3.5.53 \(\int \frac {\sqrt {a+c x^2}}{(d+e x)^5} \, dx\)
Optimal. Leaf size=206 \[ -\frac {a c^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac {5 c d e \left (a+c x^2\right )^{3/2}}{12 (d+e x)^3 \left (a e^2+c d^2\right )^2}-\frac {c \sqrt {a+c x^2} \left (4 c d^2-a e^2\right ) (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {e \left (a+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 206, normalized size of antiderivative = 1.00,
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} \begin {gather*} -\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 )} \end {gather*}
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 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])
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 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]
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*}
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Mathematica [A] time = 0.25, size = 248, normalized size = 1.20 \begin {gather*} \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}} \end {gather*}
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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IntegrateAlgebraic [B] time = 17.54, size = 1532, normalized size = 7.44 \begin {gather*} \frac {a d^2 \tan ^{-1}\left (\frac {\sqrt {c} d}{\sqrt {-c d^2-a e^2}}+\frac {\sqrt {c} e x}{\sqrt {-c d^2-a e^2}}-\frac {e \sqrt {c x^2+a}}{\sqrt {-c d^2-a e^2}}\right ) c^3}{\left (-c d^2-a e^2\right )^{7/2}}-\frac {a^2 e^2 \tan ^{-1}\left (\frac {\sqrt {c} d}{\sqrt {-c d^2-a e^2}}+\frac {\sqrt {c} e x}{\sqrt {-c d^2-a e^2}}-\frac {e \sqrt {c x^2+a}}{\sqrt {-c d^2-a e^2}}\right ) c^2}{4 \left (-c d^2-a e^2\right )^{7/2}}+\frac {6 a^6 e^7-24 a^5 \sqrt {c} x \sqrt {c x^2+a} e^7+c \left (57 a^5 x^2 e^7+4 a^5 d x e^6+19 a^5 d^2 e^5\right )+c^{3/2} \sqrt {c x^2+a} \left (-84 a^4 x^3 e^7-16 a^4 d x^2 e^6-76 a^4 d^2 x e^5\right )+c^2 \left (123 a^4 x^4 e^7+49 a^4 d x^3 e^6+211 a^4 d^2 x^2 e^5+37 a^4 d^3 x e^4+28 a^4 d^4 e^3\right )+c^{11/2} \sqrt {c x^2+a} \left (-16 x^4 d^7-64 e x^5 d^6\right )+c^{5/2} \sqrt {c x^2+a} \left (-84 a^3 x^5 e^7-87 a^3 d x^4 e^6-336 a^3 d^2 x^3 e^5-70 a^3 d^3 x^2 e^4-60 a^3 d^4 x e^3+13 a^3 d^5 e^2\right )+c^6 \left (16 x^5 d^7+64 e x^6 d^6\right )+c^3 \left (96 a^3 x^6 e^7+129 a^3 d x^5 e^6+456 a^3 d^2 x^4 e^5+19 a^3 d^3 x^3 e^4+36 a^3 d^4 x^2 e^3-64 a^3 d^5 x e^2\right )+c^{9/2} \sqrt {c x^2+a} \left (-16 a x^2 d^7-64 a e x^3 d^6+152 a e^2 x^4 d^5+224 a e^3 x^5 d^4+336 a e^4 x^6 d^3+96 a e^5 x^7 d^2\right )+c^{7/2} \sqrt {c x^2+a} \left (-2 a^2 d^7-8 a^2 e x d^6+140 a^2 e^2 x^2 d^5+104 a^2 e^3 x^3 d^4+186 a^2 e^4 x^4 d^3-216 a^2 e^5 x^5 d^2-84 a^2 e^6 x^6 d-24 a^2 e^7 x^7\right )+c^5 \left (-96 a d^2 e^5 x^8-336 a d^3 e^4 x^7-224 a d^4 e^3 x^6-152 a