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

Optimal. Leaf size=90 \[ \frac {d (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 c^{3/2}}+\frac {x^2 \sqrt {c+\frac {d}{x^2}} (4 b c-a d)}{8 c}+\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{3/2}}{4 c} \]

[Out]

1/4*a*(c+d/x^2)^(3/2)*x^4/c+1/8*d*(-a*d+4*b*c)*arctanh((c+d/x^2)^(1/2)/c^(1/2))/c^(3/2)+1/8*(-a*d+4*b*c)*x^2*(
c+d/x^2)^(1/2)/c

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Rubi [A]  time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 47, 63, 208} \[ \frac {d (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 c^{3/2}}+\frac {x^2 \sqrt {c+\frac {d}{x^2}} (4 b c-a d)}{8 c}+\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{3/2}}{4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*b*c - a*d)*Sqrt[c + d/x^2]*x^2)/(8*c) + (a*(c + d/x^2)^(3/2)*x^4)/(4*c) + (d*(4*b*c - a*d)*ArcTanh[Sqrt[c
+ d/x^2]/Sqrt[c]])/(8*c^(3/2))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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 78

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

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]

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]]

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 100, normalized size = 1.11 \[ \frac {x \sqrt {c+\frac {d}{x^2}} \left (\sqrt {c} x \sqrt {\frac {c x^2}{d}+1} \left (a \left (2 c x^2+d\right )+4 b c\right )+\sqrt {d} (4 b c-a d) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}{8 c^{3/2} \sqrt {\frac {c x^2}{d}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d/x^2]*x*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/d]*(4*b*c + a*(d + 2*c*x^2)) + Sqrt[d]*(4*b*c - a*d)*ArcSinh[(S
qrt[c]*x)/Sqrt[d]]))/(8*c^(3/2)*Sqrt[1 + (c*x^2)/d])

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fricas [A]  time = 0.62, size = 191, normalized size = 2.12 \[ \left [-\frac {{\left (4 \, b c d - a d^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left (2 \, a c^{2} x^{4} + {\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, c^{2}}, -\frac {{\left (4 \, b c d - a d^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (2 \, a c^{2} x^{4} + {\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*((4*b*c*d - a*d^2)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d) - 2*(2*a*c^2*x^4 + (
4*b*c^2 + a*c*d)*x^2)*sqrt((c*x^2 + d)/x^2))/c^2, -1/8*((4*b*c*d - a*d^2)*sqrt(-c)*arctan(sqrt(-c)*x^2*sqrt((c
*x^2 + d)/x^2)/(c*x^2 + d)) - (2*a*c^2*x^4 + (4*b*c^2 + a*c*d)*x^2)*sqrt((c*x^2 + d)/x^2))/c^2]

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giac [A]  time = 0.26, size = 105, normalized size = 1.17 \[ \frac {1}{8} \, {\left (2 \, a x^{2} \mathrm {sgn}\relax (x) + \frac {4 \, b c^{2} \mathrm {sgn}\relax (x) + a c d \mathrm {sgn}\relax (x)}{c^{2}}\right )} \sqrt {c x^{2} + d} x - \frac {{\left (4 \, b c d \mathrm {sgn}\relax (x) - a d^{2} \mathrm {sgn}\relax (x)\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right )}{8 \, c^{\frac {3}{2}}} + \frac {{\left (4 \, b c d \log \left ({\left | d \right |}\right ) - a d^{2} \log \left ({\left | d \right |}\right )\right )} \mathrm {sgn}\relax (x)}{16 \, c^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*a*x^2*sgn(x) + (4*b*c^2*sgn(x) + a*c*d*sgn(x))/c^2)*sqrt(c*x^2 + d)*x - 1/8*(4*b*c*d*sgn(x) - a*d^2*sgn
(x))*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))/c^(3/2) + 1/16*(4*b*c*d*log(abs(d)) - a*d^2*log(abs(d)))*sgn(x)/c^
(3/2)

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maple [A]  time = 0.06, size = 122, normalized size = 1.36 \[ \frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (-a \,d^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+4 b c d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )-\sqrt {c \,x^{2}+d}\, a \sqrt {c}\, d x +4 \sqrt {c \,x^{2}+d}\, b \,c^{\frac {3}{2}} x +2 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \sqrt {c}\, x \right ) x}{8 \sqrt {c \,x^{2}+d}\, c^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((c*x^2+d)/x^2)^(1/2)*x*(2*(c*x^2+d)^(3/2)*c^(1/2)*x*a-(c*x^2+d)^(1/2)*c^(1/2)*x*a*d+4*(c*x^2+d)^(1/2)*c^(
3/2)*x*b-ln(c^(1/2)*x+(c*x^2+d)^(1/2))*a*d^2+4*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*b*c*d)/(c*x^2+d)^(1/2)/c^(3/2)

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maxima [B]  time = 1.24, size = 159, normalized size = 1.77 \[ \frac {1}{16} \, {\left (\frac {d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} + \sqrt {c + \frac {d}{x^{2}}} c d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} c - 2 \, {\left (c + \frac {d}{x^{2}}\right )} c^{2} + c^{3}}\right )} a + \frac {1}{4} \, {\left (2 \, \sqrt {c + \frac {d}{x^{2}}} x^{2} - \frac {d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{\sqrt {c}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(d^2*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/c^(3/2) + 2*((c + d/x^2)^(3/2)*d^2 + sq
rt(c + d/x^2)*c*d^2)/((c + d/x^2)^2*c - 2*(c + d/x^2)*c^2 + c^3))*a + 1/4*(2*sqrt(c + d/x^2)*x^2 - d*log((sqrt
(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/sqrt(c))*b

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mupad [B]  time = 5.31, size = 93, normalized size = 1.03 \[ \frac {a\,x^4\,\sqrt {c+\frac {d}{x^2}}}{8}+\frac {b\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2}+\frac {a\,x^4\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{8\,c}+\frac {b\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2\,\sqrt {c}}-\frac {a\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8\,c^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x^4*(c + d/x^2)^(1/2))/8 + (b*x^2*(c + d/x^2)^(1/2))/2 + (a*x^4*(c + d/x^2)^(3/2))/(8*c) + (b*d*atanh((c +
d/x^2)^(1/2)/c^(1/2)))/(2*c^(1/2)) - (a*d^2*atanh((c + d/x^2)^(1/2)/c^(1/2)))/(8*c^(3/2))

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sympy [A]  time = 59.05, size = 144, normalized size = 1.60 \[ \frac {a c x^{5}}{4 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 a \sqrt {d} x^{3}}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {a d^{\frac {3}{2}} x}{8 c \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 c^{\frac {3}{2}}} + \frac {b \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} + \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2 \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x**5/(4*sqrt(d)*sqrt(c*x**2/d + 1)) + 3*a*sqrt(d)*x**3/(8*sqrt(c*x**2/d + 1)) + a*d**(3/2)*x/(8*c*sqrt(c*x
**2/d + 1)) - a*d**2*asinh(sqrt(c)*x/sqrt(d))/(8*c**(3/2)) + b*sqrt(d)*x*sqrt(c*x**2/d + 1)/2 + b*d*asinh(sqrt
(c)*x/sqrt(d))/(2*sqrt(c))

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