3.533 \(\int \frac{\sqrt{a+c x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=206 \[ -\frac{a c^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac{5 c d e \left (a+c x^2\right )^{3/2}}{12 (d+e x)^3 \left (a e^2+c d^2\right )^2}-\frac{c \sqrt{a+c x^2} \left (4 c d^2-a e^2\right ) (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]

[Out]

-(c*(4*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (e*(a + c*x^2)^(3/2))
/(4*(c*d^2 + a*e^2)*(d + e*x)^4) - (5*c*d*e*(a + c*x^2)^(3/2))/(12*(c*d^2 + a*e^2)^2*(d + e*x)^3) - (a*c^2*(4*
c*d^2 - a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(7/2))

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Rubi [A]  time = 0.126301, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {745, 807, 721, 725, 206} \[ -\frac{a c^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac{5 c d e \left (a+c x^2\right )^{3/2}}{12 (d+e x)^3 \left (a e^2+c d^2\right )^2}-\frac{c \sqrt{a+c x^2} \left (4 c d^2-a e^2\right ) (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^2]/(d + e*x)^5,x]

[Out]

-(c*(4*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (e*(a + c*x^2)^(3/2))
/(4*(c*d^2 + a*e^2)*(d + e*x)^4) - (5*c*d*e*(a + c*x^2)^(3/2))/(12*(c*d^2 + a*e^2)^2*(d + e*x)^3) - (a*c^2*(4*
c*d^2 - a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(7/2))

Rule 745

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)*(a + c*x^2)^(p
+ 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

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])

Rubi steps

\begin{align*} \int \frac{\sqrt{a+c x^2}}{(d+e x)^5} \, dx &=-\frac{e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{c \int \frac{(-4 d+e x) \sqrt{a+c x^2}}{(d+e x)^4} \, dx}{4 \left (c d^2+a e^2\right )}\\ &=-\frac{e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{5 c d e \left (a+c x^2\right )^{3/2}}{12 \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac{\left (c \left (4 c d^2-a e^2\right )\right ) \int \frac{\sqrt{a+c x^2}}{(d+e x)^3} \, dx}{4 \left (c d^2+a e^2\right )^2}\\ &=-\frac{c \left (4 c d^2-a e^2\right ) (a e-c d x) \sqrt{a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{5 c d e \left (a+c x^2\right )^{3/2}}{12 \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac{\left (a c^2 \left (4 c d^2-a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac{c \left (4 c d^2-a e^2\right ) (a e-c d x) \sqrt{a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{5 c d e \left (a+c x^2\right )^{3/2}}{12 \left (c d^2+a e^2\right )^2 (d+e x)^3}-\frac{\left (a c^2 \left (4 c d^2-a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d^2+a e^2-x^2} \, dx,x,\frac{a e-c d x}{\sqrt{a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac{c \left (4 c d^2-a e^2\right ) (a e-c d x) \sqrt{a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{e \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{5 c d e \left (a+c x^2\right )^{3/2}}{12 \left (c d^2+a e^2\right )^2 (d+e x)^3}-\frac{a c^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.23641, size = 248, normalized size = 1.2 \[ \frac{\sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left (c^2 d (d+e x)^3 \left (2 c d^2-13 a e^2\right )+2 c d (d+e x) \left (a e^2+c d^2\right )^2+c (d+e x)^2 \left (2 c d^2-3 a e^2\right ) \left (a e^2+c d^2\right )-6 \left (a e^2+c d^2\right )^3\right )+3 a c^2 e (d+e x)^4 \left (a e^2-4 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )-3 a c^2 e (d+e x)^4 \left (a e^2-4 c d^2\right ) \log (d+e x)}{24 e (d+e x)^4 \left (a e^2+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(-6*(c*d^2 + a*e^2)^3 + 2*c*d*(c*d^2 + a*e^2)^2*(d + e*x) + c*(2*c*d^2 -
3*a*e^2)*(c*d^2 + a*e^2)*(d + e*x)^2 + c^2*d*(2*c*d^2 - 13*a*e^2)*(d + e*x)^3) - 3*a*c^2*e*(-4*c*d^2 + a*e^2)*
(d + e*x)^4*Log[d + e*x] + 3*a*c^2*e*(-4*c*d^2 + a*e^2)*(d + e*x)^4*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt
[a + c*x^2]])/(24*e*(c*d^2 + a*e^2)^(7/2)*(d + e*x)^4)

