\(\int \frac {1}{(d+e x)^2 (a+c x^2)^{5/2}} \, dx\) [583]

Optimal result

Integrand size = 19, antiderivative size = 244 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{5/2}} \, dx=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {5 c d e^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{7/2}} \]

[Out]

1/3*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)/(c*x^2+a)^(3/2)-5*c*d*e^4*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*
x^2+a)^(1/2))/(a*e^2+c*d^2)^(7/2)+1/3*(-a*e*(-4*a*e^2+c*d^2)+c*d*(7*a*e^2+2*c*d^2)*x)/a^2/(a*e^2+c*d^2)^2/(e*x
+d)/(c*x^2+a)^(1/2)+1/3*e*(-8*a^2*e^4+9*a*c*d^2*e^2+2*c^2*d^4)*(c*x^2+a)^(1/2)/a^2/(a*e^2+c*d^2)^3/(e*x+d)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {755, 837, 821, 739, 212} \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{5/2}} \, dx=\frac {e \sqrt {a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac {a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt {a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}-\frac {5 c d e^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )} \]

[In]

Int[1/((d + e*x)^2*(a + c*x^2)^(5/2)),x]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^(3/2)) - (a*e*(c*d^2 - 4*a*e^2) - c*d*(2*c*d^2 + 7*a*
e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)*Sqrt[a + c*x^2]) + (e*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4)*Sqrt[
a + c*x^2])/(3*a^2*(c*d^2 + a*e^2)^3*(d + e*x)) - (5*c*d*e^4*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a
+ c*x^2])])/(c*d^2 + a*e^2)^(7/2)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

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 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 821

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 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps \begin{align*} \text {integral}& = \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-2 \left (c d^2+2 a e^2\right )-3 c d e x}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx}{3 a \left (c d^2+a e^2\right )} \\ & = \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {\int \frac {-2 a c e^2 \left (c d^2-4 a e^2\right )+c^2 d e \left (2 c d^2+7 a e^2\right ) x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )^2} \\ & = \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {\left (5 c d e^4\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{\left (c d^2+a e^2\right )^3} \\ & = \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {\left (5 c d e^4\right ) \text {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 )}{\left (c d^2+a e^2\right )^3} \\ & = \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-4 a e^2\right )-c d \left (2 c d^2+7 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \sqrt {a+c x^2}}+\frac {e \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right ) \sqrt {a+c x^2}}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {5 c d e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{5/2}} \, dx=\frac {-3 a^4 e^5+2 c^4 d^4 x^3 (d+e x)+2 a^3 c e^3 \left (7 d^2+4 d e x-6 e^2 x^2\right )+3 a c^3 d^2 x \left (d^3+d^2 e x+3 d e^2 x^2+3 e^3 x^3\right )+a^2 c^2 e \left (2 d^4+11 d^3 e x+21 d^2 e^2 x^2+7 d e^3 x^3-8 e^4 x^4\right )}{3 a^2 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )^{3/2}}+\frac {10 c d e^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{7/2}} \]

[In]

Integrate[1/((d + e*x)^2*(a + c*x^2)^(5/2)),x]

[Out]

(-3*a^4*e^5 + 2*c^4*d^4*x^3*(d + e*x) + 2*a^3*c*e^3*(7*d^2 + 4*d*e*x - 6*e^2*x^2) + 3*a*c^3*d^2*x*(d^3 + d^2*e
*x + 3*d*e^2*x^2 + 3*e^3*x^3) + a^2*c^2*e*(2*d^4 + 11*d^3*e*x + 21*d^2*e^2*x^2 + 7*d*e^3*x^3 - 8*e^4*x^4))/(3*
a^2*(c*d^2 + a*e^2)^3*(d + e*x)*(a + c*x^2)^(3/2)) + (10*c*d*e^4*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2]
)/Sqrt[-(c*d^2) - a*e^2]])/(-(c*d^2) - a*e^2)^(7/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(898\) vs. \(2(226)=452\).

