\(\int \frac {1}{(d+e x)^4 (a+c x^2)^{3/2}} \, dx\) [576]

Optimal result

Integrand size = 19, antiderivative size = 293 \[ \int \frac {1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c e \left (6 c^2 d^4-83 a c d^2 e^2+16 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {5 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{9/2}} \]

[Out]

-5/2*c^2*d*e^2*(-3*a*e^2+4*c*d^2)*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)^(9/2
)+(c*d*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)^3/(c*x^2+a)^(1/2)+1/3*e*(-4*a*e^2+3*c*d^2)*(c*x^2+a)^(1/2)/a/(a*e^2+c*d^
2)^2/(e*x+d)^3+1/6*c*d*e*(-29*a*e^2+6*c*d^2)*(c*x^2+a)^(1/2)/a/(a*e^2+c*d^2)^3/(e*x+d)^2+1/6*c*e*(16*a^2*e^4-8
3*a*c*d^2*e^2+6*c^2*d^4)*(c*x^2+a)^(1/2)/a/(a*e^2+c*d^2)^4/(e*x+d)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {755, 849, 821, 739, 212} \[ \int \frac {1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx=\frac {c e \sqrt {a+c x^2} \left (16 a^2 e^4-83 a c d^2 e^2+6 c^2 d^4\right )}{6 a (d+e x) \left (a e^2+c d^2\right )^4}-\frac {5 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}}+\frac {c d e \sqrt {a+c x^2} \left (6 c d^2-29 a e^2\right )}{6 a (d+e x)^2 \left (a e^2+c d^2\right )^3}+\frac {e \sqrt {a+c x^2} \left (3 c d^2-4 a e^2\right )}{3 a (d+e x)^3 \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{a \sqrt {a+c x^2} (d+e x)^3 \left (a e^2+c d^2\right )} \]

[In]

Int[1/((d + e*x)^4*(a + c*x^2)^(3/2)),x]

[Out]

(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*(d + e*x)^3*Sqrt[a + c*x^2]) + (e*(3*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/(3*a*(
c*d^2 + a*e^2)^2*(d + e*x)^3) + (c*d*e*(6*c*d^2 - 29*a*e^2)*Sqrt[a + c*x^2])/(6*a*(c*d^2 + a*e^2)^3*(d + e*x)^
2) + (c*e*(6*c^2*d^4 - 83*a*c*d^2*e^2 + 16*a^2*e^4)*Sqrt[a + c*x^2])/(6*a*(c*d^2 + a*e^2)^4*(d + e*x)) - (5*c^
2*d*e^2*(4*c*d^2 - 3*a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + a*e^2)^(
9/2))

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 849

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)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rubi steps \begin{align*} \text {integral}& = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}-\frac {\int \frac {-4 a e^2-3 c d e x}{(d+e x)^4 \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {\int \frac {21 a c d e^2+2 c e \left (3 c d^2-4 a e^2\right ) x}{(d+e x)^3 \sqrt {a+c x^2}} \, dx}{3 a \left (c d^2+a e^2\right )^2} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {\int \frac {-2 a c e^2 \left (27 c d^2-8 a e^2\right )-c^2 d e \left (6 c d^2-29 a e^2\right ) x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{6 a \left (c d^2+a e^2\right )^3} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c e \left (6 c^2 d^4-83 a c d^2 e^2+16 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^4 (d+e x)}+\frac {\left (5 c^2 d e^2 \left (4 c d^2-3 a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^4} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c e \left (6 c^2 d^4-83 a c d^2 e^2+16 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {\left (5 c^2 d e^2 \left (4 c d^2-3 a e^2\right )\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 )}{2 \left (c d^2+a e^2\right )^4} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c e \left (6 c^2 d^4-83 a c d^2 e^2+16 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {5 c^2 d e^2 \left (4 c d^2-3 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 )}{2 \left (c d^2+a e^2\right )^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.69 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx=\frac {1}{6} \left (\frac {\sqrt {a+c x^2} \left (-\frac {2 e^3 \left (c d^2+a e^2\right )^2}{(d+e x)^3}-\frac {11 c d e^3 \left (c d^2+a e^2\right )}{(d+e x)^2}+\frac {c e^3 \left (-47 c d^2+10 a e^2\right )}{d+e x}+\frac {6 c^2 \left (c^2 d^4 x+2 a c d^2 e (2 d-3 e x)+a^2 e^3 (-4 d+e x)\right )}{a \left (a+c x^2\right )}\right )}{\left (c d^2+a e^2\right )^4}+\frac {15 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{9/2}}-\frac {15 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{9/2}}\right ) \]

