3.367 \(\int \frac{a+b x^2}{x^5 \sqrt{-c+d x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=123 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+4 b c^2\right )}{8 c^4 x^2}+\frac{d^2 \left (3 a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^5}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{4 c^2 x^4} \]

[Out]

(a*Sqrt[-c + d*x]*Sqrt[c + d*x])/(4*c^2*x^4) + ((4*b*c^2 + 3*a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(8*c^4*x^2)
+ (d^2*(4*b*c^2 + 3*a*d^2)*ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*x])/c])/(8*c^5)

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Rubi [A]  time = 0.0953812, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {454, 103, 12, 92, 205} \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+4 b c^2\right )}{8 c^4 x^2}+\frac{d^2 \left (3 a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^5}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{4 c^2 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/(x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]

[Out]

(a*Sqrt[-c + d*x]*Sqrt[c + d*x])/(4*c^2*x^4) + ((4*b*c^2 + 3*a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(8*c^4*x^2)
+ (d^2*(4*b*c^2 + 3*a*d^2)*ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*x])/c])/(8*c^5)

Rule 454

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2*e*
(m + 1)), x] + Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*e^n*(m + 1)), Int[(e*x)^(m + n)*(a1
+ b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, p}, x] && EqQ[non2, n/2] && Eq
Q[a2*b1 + a1*b2, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1
])) &&  !ILtQ[p, -1]

Rule 103

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{a+b x^2}{x^5 \sqrt{-c+d x} \sqrt{c+d x}} \, dx &=\frac{a \sqrt{-c+d x} \sqrt{c+d x}}{4 c^2 x^4}+\frac{1}{4} \left (4 b+\frac{3 a d^2}{c^2}\right ) \int \frac{1}{x^3 \sqrt{-c+d x} \sqrt{c+d x}} \, dx\\ &=\frac{a \sqrt{-c+d x} \sqrt{c+d x}}{4 c^2 x^4}+\frac{\left (4 b c^2+3 a d^2\right ) \sqrt{-c+d x} \sqrt{c+d x}}{8 c^4 x^2}+\frac{\left (4 b c^2+3 a d^2\right ) \int \frac{d^2}{x \sqrt{-c+d x} \sqrt{c+d x}} \, dx}{8 c^4}\\ &=\frac{a \sqrt{-c+d x} \sqrt{c+d x}}{4 c^2 x^4}+\frac{\left (4 b c^2+3 a d^2\right ) \sqrt{-c+d x} \sqrt{c+d x}}{8 c^4 x^2}+\frac{\left (d^2 \left (4 b c^2+3 a d^2\right )\right ) \int \frac{1}{x \sqrt{-c+d x} \sqrt{c+d x}} \, dx}{8 c^4}\\ &=\frac{a \sqrt{-c+d x} \sqrt{c+d x}}{4 c^2 x^4}+\frac{\left (4 b c^2+3 a d^2\right ) \sqrt{-c+d x} \sqrt{c+d x}}{8 c^4 x^2}+\frac{\left (d^3 \left (4 b c^2+3 a d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c^2 d+d x^2} \, dx,x,\sqrt{-c+d x} \sqrt{c+d x}\right )}{8 c^4}\\ &=\frac{a \sqrt{-c+d x} \sqrt{c+d x}}{4 c^2 x^4}+\frac{\left (4 b c^2+3 a d^2\right ) \sqrt{-c+d x} \sqrt{c+d x}}{8 c^4 x^2}+\frac{d^2 \left (4 b c^2+3 a d^2\right ) \tan ^{-1}\left (\frac{\sqrt{-c+d x} \sqrt{c+d x}}{c}\right )}{8 c^5}\\ \end{align*}

Mathematica [A]  time = 0.0936209, size = 144, normalized size = 1.17 \[ -\frac{\left (c^2-d^2 x^2\right ) \left (c^2 \sqrt{1-\frac{d^2 x^2}{c^2}} \left (2 a c^2+3 a d^2 x^2+4 b c^2 x^2\right )+d^2 x^4 \left (3 a d^2+4 b c^2\right ) \tanh ^{-1}\left (\sqrt{1-\frac{d^2 x^2}{c^2}}\right )\right )}{8 c^6 x^4 \sqrt{d x-c} \sqrt{c+d x} \sqrt{1-\frac{d^2 x^2}{c^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/(x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]