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Maple [B]  time = 0.205, size = 2073, normalized size = 10.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/8/e*c^4*d^4/(a*e^2+c*d^2)^4/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^
2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a+5/8*c^(7/2)*d^3/(a*e^2+c*d^2)^
4*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a-3/4/e*c^3*d^2/(a*e^2+
c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e
+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a+1/8/(a*e^2+c*d^2)^3*c^2*d/(d/e+x)*(c*(d/e+x)^2-2*c*
d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-1/8/(a*e^2+c*d^2)^3*c^3*d*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)
^(1/2)*x-1/4/e^3/(a*e^2+c*d^2)/(d/e+x)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-1/8/e/(a*e^2+c*
d^2)^2*c^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+5/8/e^3*c^5*d^6/(a*e^2+c*d^2)^4/((a*e^2+c*d^2
)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+
(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))+5/8*c^4*d^3/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)
^(1/2)*x-3/4/e^2*c^(7/2)*d^3/(a*e^2+c*d^2)^3*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2
+c*d^2)/e^2)^(1/2))-3/4/e^3*c^4*d^4/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*
(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))-5/8/e*c^4*
d^4/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+3/4/e*c^3*d^2/(a*e^2+c*d^2)^3*(c*(d/
e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)-5/12/e^2*c*d/(a*e^2+c*d^2)^2/(d/e+x)^3*(c*(d/e+x)^2-2*c*d/e*(d
/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-5/8/e*c^2*d^2/(a*e^2+c*d^2)^3/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d
^2)/e^2)^(3/2)-5/8*c^3*d^3/(a*e^2+c*d^2)^4/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)+5/8/e
^2*c^(9/2)*d^5/(a*e^2+c*d^2)^4*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(
1/2))+1/8/e/(a*e^2+c*d^2)^2*c/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-1/8/(a*e^2+c*d^2
)^3*c^(5/2)*d*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a+1/8/e^2/(
a*e^2+c*d^2)^2*c^(5/2)*d*ln((-c*d/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))+
1/8/e/(a*e^2+c*d^2)^2*c^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e
^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a+1/8/e^3/(a*e^2+c*d^2)^2*c^3/((a*e^
2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(
d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*d^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((c*x^2+a)^(1/2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 24.853, size = 2958, normalized size = 14.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(3*(4*a*c^3*d^6 - a^2*c^2*d^4*e^2 + (4*a*c^3*d^2*e^4 - a^2*c^2*e^6)*x^4 + 4*(4*a*c^3*d^3*e^3 - a^2*c^2*
d*e^5)*x^3 + 6*(4*a*c^3*d^4*e^2 - a^2*c^2*d^2*e^4)*x^2 + 4*(4*a*c^3*d^5*e - a^2*c^2*d^3*e^3)*x)*sqrt(c*d^2 + a
*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)
*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(28*a*c^3*d^6*e + 47*a^2*c^2*d^4*e^3 + 25*a^3*c*d^2*e^5 + 6*a
^4*e^7 - (2*c^4*d^5*e^2 - 11*a*c^3*d^3*e^4 - 13*a^2*c^2*d*e^6)*x^3 - (8*c^4*d^6*e - 32*a*c^3*d^4*e^3 - 43*a^2*
c^2*d^2*e^5 - 3*a^3*c*e^7)*x^2 - (12*c^4*d^7 - 25*a*c^3*d^5*e^2 - 41*a^2*c^2*d^3*e^4 - 4*a^3*c*d*e^6)*x)*sqrt(
c*x^2 + a))/(c^4*d^12 + 4*a*c^3*d^10*e^2 + 6*a^2*c^2*d^8*e^4 + 4*a^3*c*d^6*e^6 + a^4*d^4*e^8 + (c^4*d^8*e^4 +
4*a*c^3*d^6*e^6 + 6*a^2*c^2*d^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12)*x^4 + 4*(c^4*d^9*e^3 + 4*a*c^3*d^7*e^5 + 6*
a^2*c^2*d^5*e^7 + 4*a^3*c*d^3*e^9 + a^4*d*e^11)*x^3 + 6*(c^4*d^10*e^2 + 4*a*c^3*d^8*e^4 + 6*a^2*c^2*d^6*e^6 +
4*a^3*c*d^4*e^8 + a^4*d^2*e^10)*x^2 + 4*(c^4*d^11*e + 4*a*c^3*d^9*e^3 + 6*a^2*c^2*d^7*e^5 + 4*a^3*c*d^5*e^7 +
a^4*d^3*e^9)*x), -1/24*(3*(4*a*c^3*d^6 - a^2*c^2*d^4*e^2 + (4*a*c^3*d^2*e^4 - a^2*c^2*e^6)*x^4 + 4*(4*a*c^3*d^
3*e^3 - a^2*c^2*d*e^5)*x^3 + 6*(4*a*c^3*d^4*e^2 - a^2*c^2*d^2*e^4)*x^2 + 4*(4*a*c^3*d^5*e - a^2*c^2*d^3*e^3)*x
)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 + (c^2*d^2
+ a*c*e^2)*x^2)) + (28*a*c^3*d^6*e + 47*a^2*c^2*d^4*e^3 + 25*a^3*c*d^2*e^5 + 6*a^4*e^7 - (2*c^4*d^5*e^2 - 11*
a*c^3*d^3*e^4 - 13*a^2*c^2*d*e^6)*x^3 - (8*c^4*d^6*e - 32*a*c^3*d^4*e^3 - 43*a^2*c^2*d^2*e^5 - 3*a^3*c*e^7)*x^
2 - (12*c^4*d^7 - 25*a*c^3*d^5*e^2 - 41*a^2*c^2*d^3*e^4 - 4*a^3*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^12 + 4*a*c
^3*d^10*e^2 + 6*a^2*c^2*d^8*e^4 + 4*a^3*c*d^6*e^6 + a^4*d^4*e^8 + (c^4*d^8*e^4 + 4*a*c^3*d^6*e^6 + 6*a^2*c^2*d
^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12)*x^4 + 4*(c^4*d^9*e^3 + 4*a*c^3*d^7*e^5 + 6*a^2*c^2*d^5*e^7 + 4*a^3*c*d^3
*e^9 + a^4*d*e^11)*x^3 + 6*(c^4*d^10*e^2 + 4*a*c^3*d^8*e^4 + 6*a^2*c^2*d^6*e^6 + 4*a^3*c*d^4*e^8 + a^4*d^2*e^1
0)*x^2 + 4*(c^4*d^11*e + 4*a*c^3*d^9*e^3 + 6*a^2*c^2*d^7*e^5 + 4*a^3*c*d^5*e^7 + a^4*d^3*e^9)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + c x^{2}}}{\left (d + e x\right )^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + c*x**2)/(d + e*x)**5, x)