Time = 2.52 (sec) , antiderivative size = 899, normalized size of antiderivative = 3.68

method	result	size
default	\(\frac {-\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {5 c d e \left (\frac {e^{2}}{3 \left (e^{2} a +c \,d^{2}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {c d e \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}-\frac {4 c d}{3 e}}{\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{3 {\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right )}^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}+\frac {e^{2} \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 c d e \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}\right )}{e^{2} a +c \,d^{2}}-\frac {4 c \,e^{2} \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}-\frac {4 c d}{3 e}}{\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{3 {\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right )}^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}}{e^{2}}\)	\(899\)

[In]

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

[Out]

1/e^2*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+5*c*d*e/(a*e^2+c*d^2
)*(1/3/(a*e^2+c*d^2)*e^2/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+c*d*e/(a*e^2+c*d^2)*(2/3*(2*c*(
x+d/e)-2*c*d/e)/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(3/2)+16
/3*c/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)^2*(2*c*(x+d/e)-2*c*d/e)/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/
e^2)^(1/2))+1/(a*e^2+c*d^2)*e^2*(1/(a*e^2+c*d^2)*e^2/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+2*c
*d*e/(a*e^2+c*d^2)*(2*c*(x+d/e)-2*c*d/e)/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a
*e^2+c*d^2)/e^2)^(1/2)-1/(a*e^2+c*d^2)*e^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*(x+d/e)+2
*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))))-4*c/(a*e^2+c*d^2)
*e^2*(2/3*(2*c*(x+d/e)-2*c*d/e)/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)/(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^
2)/e^2)^(3/2)+16/3*c/(4*c*(a*e^2+c*d^2)/e^2-4*c^2*d^2/e^2)^2*(2*c*(x+d/e)-2*c*d/e)/(c*(x+d/e)^2-2*c*d/e*(x+d/e
)+(a*e^2+c*d^2)/e^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 826 vs. \(2 (227) = 454\).