[In]

Integrate[1/((d + e*x)^4*(a + c*x^2)^(3/2)),x]

[Out]

((Sqrt[a + c*x^2]*((-2*e^3*(c*d^2 + a*e^2)^2)/(d + e*x)^3 - (11*c*d*e^3*(c*d^2 + a*e^2))/(d + e*x)^2 + (c*e^3*
(-47*c*d^2 + 10*a*e^2))/(d + e*x) + (6*c^2*(c^2*d^4*x + 2*a*c*d^2*e*(2*d - 3*e*x) + a^2*e^3*(-4*d + e*x)))/(a*
(a + c*x^2))))/(c*d^2 + a*e^2)^4 + (15*c^2*d*e^2*(4*c*d^2 - 3*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(9/2) - (15
*c^2*d*e^2*(4*c*d^2 - 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(9/2))/
6

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1550\) vs. \(2(271)=542\).

Time = 2.23 (sec) , antiderivative size = 1551, normalized size of antiderivative = 5.29

method	result	size
default	\(\text {Expression too large to display}\)	\(1551\)

[In]

int(1/(e*x+d)^4/(c*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1100 vs. \(2 (272) = 544\).

Time = 1.14 (sec) , antiderivative size = 2226, normalized size of antiderivative = 7.60 \[ \int \frac {1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^4/(c*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