d^5 e^2 x^5+96 a d^6 e x^4+24 a d^7 x^3\right )+c^4 \left (24 a^2 e^7 x^8+84 a^2 d e^6 x^7+168 a^2 d^2 e^5 x^6-354 a^2 d^3 e^4 x^5-216 a^2 d^4 e^3 x^4-216 a^2 d^5 e^2 x^3+32 a^2 d^6 e x^2+8 a^2 d^7 x\right )}{96 a^5 \sqrt {c} x (d+e x)^4 e^8-24 a^5 (d+e x)^4 \sqrt {c x^2+a} e^8+192 c^{11/2} d^6 x^5 (d+e x)^4 e^2+24 c (d+e x)^4 \sqrt {c x^2+a} \left (-8 a^4 x^2 e^6-3 a^4 d^2 e^4\right ) e^2+24 c^{3/2} (d+e x)^4 \left (12 a^4 x^3 e^6+12 a^4 d^2 x e^4\right ) e^2+24 c^4 (d+e x)^4 \sqrt {c x^2+a} \left (-8 a x^2 d^6-24 a e^2 x^4 d^4\right ) e^2+24 c^3 (d+e x)^4 \sqrt {c x^2+a} \left (-a^2 d^6-24 a^2 e^2 x^2 d^4-24 a^2 e^4 x^4 d^2\right ) e^2+24 c^2 (d+e x)^4 \sqrt {c x^2+a} \left (-8 a^3 x^4 e^6-24 a^3 d^2 x^2 e^4-3 a^3 d^4 e^2\right ) e^2+24 c^{9/2} (d+e x)^4 \left (12 a x^3 d^6+24 a e^2 x^5 d^4\right ) e^2+24 c^{7/2} (d+e x)^4 \left (4 a^2 x d^6+36 a^2 e^2 x^3 d^4+24 a^2 e^4 x^5 d^2\right ) e^2+24 c^{5/2} (d+e x)^4 \left (8 a^3 x^5 e^6+36 a^3 d^2 x^3 e^4+12 a^3 d^4 x e^2\right ) e^2-192 c^5 d^6 x^4 (d+e x)^4 \sqrt {c x^2+a} e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[a + c*x^2]/(d + e*x)^5,x]
[Out]
(6*a^6*e^7 - 24*a^5*Sqrt[c]*e^7*x*Sqrt[a + c*x^2] + c*(19*a^5*d^2*e^5 + 4*a^5*d*e^6*x + 57*a^5*e^7*x^2) + c^(3
/2)*Sqrt[a + c*x^2]*(-76*a^4*d^2*e^5*x - 16*a^4*d*e^6*x^2 - 84*a^4*e^7*x^3) + c^2*(28*a^4*d^4*e^3 + 37*a^4*d^3
*e^4*x + 211*a^4*d^2*e^5*x^2 + 49*a^4*d*e^6*x^3 + 123*a^4*e^7*x^4) + c^(11/2)*Sqrt[a + c*x^2]*(-16*d^7*x^4 - 6
4*d^6*e*x^5) + c^(5/2)*Sqrt[a + c*x^2]*(13*a^3*d^5*e^2 - 60*a^3*d^4*e^3*x - 70*a^3*d^3*e^4*x^2 - 336*a^3*d^2*e
^5*x^3 - 87*a^3*d*e^6*x^4 - 84*a^3*e^7*x^5) + c^6*(16*d^7*x^5 + 64*d^6*e*x^6) + c^3*(-64*a^3*d^5*e^2*x + 36*a^
3*d^4*e^3*x^2 + 19*a^3*d^3*e^4*x^3 + 456*a^3*d^2*e^5*x^4 + 129*a^3*d*e^6*x^5 + 96*a^3*e^7*x^6) + c^(9/2)*Sqrt[
a + c*x^2]*(-16*a*d^7*x^2 - 64*a*d^6*e*x^3 + 152*a*d^5*e^2*x^4 + 224*a*d^4*e^3*x^5 + 336*a*d^3*e^4*x^6 + 96*a*
d^2*e^5*x^7) + c^(7/2)*Sqrt[a + c*x^2]*(-2*a^2*d^7 - 8*a^2*d^6*e*x + 140*a^2*d^5*e^2*x^2 + 104*a^2*d^4*e^3*x^3
+ 186*a^2*d^3*e^4*x^4 - 216*a^2*d^2*e^5*x^5 - 84*a^2*d*e^6*x^6 - 24*a^2*e^7*x^7) + c^5*(24*a*d^7*x^3 + 96*a*d
^6*e*x^4 - 152*a*d^5*e^2*x^5 - 224*a*d^4*e^3*x^6 - 336*a*d^3*e^4*x^7 - 96*a*d^2*e^5*x^8) + c^4*(8*a^2*d^7*x +
32*a^2*d^6*e*x^2 - 216*a^2*d^5*e^2*x^3 - 216*a^2*d^4*e^3*x^4 - 354*a^2*d^3*e^4*x^5 + 168*a^2*d^2*e^5*x^6 + 84*
a^2*d*e^6*x^7 + 24*a^2*e^7*x^8))/(96*a^5*Sqrt[c]*e^8*x*(d + e*x)^4 + 192*c^(11/2)*d^6*e^2*x^5*(d + e*x)^4 - 24
*a^5*e^8*(d + e*x)^4*Sqrt[a + c*x^2] - 192*c^5*d^6*e^2*x^4*(d + e*x)^4*Sqrt[a + c*x^2] + 24*c*e^2*(d + e*x)^4*
Sqrt[a + c*x^2]*(-3*a^4*d^2*e^4 - 8*a^4*e^6*x^2) + 24*c^(3/2)*e^2*(d + e*x)^4*(12*a^4*d^2*e^4*x + 12*a^4*e^6*x
^3) + 24*c^4*e^2*(d + e*x)^4*Sqrt[a + c*x^2]*(-8*a*d^6*x^2 - 24*a*d^4*e^2*x^4) + 24*c^3*e^2*(d + e*x)^4*Sqrt[a