Time = 0.54 (sec) , antiderivative size = 1678, normalized size of antiderivative = 6.88 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/6*(15*(a^2*c^3*d*e^5*x^5 + a^2*c^3*d^2*e^4*x^4 + 2*a^3*c^2*d*e^5*x^3 + 2*a^3*c^2*d^2*e^4*x^2 + a^4*c*d*e^5*
x + a^4*c*d^2*e^4)*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*(2*a^2*c^3*d^6*e + 16*a^3*c^
2*d^4*e^3 + 11*a^4*c*d^2*e^5 - 3*a^5*e^7 + (2*c^5*d^6*e + 11*a*c^4*d^4*e^3 + a^2*c^3*d^2*e^5 - 8*a^3*c^2*e^7)*
x^4 + (2*c^5*d^7 + 11*a*c^4*d^5*e^2 + 16*a^2*c^3*d^3*e^4 + 7*a^3*c^2*d*e^6)*x^3 + 3*(a*c^4*d^6*e + 8*a^2*c^3*d
^4*e^3 + 3*a^3*c^2*d^2*e^5 - 4*a^4*c*e^7)*x^2 + (3*a*c^4*d^7 + 14*a^2*c^3*d^5*e^2 + 19*a^3*c^2*d^3*e^4 + 8*a^4
*c*d*e^6)*x)*sqrt(c*x^2 + a))/(a^4*c^4*d^9 + 4*a^5*c^3*d^7*e^2 + 6*a^6*c^2*d^5*e^4 + 4*a^7*c*d^3*e^6 + a^8*d*e
^8 + (a^2*c^6*d^8*e + 4*a^3*c^5*d^6*e^3 + 6*a^4*c^4*d^4*e^5 + 4*a^5*c^3*d^2*e^7 + a^6*c^2*e^9)*x^5 + (a^2*c^6*
d^9 + 4*a^3*c^5*d^7*e^2 + 6*a^4*c^4*d^5*e^4 + 4*a^5*c^3*d^3*e^6 + a^6*c^2*d*e^8)*x^4 + 2*(a^3*c^5*d^8*e + 4*a^
4*c^4*d^6*e^3 + 6*a^5*c^3*d^4*e^5 + 4*a^6*c^2*d^2*e^7 + a^7*c*e^9)*x^3 + 2*(a^3*c^5*d^9 + 4*a^4*c^4*d^7*e^2 +
6*a^5*c^3*d^5*e^4 + 4*a^6*c^2*d^3*e^6 + a^7*c*d*e^8)*x^2 + (a^4*c^4*d^8*e + 4*a^5*c^3*d^6*e^3 + 6*a^6*c^2*d^4*
e^5 + 4*a^7*c*d^2*e^7 + a^8*e^9)*x), -1/3*(15*(a^2*c^3*d*e^5*x^5 + a^2*c^3*d^2*e^4*x^4 + 2*a^3*c^2*d*e^5*x^3 +
2*a^3*c^2*d^2*e^4*x^2 + a^4*c*d*e^5*x + a^4*c*d^2*e^4)*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)) - (2*a^2*c^3*d^6*e + 16*a^3*c^2*d^4*e^
3 + 11*a^4*c*d^2*e^5 - 3*a^5*e^7 + (2*c^5*d^6*e + 11*a*c^4*d^4*e^3 + a^2*c^3*d^2*e^5 - 8*a^3*c^2*e^7)*x^4 + (2
*c^5*d^7 + 11*a*c^4*d^5*e^2 + 16*a^2*c^3*d^3*e^4 + 7*a^3*c^2*d*e^6)*x^3 + 3*(a*c^4*d^6*e + 8*a^2*c^3*d^4*e^3 +
3*a^3*c^2*d^2*e^5 - 4*a^4*c*e^7)*x^2 + (3*a*c^4*d^7 + 14*a^2*c^3*d^5*e^2 + 19*a^3*c^2*d^3*e^4 + 8*a^4*c*d*e^6
)*x)*sqrt(c*x^2 + a))/(a^4*c^4*d^9 + 4*a^5*c^3*d^7*e^2 + 6*a^6*c^2*d^5*e^4 + 4*a^7*c*d^3*e^6 + a^8*d*e^8 + (a^
2*c^6*d^8*e + 4*a^3*c^5*d^6*e^3 + 6*a^4*c^4*d^4*e^5 + 4*a^5*c^3*d^2*e^7 + a^6*c^2*e^9)*x^5 + (a^2*c^6*d^9 + 4*
a^3*c^5*d^7*e^2 + 6*a^4*c^4*d^5*e^4 + 4*a^5*c^3*d^3*e^6 + a^6*c^2*d*e^8)*x^4 + 2*(a^3*c^5*d^8*e + 4*a^4*c^4*d^
6*e^3 + 6*a^5*c^3*d^4*e^5 + 4*a^6*c^2*d^2*e^7 + a^7*c*e^9)*x^3 + 2*(a^3*c^5*d^9 + 4*a^4*c^4*d^7*e^2 + 6*a^5*c^
3*d^5*e^4 + 4*a^6*c^2*d^3*e^6 + a^7*c*d*e^8)*x^2 + (a^4*c^4*d^8*e + 4*a^5*c^3*d^6*e^3 + 6*a^6*c^2*d^4*e^5 + 4*
a^7*c*d^2*e^7 + a^8*e^9)*x)]

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (227) = 454\).