[1/12*(15*(4*a^2*c^3*d^6*e^2 - 3*a^3*c^2*d^4*e^4 + (4*a*c^4*d^3*e^5 - 3*a^2*c^3*d*e^7)*x^5 + 3*(4*a*c^4*d^4*e^
4 - 3*a^2*c^3*d^2*e^6)*x^4 + (12*a*c^4*d^5*e^3 - 5*a^2*c^3*d^3*e^5 - 3*a^3*c^2*d*e^7)*x^3 + (4*a*c^4*d^6*e^2 +
9*a^2*c^3*d^4*e^4 - 9*a^3*c^2*d^2*e^6)*x^2 + 3*(4*a^2*c^3*d^5*e^3 - 3*a^3*c^2*d^3*e^5)*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*(24*a*c^4*d^8*e - 60*a^2*c^3*d^6*e^3 - 89*a^3*c^2*d^4*e^5 - 7*a^4*
c*d^2*e^7 - 2*a^5*e^9 + (6*c^5*d^6*e^3 - 77*a*c^4*d^4*e^5 - 67*a^2*c^3*d^2*e^7 + 16*a^3*c^2*e^9)*x^4 + 3*(6*c^
5*d^7*e^2 - 57*a*c^4*d^5*e^4 - 62*a^2*c^3*d^3*e^6 + a^3*c^2*d*e^8)*x^3 + 2*(9*c^5*d^8*e - 39*a*c^4*d^6*e^3 - 1
01*a^2*c^3*d^4*e^5 - 49*a^3*c^2*d^2*e^7 + 4*a^4*c*e^9)*x^2 + 3*(2*c^5*d^9 + 14*a*c^4*d^7*e^2 - 45*a^2*c^3*d^5*
e^4 - 54*a^3*c^2*d^3*e^6 + 3*a^4*c*d*e^8)*x)*sqrt(c*x^2 + a))/(a^2*c^5*d^13 + 5*a^3*c^4*d^11*e^2 + 10*a^4*c^3*
d^9*e^4 + 10*a^5*c^2*d^7*e^6 + 5*a^6*c*d^5*e^8 + a^7*d^3*e^10 + (a*c^6*d^10*e^3 + 5*a^2*c^5*d^8*e^5 + 10*a^3*c
^4*d^6*e^7 + 10*a^4*c^3*d^4*e^9 + 5*a^5*c^2*d^2*e^11 + a^6*c*e^13)*x^5 + 3*(a*c^6*d^11*e^2 + 5*a^2*c^5*d^9*e^4
+ 10*a^3*c^4*d^7*e^6 + 10*a^4*c^3*d^5*e^8 + 5*a^5*c^2*d^3*e^10 + a^6*c*d*e^12)*x^4 + (3*a*c^6*d^12*e + 16*a^2
*c^5*d^10*e^3 + 35*a^3*c^4*d^8*e^5 + 40*a^4*c^3*d^6*e^7 + 25*a^5*c^2*d^4*e^9 + 8*a^6*c*d^2*e^11 + a^7*e^13)*x^
3 + (a*c^6*d^13 + 8*a^2*c^5*d^11*e^2 + 25*a^3*c^4*d^9*e^4 + 40*a^4*c^3*d^7*e^6 + 35*a^5*c^2*d^5*e^8 + 16*a^6*c
*d^3*e^10 + 3*a^7*d*e^12)*x^2 + 3*(a^2*c^5*d^12*e + 5*a^3*c^4*d^10*e^3 + 10*a^4*c^3*d^8*e^5 + 10*a^5*c^2*d^6*e
