3.602 \(\int \frac{(a+b x^2)^2 \sqrt{c+d x^2}}{x^3} \, dx\)

Optimal. Leaf size=109 \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}+\frac{a \sqrt{c+d x^2} (a d+4 b c)}{2 c}-\frac{a (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 \sqrt{c}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d} \]

[Out]

(a*(4*b*c + a*d)*Sqrt[c + d*x^2])/(2*c) + (b^2*(c + d*x^2)^(3/2))/(3*d) - (a^2*(c + d*x^2)^(3/2))/(2*c*x^2) -
(a*(4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*Sqrt[c])

________________________________________________________________________________________

Rubi [A]  time = 0.0873646, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 89, 80, 50, 63, 208} \[ -\frac{a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}+\frac{a \sqrt{c+d x^2} (a d+4 b c)}{2 c}-\frac{a (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 \sqrt{c}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^3,x]

[Out]

(a*(4*b*c + a*d)*Sqrt[c + d*x^2])/(2*c) + (b^2*(c + d*x^2)^(3/2))/(3*d) - (a^2*(c + d*x^2)^(3/2))/(2*c*x^2) -
(a*(4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*Sqrt[c])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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

Mathematica [A]  time = 0.0773352, size = 87, normalized size = 0.8 \[ \frac{1}{6} \left (\frac{\sqrt{c+d x^2} \left (-3 a^2 d+12 a b d x^2+2 b^2 x^2 \left (c+d x^2\right )\right )}{d x^2}-\frac{3 a (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^3,x]

[Out]

((Sqrt[c + d*x^2]*(-3*a^2*d + 12*a*b*d*x^2 + 2*b^2*x^2*(c + d*x^2)))/(d*x^2) - (3*a*(4*b*c + a*d)*ArcTanh[Sqrt
[c + d*x^2]/Sqrt[c]])/Sqrt[c])/6

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 132, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-2\,\sqrt{c}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) ab+2\,\sqrt{d{x}^{2}+c}ab-{\frac{{a}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{{a}^{2}d}{2\,c}\sqrt{d{x}^{2}+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^3,x)

[Out]

1/3*b^2*(d*x^2+c)^(3/2)/d-2*c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)*a*b+2*(d*x^2+c)^(1/2)*a*b-1/2*a^2*(d
*x^2+c)^(3/2)/c/x^2-1/2*a^2*d/c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+1/2*a^2*d/c*(d*x^2+c)^(1/2)

________________________________________________________________________________________

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^2+a)^2*(d*x^2+c)^(1/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.69558, size = 474, normalized size = 4.35 \begin{align*} \left [\frac{3 \,{\left (4 \, a b c d + a^{2} d^{2}\right )} \sqrt{c} x^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (2 \, b^{2} c d x^{4} - 3 \, a^{2} c d + 2 \,{\left (b^{2} c^{2} + 6 \, a b c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{12 \, c d x^{2}}, \frac{3 \,{\left (4 \, a b c d + a^{2} d^{2}\right )} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) +{\left (2 \, b^{2} c d x^{4} - 3 \, a^{2} c d + 2 \,{\left (b^{2} c^{2} + 6 \, a b c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{6 \, c d x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^3,x, algorithm="fricas")

[Out]

[1/12*(3*(4*a*b*c*d + a^2*d^2)*sqrt(c)*x^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(2*b^2*c*d*
x^4 - 3*a^2*c*d + 2*(b^2*c^2 + 6*a*b*c*d)*x^2)*sqrt(d*x^2 + c))/(c*d*x^2), 1/6*(3*(4*a*b*c*d + a^2*d^2)*sqrt(-
c)*x^2*arctan(sqrt(-c)/sqrt(d*x^2 + c)) + (2*b^2*c*d*x^4 - 3*a^2*c*d + 2*(b^2*c^2 + 6*a*b*c*d)*x^2)*sqrt(d*x^2
+ c))/(c*d*x^2)]

________________________________________________________________________________________

Sympy [A]  time = 30.8147, size = 148, normalized size = 1.36 \begin{align*} - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} - \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2 \sqrt{c}} - 2 a b \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} + \frac{2 a b c}{\sqrt{d} x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{2 a b \sqrt{d} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + b^{2} \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**3,x)

[Out]

-a**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(2*x) - a**2*d*asinh(sqrt(c)/(sqrt(d)*x))/(2*sqrt(c)) - 2*a*b*sqrt(c)*asinh
(sqrt(c)/(sqrt(d)*x)) + 2*a*b*c/(sqrt(d)*x*sqrt(c/(d*x**2) + 1)) + 2*a*b*sqrt(d)*x/sqrt(c/(d*x**2) + 1) + b**2
*Piecewise((sqrt(c)*x**2/2, Eq(d, 0)), ((c + d*x**2)**(3/2)/(3*d), True))

________________________________________________________________________________________

Giac [A]  time = 1.10147, size = 120, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} + 12 \, \sqrt{d x^{2} + c} a b d - \frac{3 \, \sqrt{d x^{2} + c} a^{2} d}{x^{2}} + \frac{3 \,{\left (4 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^3,x, algorithm="giac")

[Out]

1/6*(2*(d*x^2 + c)^(3/2)*b^2 + 12*sqrt(d*x^2 + c)*a*b*d - 3*sqrt(d*x^2 + c)*a^2*d/x^2 + 3*(4*a*b*c*d + a^2*d^2
)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c))/d