Time = 0.25 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{5/2}} \, dx=\frac {5 \, c^{2} d^{2} x}{3 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} a c^{2} d^{4} + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} c d^{2} e^{2} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3} e^{4}\right )}} + \frac {5 \, c^{2} d^{2} x}{3 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} + \frac {\sqrt {c x^{2} + a} a c^{3} d^{6}}{e^{2}} + 3 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{4} e^{4}} + \frac {10 \, c^{2} d^{2} x}{3 \, {\left (\sqrt {c x^{2} + a} a^{2} c^{2} d^{4} + 2 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{4} e^{4}\right )}} + \frac {5 \, c d}{\frac {\sqrt {c x^{2} + a} c^{3} d^{6}}{e^{3}} + \frac {3 \, \sqrt {c x^{2} + a} a c^{2} d^{4}}{e} + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e + \sqrt {c x^{2} + a} a^{3} e^{3}} + \frac {5 \, c d}{3 \, {\left (\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2} d^{4}}{e} + 2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a c d^{2} e + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{3}\right )}} - \frac {4 \, c x}{3 \, {\left ({\left (c x^{2} + a\right )}^{\frac {3}{2}} a c d^{2} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{2}\right )}} - \frac {8 \, c x}{3 \, {\left (\sqrt {c x^{2} + a} a^{2} c d^{2} + \sqrt {c x^{2} + a} a^{3} e^{2}\right )}} - \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{2} x + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a e^{2} x + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c d^{3}}{e} + {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d e} + \frac {5 \, c d \operatorname {arsinh}\left (\frac {c d x}{e \sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}} - \frac {a}{\sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {7}{2}} e^{3}} \]

[In]

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

[Out]

5/3*c^2*d^2*x/((c*x^2 + a)^(3/2)*a*c^2*d^4 + 2*(c*x^2 + a)^(3/2)*a^2*c*d^2*e^2 + (c*x^2 + a)^(3/2)*a^3*e^4) +
5*c^2*d^2*x/(3*sqrt(c*x^2 + a)*a^2*c^2*d^4 + sqrt(c*x^2 + a)*a*c^3*d^6/e^2 + 3*sqrt(c*x^2 + a)*a^3*c*d^2*e^2 +
sqrt(c*x^2 + a)*a^4*e^4) + 10/3*c^2*d^2*x/(sqrt(c*x^2 + a)*a^2*c^2*d^4 + 2*sqrt(c*x^2 + a)*a^3*c*d^2*e^2 + sq
rt(c*x^2 + a)*a^4*e^4) + 5*c*d/(sqrt(c*x^2 + a)*c^3*d^6/e^3 + 3*sqrt(c*x^2 + a)*a*c^2*d^4/e + 3*sqrt(c*x^2 + a
)*a^2*c*d^2*e + sqrt(c*x^2 + a)*a^3*e^3) + 5/3*c*d/((c*x^2 + a)^(3/2)*c^2*d^4/e + 2*(c*x^2 + a)^(3/2)*a*c*d^2*
e + (c*x^2 + a)^(3/2)*a^2*e^3) - 4/3*c*x/((c*x^2 + a)^(3/2)*a*c*d^2 + (c*x^2 + a)^(3/2)*a^2*e^2) - 8/3*c*x/(sq
rt(c*x^2 + a)*a^2*c*d^2 + sqrt(c*x^2 + a)*a^3*e^2) - 1/((c*x^2 + a)^(3/2)*c*d^2*x + (c*x^2 + a)^(3/2)*a*e^2*x
+ (c*x^2 + a)^(3/2)*c*d^3/e + (c*x^2 + a)^(3/2)*a*d*e) + 5*c*d*arcsinh(c*d*x/(e*sqrt(a*c/e^2)*abs(e*x + d)) -
a/(sqrt(a*c/e^2)*abs(e*x + d)))/((a + c*d^2/e^2)^(7/2)*e^3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1180 vs. \(2 (227) = 454\).