^7 + 5*a^6*c*d^4*e^9 + a^7*d^2*e^11)*x), -1/6*(15*(4*a^2*c^3*d^6*e^2 - 3*a^3*c^2*d^4*e^4 + (4*a*c^4*d^3*e^5 -
3*a^2*c^3*d*e^7)*x^5 + 3*(4*a*c^4*d^4*e^4 - 3*a^2*c^3*d^2*e^6)*x^4 + (12*a*c^4*d^5*e^3 - 5*a^2*c^3*d^3*e^5 - 3
*a^3*c^2*d*e^7)*x^3 + (4*a*c^4*d^6*e^2 + 9*a^2*c^3*d^4*e^4 - 9*a^3*c^2*d^2*e^6)*x^2 + 3*(4*a^2*c^3*d^5*e^3 - 3
*a^3*c^2*d^3*e^5)*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)) - (24*a*c^4*d^8*e - 60*a^2*c^3*d^6*e^3 - 89*a^3*c^2*d^4*e^5 - 7*a^4*c*d^2
*e^7 - 2*a^5*e^9 + (6*c^5*d^6*e^3 - 77*a*c^4*d^4*e^5 - 67*a^2*c^3*d^2*e^7 + 16*a^3*c^2*e^9)*x^4 + 3*(6*c^5*d^7
*e^2 - 57*a*c^4*d^5*e^4 - 62*a^2*c^3*d^3*e^6 + a^3*c^2*d*e^8)*x^3 + 2*(9*c^5*d^8*e - 39*a*c^4*d^6*e^3 - 101*a^
2*c^3*d^4*e^5 - 49*a^3*c^2*d^2*e^7 + 4*a^4*c*e^9)*x^2 + 3*(2*c^5*d^9 + 14*a*c^4*d^7*e^2 - 45*a^2*c^3*d^5*e^4 -
54*a^3*c^2*d^3*e^6 + 3*a^4*c*d*e^8)*x)*sqrt(c*x^2 + a))/(a^2*c^5*d^13 + 5*a^3*c^4*d^11*e^2 + 10*a^4*c^3*d^9*e
^4 + 10*a^5*c^2*d^7*e^6 + 5*a^6*c*d^5*e^8 + a^7*d^3*e^10 + (a*c^6*d^10*e^3 + 5*a^2*c^5*d^8*e^5 + 10*a^3*c^4*d^
6*e^7 + 10*a^4*c^3*d^4*e^9 + 5*a^5*c^2*d^2*e^11 + a^6*c*e^13)*x^5 + 3*(a*c^6*d^11*e^2 + 5*a^2*c^5*d^9*e^4 + 10
*a^3*c^4*d^7*e^6 + 10*a^4*c^3*d^5*e^8 + 5*a^5*c^2*d^3*e^10 + a^6*c*d*e^12)*x^4 + (3*a*c^6*d^12*e + 16*a^2*c^5*
d^10*e^3 + 35*a^3*c^4*d^8*e^5 + 40*a^4*c^3*d^6*e^7 + 25*a^5*c^2*d^4*e^9 + 8*a^6*c*d^2*e^11 + a^7*e^13)*x^3 + (
a*c^6*d^13 + 8*a^2*c^5*d^11*e^2 + 25*a^3*c^4*d^9*e^4 + 40*a^4*c^3*d^7*e^6 + 35*a^5*c^2*d^5*e^8 + 16*a^6*c*d^3*
e^10 + 3*a^7*d*e^12)*x^2 + 3*(a^2*c^5*d^12*e + 5*a^3*c^4*d^10*e^3 + 10*a^4*c^3*d^8*e^5 + 10*a^5*c^2*d^6*e^7 +
5*a^6*c*d^4*e^9 + a^7*d^2*e^11)*x)]