+ c*x^2]*(-(a^2*d^6) - 24*a^2*d^4*e^2*x^2 - 24*a^2*d^2*e^4*x^4) + 24*c^2*e^2*(d + e*x)^4*Sqrt[a + c*x^2]*(-3*
a^3*d^4*e^2 - 24*a^3*d^2*e^4*x^2 - 8*a^3*e^6*x^4) + 24*c^(9/2)*e^2*(d + e*x)^4*(12*a*d^6*x^3 + 24*a*d^4*e^2*x^
5) + 24*c^(7/2)*e^2*(d + e*x)^4*(4*a^2*d^6*x + 36*a^2*d^4*e^2*x^3 + 24*a^2*d^2*e^4*x^5) + 24*c^(5/2)*e^2*(d +
e*x)^4*(12*a^3*d^4*e^2*x + 36*a^3*d^2*e^4*x^3 + 8*a^3*e^6*x^5)) + (a*c^3*d^2*ArcTan[(Sqrt[c]*d)/Sqrt[-(c*d^2)
- a*e^2] + (Sqrt[c]*e*x)/Sqrt[-(c*d^2) - a*e^2] - (e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(-(c*d^2) - a*e
^2)^(7/2) - (a^2*c^2*e^2*ArcTan[(Sqrt[c]*d)/Sqrt[-(c*d^2) - a*e^2] + (Sqrt[c]*e*x)/Sqrt[-(c*d^2) - a*e^2] - (e
*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/(4*(-(c*d^2) - a*e^2)^(7/2))
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fricas [B] time = 2.42, size = 1485, normalized size = 7.21
result too large to display
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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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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]
Timed out
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maple [B] time = 0.06, size = 2073, normalized size = 10.06
result too large to display
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^3*c^5*d^6/(a*e^2+c*d^2)^4/((a*e^2+c*d^2)/e^2)^(1/2)*ln((-2*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c
*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))+5/8*c^4*d^3/(a*e^2+c*d^2)^4*
(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(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+(x+d/e
)*c)/c^(1/2)+(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(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*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e
)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))+5/8/e^2*c^(9/2)*d^5/(a*e^2+c*d^2)^4*ln((-c*d/e+(x+d/e)*c)/c^(1/2)+(-2
*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))-5/8*c^3*d^3/(a*e^2+c*d^2)^4/(x+d/e)*(-2*(x+d/e)*c*d/e+(x+
d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-5/12/e^2*c*d/(a*e^2+c*d^2)^2/(x+d/e)^3*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+
c*d^2)/e^2)^(3/2)-5/8/e*c^2*d^2/(a*e^2+c*d^2)^3/(x+d/e)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/
2)-1/4/e^3/(a*e^2+c*d^2)/(x+d/e)^4*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)-5/8/e*c^4*d^4/(a*e^2
+c*d^2)^4*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)+3/4/e*c^3*d^2/(a*e^2+c*d^2)^3*(-2*(x+d/e)*c*d
/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)-1/8/e/(a*e^2+c*d^2)^2*c^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/
