3.576 $$\int \frac{1}{(d+e x)^4 (a+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=293 $\frac{c e \sqrt{a+c x^2} \left (16 a^2 e^4-83 a c d^2 e^2+6 c^2 d^4\right )}{6 a (d+e x) \left (a e^2+c d^2\right )^4}-\frac{5 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}}+\frac{c d e \sqrt{a+c x^2} \left (6 c d^2-29 a e^2\right )}{6 a (d+e x)^2 \left (a e^2+c d^2\right )^3}+\frac{e \sqrt{a+c x^2} \left (3 c d^2-4 a e^2\right )}{3 a (d+e x)^3 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^3 \left (a e^2+c d^2\right )}$

[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))

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Rubi [A]  time = 0.373861, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.263, Rules used = {741, 835, 807, 725, 206} $\frac{c e \sqrt{a+c x^2} \left (16 a^2 e^4-83 a c d^2 e^2+6 c^2 d^4\right )}{6 a (d+e x) \left (a e^2+c d^2\right )^4}-\frac{5 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}}+\frac{c d e \sqrt{a+c x^2} \left (6 c d^2-29 a e^2\right )}{6 a (d+e x)^2 \left (a e^2+c d^2\right )^3}+\frac{e \sqrt{a+c x^2} \left (3 c d^2-4 a e^2\right )}{3 a (d+e x)^3 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^3 \left (a e^2+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[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 741

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 835

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])

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 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{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{\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 ) \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 )}{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]  time = 0.651784, size = 279, normalized size = 0.95 $\frac{1}{6} \left (\frac{\sqrt{a+c x^2} \left (\frac{6 c^2 \left (a^2 e^3 (e x-4 d)+2 a c d^2 e (2 d-3 e x)+c^2 d^4 x\right )}{a \left (a+c x^2\right )}+\frac{c e^3 \left (10 a e^2-47 c d^2\right )}{d+e x}-\frac{11 c d e^3 \left (a e^2+c d^2\right )}{(d+e x)^2}-\frac{2 e^3 \left (a e^2+c d^2\right )^2}{(d+e x)^3}\right )}{\left (a e^2+c d^2\right )^4}-\frac{15 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{9/2}}+\frac{15 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{9/2}}\right )$

Antiderivative was successfully veriﬁed.

[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

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Maple [B]  time = 0.201, size = 898, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 18.5289, size = 4508, normalized size = 15.39 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.96978, size = 1384, normalized size = 4.72 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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*(188*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^4*d^5*e^2 + 1
62*(sqrt(c)*x - sqrt(c*x^2 + a))^4*c^(7/2)*d^4*e^3 + 36*(sqrt(c)*x - sqrt(c*x^2 + a))^5*c^3*d^3*e^4 - 402*(sqr
t(c)*x - sqrt(c*x^2 + a))^2*a*c^(7/2)*d^4*e^3 - 322*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^3*d^3*e^4 - 117*(sqrt(
c)*x - sqrt(c*x^2 + a))^4*a*c^(5/2)*d^2*e^5 - 21*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d*e^6 + 246*(sqrt(c)*x
- sqrt(c*x^2 + a))*a^2*c^3*d^3*e^4 + 144*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c^(5/2)*d^2*e^5 + 60*(sqrt(c)*x -
sqrt(c*x^2 + a))^3*a^2*c^2*d*e^6 + 6*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(3/2)*e^7 - 47*a^3*c^(5/2)*d^2*e^5
- 39*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*c^2*d*e^6 - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*c^(3/2)*e^7 + 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)