Sympy [F]

\[ \int \frac {1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{4}}\, dx \]

[In]

integrate(1/(e*x+d)**4/(c*x**2+a)**(3/2),x)

[Out]

Integral(1/((a + c*x**2)**(3/2)*(d + e*x)**4), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1195 vs. \(2 (272) = 544\).

Time = 0.25 (sec) , antiderivative size = 1195, normalized size of antiderivative = 4.08 \[ \int \frac {1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx=\frac {35 \, c^{4} d^{4} x}{2 \, {\left (\sqrt {c x^{2} + a} a c^{4} d^{8} + 4 \, \sqrt {c x^{2} + a} a^{2} c^{3} d^{6} e^{2} + 6 \, \sqrt {c x^{2} + a} a^{3} c^{2} d^{4} e^{4} + 4 \, \sqrt {c x^{2} + a} a^{4} c d^{2} e^{6} + \sqrt {c x^{2} + a} a^{5} e^{8}\right )}} + \frac {35 \, c^{3} d^{3}}{2 \, {\left (\frac {\sqrt {c x^{2} + a} c^{4} d^{8}}{e} + 4 \, \sqrt {c x^{2} + a} a c^{3} d^{6} e + 6 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} e^{3} + 4 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{5} + \sqrt {c x^{2} + a} a^{4} e^{7}\right )}} - \frac {115 \, c^{3} d^{2} x}{6 \, {\left (\sqrt {c x^{2} + a} a c^{3} d^{6} + 3 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} e^{2} + 3 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{4} + \sqrt {c x^{2} + a} a^{4} e^{6}\right )}} - \frac {35 \, c^{2} d^{2}}{6 \, {\left (\sqrt {c x^{2} + a} c^{3} d^{6} x + 3 \, \sqrt {c x^{2} + a} a c^{2} d^{4} e^{2} x + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{4} x + \sqrt {c x^{2} + a} a^{3} e^{6} x + \frac {\sqrt {c x^{2} + a} c^{3} d^{7}}{e} + 3 \, \sqrt {c x^{2} + a} a c^{2} d^{5} e + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{3} e^{3} + \sqrt {c x^{2} + a} a^{3} d e^{5}\right )}} - \frac {15 \, c^{2} d}{2 \, {\left (\frac {\sqrt {c x^{2} + a} c^{3} d^{6}}{e} + 3 \, \sqrt {c x^{2} + a} a c^{2} d^{4} e + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{3} + \sqrt {c x^{2} + a} a^{3} e^{5}\right )}} + \frac {8 \, c^{2} x}{3 \, {\left (\sqrt {c x^{2} + a} a c^{2} d^{4} + 2 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{3} e^{4}\right )}} - \frac {7 \, c d}{6 \, {\left (\sqrt {c x^{2} + a} c^{2} d^{4} e x^{2} + 2 \, \sqrt {c x^{2} + a} a c d^{2} e^{3} x^{2} + \sqrt {c x^{2} + a} a^{2} e^{5} x^{2} + 2 \, \sqrt {c x^{2} + a} c^{2} d^{5} x + 4 \, \sqrt {c x^{2} + a} a c d^{3} e^{2} x + 2 \, \sqrt {c x^{2} + a} a^{2} d e^{4} x + \frac {\sqrt {c x^{2} + a} c^{2} d^{6}}{e} + 2 \, \sqrt {c x^{2} + a} a c d^{4} e + \sqrt {c x^{2} + a} a^{2} d^{2} e^{3}\right )}} + \frac {4 \, c}{3 \, {\left (\sqrt {c x^{2} + a} c^{2} d^{4} x + 2 \, \sqrt {c x^{2} + a} a c d^{2} e^{2} x + \sqrt {c x^{2} + a} a^{2} e^{4} x + \frac {\sqrt {c x^{2} + a} c^{2} d^{5}}{e} + 2 \, \sqrt {c x^{2} + a} a c d^{3} e + \sqrt {c x^{2} + a} a^{2} d e^{3}\right )}} - \frac {1}{3 \, {\left (\sqrt {c x^{2} + a} c d^{2} e^{2} x^{3} + \sqrt {c x^{2} + a} a e^{4} x^{3} + 3 \, \sqrt {c x^{2} + a} c d^{3} e x^{2} + 3 \, \sqrt {c x^{2} + a} a d e^{3} x^{2} + 3 \, \sqrt {c x^{2} + a} c d^{4} x + 3 \, \sqrt {c x^{2} + a} a d^{2} e^{2} x + \frac {\sqrt {c x^{2} + a} c d^{5}}{e} + \sqrt {c x^{2} + a} a d^{3} e\right )}} + \frac {35 \, c^{3} d^{3} \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 )}{2 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {9}{2}} e^{7}} - \frac {15 \, c^{2} 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 )}{2 \, {\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {7}{2}} e^{5}} \]