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Giac [B]  time = 4.21485, size = 1297, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)*((2*((c^3*d^5*e^6*sgn(1/(x*e + d)) +
2*a*c^2*d^3*e^8*sgn(1/(x*e + d)) + a^2*c*d*e^10*sgn(1/(x*e + d)))/(c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^2*c^2*
d^4*e^12 + 4*a^3*c*d^2*e^14 + a^4*e^16) - 3*(c^3*d^6*e^7*sgn(1/(x*e + d)) + 3*a*c^2*d^4*e^9*sgn(1/(x*e + d)) +
3*a^2*c*d^2*e^11*sgn(1/(x*e + d)) + a^3*e^13*sgn(1/(x*e + d)))*e^(-1)/((c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^
2*c^2*d^4*e^12 + 4*a^3*c*d^2*e^14 + a^4*e^16)*(x*e + d)))*e^(-1)/(x*e + d) + (2*c^3*d^4*e^5*sgn(1/(x*e + d)) -
a*c^2*d^2*e^7*sgn(1/(x*e + d)) - 3*a^2*c*e^9*sgn(1/(x*e + d)))/(c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^2*c^2*d^
4*e^12 + 4*a^3*c*d^2*e^14 + a^4*e^16))*e^(-1)/(x*e + d) + (2*c^3*d^3*e^4*sgn(1/(x*e + d)) - 13*a*c^2*d*e^6*sgn
(1/(x*e + d)))/(c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^2*c^2*d^4*e^12 + 4*a^3*c*d^2*e^14 + a^4*e^16)) - 3*(4*a*c
^3*d^2*sgn(1/(x*e + d)) - a^2*c^2*e^2*sgn(1/(x*e + d)))*sqrt(c*d^2 + a*e^2)*log(abs(sqrt(c*d^2 + a*e^2)*c*d -
(c*d^2 + a*e^2)*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2 + a*e^4)*e
^(-1)/(x*e + d))))/(c^5*d^10*e^2 + 5*a*c^4*d^8*e^4 + 10*a^2*c^3*d^6*e^6 + 10*a^3*c^2*d^4*e^8 + 5*a^4*c*d^2*e^1
0 + a^5*e^12) - (2*c^(9/2)*d^5 - 11*a*c^(7/2)*d^3*e^2 - 12*sqrt(c*d^2 + a*e^2)*a*c^3*d^2*e^2*log(abs(-c^(3/2)*
d^2 + sqrt(c*d^2 + a*e^2)*c*d - a*sqrt(c)*e^2)) - 13*a^2*c^(5/2)*d*e^4 + 3*sqrt(c*d^2 + a*e^2)*a^2*c^2*e^4*log
(abs(-c^(3/2)*d^2 + sqrt(c*d^2 + a*e^2)*c*d - a*sqrt(c)*e^2)))*sgn(1/(x*e + d))/(c^5*d^10*e^4 + 5*a*c^4*d^8*e^
6 + 10*a^2*c^3*d^6*e^8 + 10*a^3*c^2*d^4*e^10 + 5*a^4*c*d^2*e^12 + a^5*e^14))*e^2