Time = 0.46 (sec) , antiderivative size = 1180, normalized size of antiderivative = 4.84 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/3*(15*c*d*e^7*log(abs(-c*d*e + sqrt(c*d^2 + a*e^2)*(sqrt(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + a*e^2/(e
*x + d)^2) + sqrt(c*d^2*e^2 + a*e^4)/((e*x + d)*e))*abs(e)))/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a
^3*e^6)*sqrt(c*d^2 + a*e^2)*abs(e)*sgn(1/(e*x + d))*sgn(e)) - (15*a^2*c^(3/2)*d*e^7*log(abs(-c*d*e + sqrt(c*d^
2 + a*e^2)*sqrt(c)*abs(e))) - 2*sqrt(c*d^2 + a*e^2)*c^3*d^4*e^2*abs(e) - 9*sqrt(c*d^2 + a*e^2)*a*c^2*d^2*e^4*a
bs(e) + 8*sqrt(c*d^2 + a*e^2)*a^2*c*e^6*abs(e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(c*d^2 + a*e^2)*a^2*c^(7/2)*d^6*a
bs(e) + 3*sqrt(c*d^2 + a*e^2)*a^3*c^(5/2)*d^4*e^2*abs(e) + 3*sqrt(c*d^2 + a*e^2)*a^4*c^(3/2)*d^2*e^4*abs(e) +
sqrt(c*d^2 + a*e^2)*a^5*sqrt(c)*e^6*abs(e)) - ((2*c^5*d^4*e^13 + 9*a*c^4*d^2*e^15 - 8*a^2*c^3*e^17)/(a^2*c^4*d
^6*e^11*sgn(1/(e*x + d))*sgn(e) + 3*a^3*c^3*d^4*e^13*sgn(1/(e*x + d))*sgn(e) + 3*a^4*c^2*d^2*e^15*sgn(1/(e*x +
d))*sgn(e) + a^5*c*e^17*sgn(1/(e*x + d))*sgn(e)) - (3*(2*c^5*d^5*e^14 + 9*a*c^4*d^3*e^16 - 13*a^2*c^3*d*e^18)
/(a^2*c^4*d^6*e^11*sgn(1/(e*x + d))*sgn(e) + 3*a^3*c^3*d^4*e^13*sgn(1/(e*x + d))*sgn(e) + 3*a^4*c^2*d^2*e^15*s
gn(1/(e*x + d))*sgn(e) + a^5*c*e^17*sgn(1/(e*x + d))*sgn(e)) - (6*(c^5*d^6*e^15 + 5*a*c^4*d^4*e^17 - 8*a^2*c^3
*d^2*e^19 - 2*a^3*c^2*e^21)/(a^2*c^4*d^6*e^11*sgn(1/(e*x + d))*sgn(e) + 3*a^3*c^3*d^4*e^13*sgn(1/(e*x + d))*sg
n(e) + 3*a^4*c^2*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + a^5*c*e^17*sgn(1/(e*x + d))*sgn(e)) - (2*(c^5*d^7*e^16 + 6
*a*c^4*d^5*e^18 - 11*a^2*c^3*d^3*e^20 - 16*a^3*c^2*d*e^22)/(a^2*c^4*d^6*e^11*sgn(1/(e*x + d))*sgn(e) + 3*a^3*c
^3*d^4*e^13*sgn(1/(e*x + d))*sgn(e) + 3*a^4*c^2*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + a^5*c*e^17*sgn(1/(e*x + d))
*sgn(e)) + 3*(a^2*c^3*d^4*e^21 + 2*a^3*c^2*d^2*e^23 + a^4*c*e^25)/((a^2*c^4*d^6*e^11*sgn(1/(e*x + d))*sgn(e) +
3*a^3*c^3*d^4*e^13*sgn(1/(e*x + d))*sgn(e) + 3*a^4*c^2*d^2*e^15*sgn(1/(e*x + d))*sgn(e) + a^5*c*e^17*sgn(1/(e
*x + d))*sgn(e))*(e*x + d)*e))/((e*x + d)*e))/((e*x + d)*e))/((e*x + d)*e))/(c - 2*c*d/(e*x + d) + c*d^2/(e*x
+ d)^2 + a*e^2/(e*x + d)^2)^(3/2))/e^2

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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