[Out]

-((c^2 - d^2*x^2)*(c^2*(2*a*c^2 + 4*b*c^2*x^2 + 3*a*d^2*x^2)*Sqrt[1 - (d^2*x^2)/c^2] + d^2*(4*b*c^2 + 3*a*d^2)
*x^4*ArcTanh[Sqrt[1 - (d^2*x^2)/c^2]]))/(8*c^6*x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]*Sqrt[1 - (d^2*x^2)/c^2])

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Maple [B]  time = 0.02, size = 227, normalized size = 1.9 \begin{align*} -{\frac{1}{8\,{c}^{4}{x}^{4}}\sqrt{dx-c}\sqrt{dx+c} \left ( 3\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}a{d}^{4}+4\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}b{c}^{2}{d}^{2}-3\,\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{x}^{2}a{d}^{2}-4\,\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{x}^{2}b{c}^{2}-2\,\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}a{c}^{2} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/x^5/(d*x-c)^(1/2)/(d*x+c)^(1/2),x)

[Out]

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

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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^2+a)/x^5/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58047, size = 219, normalized size = 1.78 \begin{align*} \frac{2 \,{\left (4 \, b c^{2} d^{2} + 3 \, a d^{4}\right )} x^{4} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right ) +{\left (2 \, a c^{3} +{\left (4 \, b c^{3} + 3 \, a c d^{2}\right )} x^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}}{8 \, c^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^5/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 61.5923, size = 172, normalized size = 1.4 \begin{align*} - \frac{a d^{4}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{11}{4}, \frac{13}{4}, 1 & 3, 3, \frac{7}{2} \\\frac{5}{2}, \frac{11}{4}, 3, \frac{13}{4}, \frac{7}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} + \frac{i a d^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} 2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3, 1 & \\\frac{9}{4}, \frac{11}{4} & 2, \frac{5}{2}, \frac{5}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} - \frac{b d^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} + \frac{i b d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)/x**5/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

-a*d**4*meijerg(((11/4, 13/4, 1), (3, 3, 7/2)), ((5/2, 11/4, 3, 13/4, 7/2), (0,)), c**2/(d**2*x**2))/(4*pi**(3
/2)*c**5) + I*a*d**4*meijerg(((2, 9/4, 5/2, 11/4, 3, 1), ()), ((9/4, 11/4), (2, 5/2, 5/2, 0)), c**2*exp_polar(
2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*c**5) - b*d**2*meijerg(((7/4, 9/4, 1), (2, 2, 5/2)), ((3/2, 7/4, 2, 9/4, 5/2
), (0,)), c**2/(d**2*x**2))/(4*pi**(3/2)*c**3) + I*b*d**2*meijerg(((1, 5/4, 3/2, 7/4, 2, 1), ()), ((5/4, 7/4),
(1, 3/2, 3/2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*c**3)

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Giac [B]  time = 1.20858, size = 439, normalized size = 3.57 \begin{align*} -\frac{\frac{{\left (4 \, b c^{2} d^{3} + 3 \, a d^{5}\right )} \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c^{5}} + \frac{2 \,{\left (4 \, b c^{2} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{14} + 3 \, a d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{14} + 16 \, b c^{4} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{10} + 44 \, a c^{2} d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{10} - 64 \, b c^{6} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 176 \, a c^{4} d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 256 \, b c^{8} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} - 192 \, a c^{6} d^{5}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{4} c^{4}}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^5/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

-1/4*((4*b*c^2*d^3 + 3*a*d^5)*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c)/c^5 + 2*(4*b*c^2*d^3*(sqrt(d*x +
c) - sqrt(d*x - c))^14 + 3*a*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^14 + 16*b*c^4*d^3*(sqrt(d*x + c) - sqrt(d*x
- c))^10 + 44*a*c^2*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^10 - 64*b*c^6*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^6 -
176*a*c^4*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^6 - 256*b*c^8*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^2 - 192*a*c^6*
d^5*(sqrt(d*x + c) - sqrt(d*x - c))^2)/(((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^4*c^4))/d