### 3.584 $$\int \frac{1}{(d+e x)^3 (a+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=327 $\frac{c d e \sqrt{a+c x^2} \left (-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{e \sqrt{a+c x^2} \left (-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{a e \left (2 c d^2-5 a e^2\right )-c d x \left (9 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{5 c e^4 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}}$

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)^2*(a + c*x^2)^(3/2)) - (a*e*(2*c*d^2 - 5*a*e^2) - c*d*(2*c*d^2 +
9*a*e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)^2*Sqrt[a + c*x^2]) + (e*(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^
4)*Sqrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^3*(d + e*x)^2) + (c*d*e*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*a^2*e^4)*S
qrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^4*(d + e*x)) - (5*c*e^4*(6*c*d^2 - 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 = 0.371794, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.316, Rules used = {741, 823, 835, 807, 725, 206} $\frac{c d e \sqrt{a+c x^2} \left (-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{e \sqrt{a+c x^2} \left (-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{a e \left (2 c d^2-5 a e^2\right )-c d x \left (9 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{5 c e^4 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^3*(a + c*x^2)^(5/2)),x]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)^2*(a + c*x^2)^(3/2)) - (a*e*(2*c*d^2 - 5*a*e^2) - c*d*(2*c*d^2 +
9*a*e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)^2*Sqrt[a + c*x^2]) + (e*(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^
4)*Sqrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^3*(d + e*x)^2) + (c*d*e*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*a^2*e^4)*S
qrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^4*(d + e*x)) - (5*c*e^4*(6*c*d^2 - 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 823

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] + Di
st[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])

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)^3 \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)^2 \left (a+c x^2\right )^{3/2}}-\frac{\int \frac{-2 c d^2-5 a e^2-4 c d e x}{(d+e x)^3 \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)^2 \left (a+c x^2\right )^{3/2}}-\frac{a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt{a+c x^2}}+\frac{\int \frac{-3 a c e^2 \left (2 c d^2-5 a e^2\right )+2 c^2 d e \left (2 c d^2+9 a e^2\right ) x}{(d+e x)^3 \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)^2 \left (a+c x^2\right )^{3/2}}-\frac{a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt{a+c x^2}}+\frac{e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt{a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{\int \frac{2 a c^2 d e^2 \left (2 c d^2-33 a e^2\right )-c^2 e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) x}{(d+e x)^2 \sqrt{a+c x^2}} \, dx}{6 a^2 c \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)^2 \left (a+c x^2\right )^{3/2}}-\frac{a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt{a+c x^2}}+\frac{e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt{a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac{c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt{a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac{\left (5 c e^4 \left (6 c d^2-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}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac{a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt{a+c x^2}}+\frac{e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt{a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac{c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt{a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}-\frac{\left (5 c e^4 \left (6 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 )}{2 \left (c d^2+a e^2\right )^4}\\ &=\frac{a e+c d x}{3 a \left (c d^2+a e^2\right ) (d+e x)^2 \left (a+c x^2\right )^{3/2}}-\frac{a e \left (2 c d^2-5 a e^2\right )-c d \left (2 c d^2+9 a e^2\right ) x}{3 a^2 \left (c d^2+a e^2\right )^2 (d+e x)^2 \sqrt{a+c x^2}}+\frac{e \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) \sqrt{a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac{c d e \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) \sqrt{a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^4 (d+e x)}-\frac{5 c e^4 \left (6 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 )}{2 \left (c d^2+a e^2\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.901353, size = 296, normalized size = 0.91 $\frac{1}{6} \left (\frac{\sqrt{a+c x^2} \left (\frac{4 c \left (3 a^2 c d e^3 (5 d-4 e x)-3 a^3 e^5+7 a c^2 d^3 e^2 x+c^3 d^5 x\right )}{a^2 \left (a+c x^2\right )}+\frac{2 c \left (a e^2+c d^2\right ) \left (-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right )}{a \left (a+c x^2\right )^2}-\frac{3 e^5 \left (a e^2+c d^2\right )}{(d+e x)^2}-\frac{33 c d e^5}{d+e x}\right )}{\left (a e^2+c d^2\right )^4}+\frac{15 c e^4 \left (a e^2-6 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{9/2}}+\frac{15 c e^4 \left (6 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{9/2}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^3*(a + c*x^2)^(5/2)),x]

[Out]

((Sqrt[a + c*x^2]*((-3*e^5*(c*d^2 + a*e^2))/(d + e*x)^2 - (33*c*d*e^5)/(d + e*x) + (4*c*(-3*a^3*e^5 + c^3*d^5*
x + 7*a*c^2*d^3*e^2*x + 3*a^2*c*d*e^3*(5*d - 4*e*x)))/(a^2*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)*(-(a^2*e^3) + c
^2*d^3*x + 3*a*c*d*e*(d - e*x)))/(a*(a + c*x^2)^2)))/(c*d^2 + a*e^2)^4 + (15*c*e^4*(6*c*d^2 - a*e^2)*Log[d + e
*x])/(c*d^2 + a*e^2)^(9/2) + (15*c*e^4*(-6*c*d^2 + 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 = 0.199, size = 1088, normalized size = 3.3 \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)^3/(c*x^2+a)^(5/2),x)

[Out]