e^2)^(1/2)+1/8/(a*e^2+c*d^2)^3*c^2*d/(x+d/e)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)+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*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(
1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*a+5/8*c^(7/2)*d^3/(a*e^2+c*d^2)^4*ln((-c
*d/e+(x+d/e)*c)/c^(1/2)+(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(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*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c
*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*a-1/8/(a*e^2+c*d^2)^3*c^(5/2)*d*ln((-c*d/e+(x+d/e)*c)/c^(1
/2)+(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))*a-1/8/(a*e^2+c*d^2)^3*c^3*d*(-2*(x+d/e)*c*d/e+(x+d
/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)*x+1/8/e^2/(a*e^2+c*d^2)^2*c^(5/2)*d*ln((-c*d/e+(x+d/e)*c)/c^(1/2)+(-2*(x+d/e)
*c*d/e+(x+d/e)^2*c+(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*(x+d/e
)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2)
)/(x+d/e))*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*(x+d/e)*c*d/e+2*(a*e^2+c*d^2)/e^2+2*
((a*e^2+c*d^2)/e^2)^(1/2)*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))*d^2+1/8/e/(a*e^2+c*
d^2)^2*c/(x+d/e)^2*(-2*(x+d/e)*c*d/e+(x+d/e)^2*c+(a*e^2+c*d^2)/e^2)^(3/2)
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maxima [B] time = 2.16, size = 977, normalized size = 4.74 \begin {gather*} -\frac {5 \, \sqrt {c x^{2} + a} c^{3} d^{3}}{8 \, {\left (c^{3} d^{6} e^{2} x + 3 \, a c^{2} d^{4} e^{4} x + 3 \, a^{2} c d^{2} e^{6} x + a^{3} e^{8} x + c^{3} d^{7} e + 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )}} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d^{2}}{8 \, {\left (c^{3} d^{6} e x^{2} + 3 \, a c^{2} d^{4} e^{3} x^{2} + 3 \, a^{2} c d^{2} e^{5} x^{2} + a^{3} e^{7} x^{2} + 2 \, c^{3} d^{7} x + 6 \, a c^{2} d^{5} e^{2} x + 6 \, a^{2} c d^{3} e^{4} x + 2 \, a^{3} d e^{6} x + \frac {c^{3} d^{8}}{e} + 3 \, a c^{2} d^{6} e + 3 \, a^{2} c d^{4} e^{3} + a^{3} d^{2} e^{5}\right )}} + \frac {5 \, \sqrt {c x^{2} + a} c^{3} d^{2}}{8 \, {\left (c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )}} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c d}{12 \, {\left (c^{2} d^{4} e^{2} x^{3} + 2 \, a c d^{2} e^{4} x^{3} + a^{2} e^{6} x^{3} + 3 \, c^{2} d^{5} e x^{2} + 6 \, a c d^{3} e^{3} x^{2} + 3 \, a^{2} d e^{5} x^{2} + 3 \, c^{2} d^{6} x + 6 \, a c d^{4} e^{2} x + 3 \, a^{2} d^{2} e^{4} x + \frac {c^{2} d^{7}}{e} + 2 \, a c d^{5} e + a^{2} d^{3} e^{3}\right )}} + \frac {\sqrt {c x^{2} + a} c^{2} d}{8 \, {\left (c^{2} d^{4} e^{2} x + 2 \, a c d^{2} e^{4} x + a^{2} e^{6} x + c^{2} d^{5} e + 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c}{8 \, {\left (c^{2} d^{4} e x^{2} + 2 \, a c d^{2} e^{3} x^{2} + a^{2} e^{5} x^{2} + 2 \, c^{2} d^{5} x + 4 \, a c d^{3} e^{2} x + 2 \, a^{2} d e^{4} x + \frac {c^{2} d^{6}}{e} + 2 \, a c