized size = 0.81 \[ \frac{A c \left (a+c x^2\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{c x^2}{a}+1\right )}{5 a^2}-\frac{a B \sqrt{a+c x^2} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x^2}{a}\right )}{x \sqrt{\frac{c x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*B*Sqrt[a + c*x^2]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x^2)/a)])/(x*Sqrt[1 + (c*x^2)/a])) + (A*c*(a +
c*x^2)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (c*x^2)/a])/(5*a^2)

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Maple [A]  time = 0.009, size = 150, normalized size = 1.4 \begin{align*} -{\frac{A}{2\,a{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ac}{2\,a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,Ac}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{3\,Ac}{2}\sqrt{c{x}^{2}+a}}-{\frac{B}{ax} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bcx}{a} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Bcx}{2}\sqrt{c{x}^{2}+a}}+{\frac{3\,aB}{2}\sqrt{c}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(3/2)/x^3,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(3/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.06997, size = 1048, normalized size = 9.44 \begin{align*} \left [\frac{3 \, B a \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 3 \, A \sqrt{a} c x^{2} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (B c x^{3} + 2 \, A c x^{2} - 2 \, B a x - A a\right )} \sqrt{c x^{2} + a}}{4 \, x^{2}}, -\frac{6 \, B a \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 3 \, A \sqrt{a} c x^{2} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (B c x^{3} + 2 \, A c x^{2} - 2 \, B a x - A a\right )} \sqrt{c x^{2} + a}}{4 \, x^{2}}, \frac{6 \, A \sqrt{-a} c x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) + 3 \, B a \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (B c x^{3} + 2 \, A c x^{2} - 2 \, B a x - A a\right )} \sqrt{c x^{2} + a}}{4 \, x^{2}}, -\frac{3 \, B a \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 3 \, A \sqrt{-a} c x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (B c x^{3} + 2 \, A c x^{2} - 2 \, B a x - A a\right )} \sqrt{c x^{2} + a}}{2 \, x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(3*B*a*sqrt(c)*x^2*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 3*A*sqrt(a)*c*x^2*log(-(c*x^2 - 2*sq
rt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(B*c*x^3 + 2*A*c*x^2 - 2*B*a*x - A*a)*sqrt(c*x^2 + a))/x^2, -1/4*(6*B*a*
sqrt(-c)*x^2*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - 3*A*sqrt(a)*c*x^2*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) +
2*a)/x^2) - 2*(B*c*x^3 + 2*A*c*x^2 - 2*B*a*x - A*a)*sqrt(c*x^2 + a))/x^2, 1/4*(6*A*sqrt(-a)*c*x^2*arctan(sqrt(
-a)/sqrt(c*x^2 + a)) + 3*B*a*sqrt(c)*x^2*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(B*c*x^3 + 2*A*c*
x^2 - 2*B*a*x - A*a)*sqrt(c*x^2 + a))/x^2, -1/2*(3*B*a*sqrt(-c)*x^2*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - 3*A*s
qrt(-a)*c*x^2*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (B*c*x^3 + 2*A*c*x^2 - 2*B*a*x - A*a)*sqrt(c*x^2 + a))/x^2]

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Sympy [A]  time = 8.3721, size = 182, normalized size = 1.64 \begin{align*} - \frac{3 A \sqrt{a} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2} - \frac{A a \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} + \frac{A a \sqrt{c}}{x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A c^{\frac{3}{2}} x}{\sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B a^{\frac{3}{2}}}{x \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{B \sqrt{a} c x \sqrt{1 + \frac{c x^{2}}{a}}}{2} - \frac{B \sqrt{a} c x}{\sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 B a \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**(3/2)/x**3,x)

[Out]

-3*A*sqrt(a)*c*asinh(sqrt(a)/(sqrt(c)*x))/2 - A*a*sqrt(c)*sqrt(a/(c*x**2) + 1)/(2*x) + A*a*sqrt(c)/(x*sqrt(a/(
c*x**2) + 1)) + A*c**(3/2)*x/sqrt(a/(c*x**2) + 1) - B*a**(3/2)/(x*sqrt(1 + c*x**2/a)) + B*sqrt(a)*c*x*sqrt(1 +
c*x**2/a)/2 - B*sqrt(a)*c*x/sqrt(1 + c*x**2/a) + 3*B*a*sqrt(c)*asinh(sqrt(c)*x/sqrt(a))/2

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Giac [B]  time = 1.17507, size = 258, normalized size = 2.32 \begin{align*} \frac{3 \, A a c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{3}{2} \, B a \sqrt{c} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{1}{2} \,{\left (B c x + 2 \, A c\right )} \sqrt{c x^{2} + a} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{2} \sqrt{c} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{2} c - 2 \, B a^{3} \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*A*a*c*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - 3/2*B*a*sqrt(c)*log(abs(-sqrt(c)*x + sqrt(c
*x^2 + a))) + 1/2*(B*c*x + 2*A*c)*sqrt(c*x^2 + a) + ((sqrt(c)*x - sqrt(c*x^2 + a))^3*A*a*c + 2*(sqrt(c)*x - sq
rt(c*x^2 + a))^2*B*a^2*sqrt(c) + (sqrt(c)*x - sqrt(c*x^2 + a))*A*a^2*c - 2*B*a^3*sqrt(c))/((sqrt(c)*x - sqrt(c
*x^2 + a))^2 - a)^2