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

Optimal. Leaf size=121 \[ -\frac{\sqrt{d x-c} \sqrt{c+d x} \left (a d^2+4 b c^2\right )}{8 c^2 x^2}+\frac{d^2 \left (a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^3}+\frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.103594, antiderivative size = 164, normalized size of antiderivative = 1.36, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {454, 94, 92, 205} \[ -\frac{\sqrt{d x-c} (c+d x)^{3/2} \left (a d^2+4 b c^2\right )}{8 c^3 x^2}+\frac{d \sqrt{d x-c} \sqrt{c+d x} \left (a d^2+4 b c^2\right )}{8 c^3 x}+\frac{d^2 \left (a d^2+4 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{8 c^3}+\frac{a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 94

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

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

Mathematica [A]  time = 0.0843731, size = 137, normalized size = 1.13 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (\left (c^2-d^2 x^2\right ) \left (2 a c^2-a d^2 x^2+4 b c^2 x^2\right )-d^2 x^4 \sqrt{1-\frac{d^2 x^2}{c^2}} \left (a d^2+4 b c^2\right ) \tanh ^{-1}\left (\sqrt{1-\frac{d^2 x^2}{c^2}}\right )\right )}{8 c^2 d^2 x^6-8 c^4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.014, size = 226, normalized size = 1.9 \begin{align*} -{\frac{1}{8\,{c}^{2}{x}^{4}}\sqrt{dx-c}\sqrt{dx+c} \left ( \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}-\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)*(d*x-c)^(1/2)*(d*x+c)^(1/2)/x^5,x)

[Out]

-1/8*(d*x-c)^(1/2)*(d*x+c)^(1/2)/c^2*(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-(-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)/x^4/(-c^2)^(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^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58017, size = 213, normalized size = 1.76 \begin{align*} \frac{2 \,{\left (4 \, b c^{2} d^{2} + 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} - a c d^{2}\right )} x^{2}\right )} \sqrt{d x + c} \sqrt{d x - c}}{8 \, c^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: MellinTransformStripError

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Giac [B]  time = 1.21363, size = 437, normalized size = 3.61 \begin{align*} -\frac{\frac{{\left (4 \, b c^{2} d^{3} + a d^{5}\right )} \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c^{3}} - \frac{2 \,{\left (4 \, b c^{2} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{14} - 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} + 28 \, 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} - 112 \, 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} + 64 \, 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^{2}}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((4*b*c^2*d^3 + a*d^5)*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c)/c^3 - 2*(4*b*c^2*d^3*(sqrt(d*x + c
) - sqrt(d*x - c))^14 - 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 + 28*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 - 112*
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 + 64*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^2))/d