[In]

integrate(1/(e*x+d)^4/(c*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

35/2*c^4*d^4*x/(sqrt(c*x^2 + a)*a*c^4*d^8 + 4*sqrt(c*x^2 + a)*a^2*c^3*d^6*e^2 + 6*sqrt(c*x^2 + a)*a^3*c^2*d^4*
e^4 + 4*sqrt(c*x^2 + a)*a^4*c*d^2*e^6 + sqrt(c*x^2 + a)*a^5*e^8) + 35/2*c^3*d^3/(sqrt(c*x^2 + a)*c^4*d^8/e + 4
*sqrt(c*x^2 + a)*a*c^3*d^6*e + 6*sqrt(c*x^2 + a)*a^2*c^2*d^4*e^3 + 4*sqrt(c*x^2 + a)*a^3*c*d^2*e^5 + sqrt(c*x^
2 + a)*a^4*e^7) - 115/6*c^3*d^2*x/(sqrt(c*x^2 + a)*a*c^3*d^6 + 3*sqrt(c*x^2 + a)*a^2*c^2*d^4*e^2 + 3*sqrt(c*x^
2 + a)*a^3*c*d^2*e^4 + sqrt(c*x^2 + a)*a^4*e^6) - 35/6*c^2*d^2/(sqrt(c*x^2 + a)*c^3*d^6*x + 3*sqrt(c*x^2 + a)*
a*c^2*d^4*e^2*x + 3*sqrt(c*x^2 + a)*a^2*c*d^2*e^4*x + sqrt(c*x^2 + a)*a^3*e^6*x + sqrt(c*x^2 + a)*c^3*d^7/e +
3*sqrt(c*x^2 + a)*a*c^2*d^5*e + 3*sqrt(c*x^2 + a)*a^2*c*d^3*e^3 + sqrt(c*x^2 + a)*a^3*d*e^5) - 15/2*c^2*d/(sqr
t(c*x^2 + a)*c^3*d^6/e + 3*sqrt(c*x^2 + a)*a*c^2*d^4*e + 3*sqrt(c*x^2 + a)*a^2*c*d^2*e^3 + sqrt(c*x^2 + a)*a^3
*e^5) + 8/3*c^2*x/(sqrt(c*x^2 + a)*a*c^2*d^4 + 2*sqrt(c*x^2 + a)*a^2*c*d^2*e^2 + sqrt(c*x^2 + a)*a^3*e^4) - 7/
6*c*d/(sqrt(c*x^2 + a)*c^2*d^4*e*x^2 + 2*sqrt(c*x^2 + a)*a*c*d^2*e^3*x^2 + sqrt(c*x^2 + a)*a^2*e^5*x^2 + 2*sqr
t(c*x^2 + a)*c^2*d^5*x + 4*sqrt(c*x^2 + a)*a*c*d^3*e^2*x + 2*sqrt(c*x^2 + a)*a^2*d*e^4*x + sqrt(c*x^2 + a)*c^2
*d^6/e + 2*sqrt(c*x^2 + a)*a*c*d^4*e + sqrt(c*x^2 + a)*a^2*d^2*e^3) + 4/3*c/(sqrt(c*x^2 + a)*c^2*d^4*x + 2*sqr
t(c*x^2 + a)*a*c*d^2*e^2*x + sqrt(c*x^2 + a)*a^2*e^4*x + sqrt(c*x^2 + a)*c^2*d^5/e + 2*sqrt(c*x^2 + a)*a*c*d^3
*e + sqrt(c*x^2 + a)*a^2*d*e^3) - 1/3/(sqrt(c*x^2 + a)*c*d^2*e^2*x^3 + sqrt(c*x^2 + a)*a*e^4*x^3 + 3*sqrt(c*x^
2 + a)*c*d^3*e*x^2 + 3*sqrt(c*x^2 + a)*a*d*e^3*x^2 + 3*sqrt(c*x^2 + a)*c*d^4*x + 3*sqrt(c*x^2 + a)*a*d^2*e^2*x
+ sqrt(c*x^2 + a)*c*d^5/e + sqrt(c*x^2 + a)*a*d^3*e) + 35/2*c^3*d^3*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)^(9/2)*e^7) - 15/2*c^2*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^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1075 vs. \(2 (272) = 544\).

Time = 0.35 (sec) , antiderivative size = 1075, normalized size of antiderivative = 3.67 \[ \int \frac {1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx=\frac {\frac {{\left (c^{8} d^{12} - 2 \, a c^{7} d^{10} e^{2} - 17 \, a^{2} c^{6} d^{8} e^{4} - 28 \, a^{3} c^{5} d^{6} e^{6} - 17 \, a^{4} c^{4} d^{4} e^{8} - 2 \, a^{5} c^{3} d^{2} e^{10} + a^{6} c^{2} e^{12}\right )} x}{a c^{8} d^{16} + 8 \, a^{2} c^{7} d^{14} e^{2} + 28 \, a^{3} c^{6} d^{12} e^{4} + 56 \, a^{4} c^{5} d^{10} e^{6} + 70 \, a^{5} c^{4} d^{8} e^{8} + 56 \, a^{6} c^{3} d^{6} e^{10} + 28 \, a^{7} c^{2} d^{4} e^{12} + 8 \, a^{8} c d^{2} e^{14} + a^{9} e^{16}} + \frac {4 \, {\left (a c^{7} d^{11} e + 3 \, a^{2} c^{6} d^{9} e^{3} + 2 \, a^{3} c^{5} d^{7} e^{5} - 2 \, a^{4} c^{4} d^{5} e^{7} - 3 \, a^{5} c^{3} d^{3} e^{9} - a^{6} c^{2} d e^{11}\right )}}{a c^{8} d^{16} + 8 \, a^{2} c^{7} d^{14} e^{2} + 28 \, a^{3} c^{6} d^{12} e^{4} + 56 \, a^{4} c^{5} d^{10} e^{6} + 70 \, a^{5} c^{4} d^{8} e^{8} + 56 \, a^{6} c^{3} d^{6} e^{10} + 28 \, a^{7} c^{2} d^{4} e^{12} + 8 \, a^{8} c d^{2} e^{14} + a^{9} e^{16}}}{\sqrt {c x^{2} + a}} + \frac {5 \, {\left (4 \, c^{3} d^{3} e^{2} - 3 \, a c^{2} d e^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{3} d^{3} e^{4} - 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d e^{6} + 162 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{\frac {7}{2}} d^{4} e^{3} - 117 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {5}{2}} d^{2} e^{5} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} e^{7} + 188 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{4} d^{5} e^{2} - 322 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{4} + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{6} - 402 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {7}{2}} d^{4} e^{3} + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c^{\frac {5}{2}} d^{2} e^{5} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} e^{7} + 246 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{4} - 39 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d e^{6} - 47 \, a^{3} c^{\frac {5}{2}} d^{2} e^{5} + 10 \, a^{4} c^{\frac {3}{2}} e^{7}}{3 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{3}} \]