d^{4} e + a^{2} d^{2} e^{3}\right )}} - \frac {\sqrt {c x^{2} + a} c^{2}}{8 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{4 \, {\left (c d^{2} e^{3} x^{4} + a e^{5} x^{4} + 4 \, c d^{3} e^{2} x^{3} + 4 \, a d e^{4} x^{3} + 6 \, c d^{4} e x^{2} + 6 \, a d^{2} e^{3} x^{2} + 4 \, c d^{5} x + 4 \, a d^{3} e^{2} x + \frac {c d^{6}}{e} + a d^{4} e\right )}} - \frac {5 \, c^{4} d^{4} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{8 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {7}{2}} e^{9}} + \frac {3 \, c^{3} d^{2} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{4 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {5}{2}} e^{7}} - \frac {c^{2} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | e x + d \right |}} - \frac {a e}{\sqrt {a c} {\left | e x + d \right |}}\right )}{8 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{5}} \end {gather*}
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]
-5/8*sqrt(c*x^2 + a)*c^3*d^3/(c^3*d^6*e^2*x + 3*a*c^2*d^4*e^4*x + 3*a^2*c*d^2*e^6*x + a^3*e^8*x + c^3*d^7*e +
3*a*c^2*d^5*e^3 + 3*a^2*c*d^3*e^5 + a^3*d*e^7) - 5/8*(c*x^2 + a)^(3/2)*c^2*d^2/(c^3*d^6*e*x^2 + 3*a*c^2*d^4*e^
3*x^2 + 3*a^2*c*d^2*e^5*x^2 + a^3*e^7*x^2 + 2*c^3*d^7*x + 6*a*c^2*d^5*e^2*x + 6*a^2*c*d^3*e^4*x + 2*a^3*d*e^6*
x + c^3*d^8/e + 3*a*c^2*d^6*e + 3*a^2*c*d^4*e^3 + a^3*d^2*e^5) + 5/8*sqrt(c*x^2 + a)*c^3*d^2/(c^3*d^6*e + 3*a*
c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7) - 5/12*(c*x^2 + a)^(3/2)*c*d/(c^2*d^4*e^2*x^3 + 2*a*c*d^2*e^4*x^3 + a
^2*e^6*x^3 + 3*c^2*d^5*e*x^2 + 6*a*c*d^3*e^3*x^2 + 3*a^2*d*e^5*x^2 + 3*c^2*d^6*x + 6*a*c*d^4*e^2*x + 3*a^2*d^2
*e^4*x + c^2*d^7/e + 2*a*c*d^5*e + a^2*d^3*e^3) + 1/8*sqrt(c*x^2 + a)*c^2*d/(c^2*d^4*e^2*x + 2*a*c*d^2*e^4*x +
a^2*e^6*x + c^2*d^5*e + 2*a*c*d^3*e^3 + a^2*d*e^5) + 1/8*(c*x^2 + a)^(3/2)*c/(c^2*d^4*e*x^2 + 2*a*c*d^2*e^3*x
^2 + a^2*e^5*x^2 + 2*c^2*d^5*x + 4*a*c*d^3*e^2*x + 2*a^2*d*e^4*x + c^2*d^6/e + 2*a*c*d^4*e + a^2*d^2*e^3) - 1/
8*sqrt(c*x^2 + a)*c^2/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) - 1/4*(c*x^2 + a)^(3/2)/(c*d^2*e^3*x^4 + a*e^5*x^4
+ 4*c*d^3*e^2*x^3 + 4*a*d*e^4*x^3 + 6*c*d^4*e*x^2 + 6*a*d^2*e^3*x^2 + 4*c*d^5*x + 4*a*d^3*e^2*x + c*d^6/e + a
*d^4*e) - 5/8*c^4*d^4*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2)^
(7/2)*e^9) + 3/4*c^3*d^2*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^
2)^(5/2)*e^7) - 1/8*c^2*arcsinh(c*d*x/(sqrt(a*c)*abs(e*x + d)) - a*e/(sqrt(a*c)*abs(e*x + d)))/((a + c*d^2/e^2
)^(3/2)*e^5)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + c*x^2)^(1/2)/(d + e*x)^5,x)
[Out]
int((a + c*x^2)^(1/2)/(d + e*x)^5, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{5}}\, dx \end {gather*}
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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