[In]

integrate(1/(e*x+d)^4/(c*x^2+a)^(3/2),x, algorithm="giac")

[Out]

((c^8*d^12 - 2*a*c^7*d^10*e^2 - 17*a^2*c^6*d^8*e^4 - 28*a^3*c^5*d^6*e^6 - 17*a^4*c^4*d^4*e^8 - 2*a^5*c^3*d^2*e
^10 + a^6*c^2*e^12)*x/(a*c^8*d^16 + 8*a^2*c^7*d^14*e^2 + 28*a^3*c^6*d^12*e^4 + 56*a^4*c^5*d^10*e^6 + 70*a^5*c^
4*d^8*e^8 + 56*a^6*c^3*d^6*e^10 + 28*a^7*c^2*d^4*e^12 + 8*a^8*c*d^2*e^14 + a^9*e^16) + 4*(a*c^7*d^11*e + 3*a^2
*c^6*d^9*e^3 + 2*a^3*c^5*d^7*e^5 - 2*a^4*c^4*d^5*e^7 - 3*a^5*c^3*d^3*e^9 - a^6*c^2*d*e^11)/(a*c^8*d^16 + 8*a^2
*c^7*d^14*e^2 + 28*a^3*c^6*d^12*e^4 + 56*a^4*c^5*d^10*e^6 + 70*a^5*c^4*d^8*e^8 + 56*a^6*c^3*d^6*e^10 + 28*a^7*
c^2*d^4*e^12 + 8*a^8*c*d^2*e^14 + a^9*e^16))/sqrt(c*x^2 + a) + 5*(4*c^3*d^3*e^2 - 3*a*c^2*d*e^4)*arctan(-((sqr
t(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4
+ 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(-c*d^2 - a*e^2)) - 1/3*(36*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*d^3*e^4 - 21
*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d*e^6 + 162*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d^4*e^3 - 117*(sqrt
(c)*x - sqrt(c*x^2 + a))^4*a*c^(5/2)*d^2*e^5 + 6*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*e^7 + 188*(sqrt(c
)*x - sqrt(c*x^2 + a))^3*c^4*d^5*e^2 - 322*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^3*d^3*e^4 + 60*(sqrt(c)*x - sqr
t(c*x^2 + a))^3*a^2*c^2*d*e^6 - 402*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d^4*e^3 + 144*(sqrt(c)*x - sqrt(
c*x^2 + a))^2*a^2*c^(5/2)*d^2*e^5 - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(3/2)*e^7 + 246*(sqrt(c)*x - sqrt
(c*x^2 + a))*a^2*c^3*d^3*e^4 - 39*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*e^6 - 47*a^3*c^(5/2)*d^2*e^5 + 10*a^
4*c^(3/2)*e^7)/((c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*((sqrt(c)*x - sqrt
(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^4} \,d x \]

[In]

int(1/((a + c*x^2)^(3/2)*(d + e*x)^4),x)

[Out]

int(1/((a + c*x^2)^(3/2)*(d + e*x)^4), x)