-1/2/e/(a*e^2+c*d^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)^(3/2)-7/2*c*d/(a*e^2+c*d^2)^2/(
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)+35/6*e*c^2*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)^(3/2)+35/6*c^3*d^3/(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)^(3/2)*x+35/3*c^3*d^3/(a*e^2+c*d^2)^3/a^2/(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^3
*c^2*d^2/(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*e^2*c^3*d^3/(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^3*c^2*d^2/(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))-11/2*c^2*d/(a*e^2+c*d^2)^2/a/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3
/2)*x-11*c^2*d/(a*e^2+c*d^2)^2/a^2/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-5/6*e/(a*e^2+c*d^2)
^2*c/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-5/2*e^3/(a*e^2+c*d^2)^3*c/(c*(d/e+x)^2-2*c*d/e*(d/e
+x)+(a*e^2+c*d^2)/e^2)^(1/2)-5/2*e^2/(a*e^2+c*d^2)^3*c^2*d/a/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(
1/2)*x+5/2*e^3/(a*e^2+c*d^2)^3*c/((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))

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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)^3/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 13.7516, size = 5269, normalized size = 16.11 \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)^3/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(15*(6*a^4*c^2*d^4*e^4 - a^5*c*d^2*e^6 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*e^8)*x^6 + 2*(6*a^2*c^4*d^3*e^5 -
a^3*c^3*d*e^7)*x^5 + (6*a^2*c^4*d^4*e^4 + 11*a^3*c^3*d^2*e^6 - 2*a^4*c^2*e^8)*x^4 + 4*(6*a^3*c^3*d^3*e^5 - a^4
*c^2*d*e^7)*x^3 + (12*a^3*c^3*d^4*e^4 + 4*a^4*c^2*d^2*e^6 - a^5*c*e^8)*x^2 + 2*(6*a^4*c^2*d^3*e^5 - a^5*c*d*e^
7)*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*(6*a^2*c^4*d^8*e + 70*a^3*c^3*d^6*e^3 + 1
4*a^4*c^2*d^4*e^5 - 53*a^5*c*d^2*e^7 - 3*a^6*e^9 + (4*c^6*d^7*e^2 + 32*a*c^5*d^5*e^4 - 53*a^2*c^4*d^3*e^6 - 81
*a^3*c^3*d*e^8)*x^5 + (8*c^6*d^8*e + 64*a*c^5*d^6*e^3 - 16*a^2*c^4*d^4*e^5 - 87*a^3*c^3*d^2*e^7 - 15*a^4*c^2*e
^9)*x^4 + 2*(2*c^6*d^9 + 19*a*c^5*d^7*e^2 + 65*a^2*c^4*d^5*e^4 - 24*a^3*c^3*d^3*e^6 - 72*a^4*c^2*d*e^8)*x^3 +
2*(6*a*c^5*d^8*e + 63*a^2*c^4*d^6*e^3 - 7*a^3*c^3*d^4*e^5 - 74*a^4*c^2*d^2*e^7 - 10*a^5*c*e^9)*x^2 + (6*a*c^5*
d^9 + 42*a^2*c^4*d^7*e^2 + 110*a^3*c^3*d^5*e^4 + 13*a^4*c^2*d^3*e^6 - 61*a^5*c*d*e^8)*x)*sqrt(c*x^2 + a))/(a^4
*c^5*d^12 + 5*a^5*c^4*d^10*e^2 + 10*a^6*c^3*d^8*e^4 + 10*a^7*c^2*d^6*e^6 + 5*a^8*c*d^4*e^8 + a^9*d^2*e^10 + (a
^2*c^7*d^10*e^2 + 5*a^3*c^6*d^8*e^4 + 10*a^4*c^5*d^6*e^6 + 10*a^5*c^4*d^4*e^8 + 5*a^6*c^3*d^2*e^10 + a^7*c^2*e
^12)*x^6 + 2*(a^2*c^7*d^11*e + 5*a^3*c^6*d^9*e^3 + 10*a^4*c^5*d^7*e^5 + 10*a^5*c^4*d^5*e^7 + 5*a^6*c^3*d^3*e^9
+ a^7*c^2*d*e^11)*x^5 + (a^2*c^7*d^12 + 7*a^3*c^6*d^10*e^2 + 20*a^4*c^5*d^8*e^4 + 30*a^5*c^4*d^6*e^6 + 25*a^6
*c^3*d^4*e^8 + 11*a^7*c^2*d^2*e^10 + 2*a^8*c*e^12)*x^4 + 4*(a^3*c^6*d^11*e + 5*a^4*c^5*d^9*e^3 + 10*a^5*c^4*d^
7*e^5 + 10*a^6*c^3*d^5*e^7 + 5*a^7*c^2*d^3*e^9 + a^8*c*d*e^11)*x^3 + (2*a^3*c^6*d^12 + 11*a^4*c^5*d^10*e^2 + 2
5*a^5*c^4*d^8*e^4 + 30*a^6*c^3*d^6*e^6 + 20*a^7*c^2*d^4*e^8 + 7*a^8*c*d^2*e^10 + a^9*e^12)*x^2 + 2*(a^4*c^5*d^
11*e + 5*a^5*c^4*d^9*e^3 + 10*a^6*c^3*d^7*e^5 + 10*a^7*c^2*d^5*e^7 + 5*a^8*c*d^3*e^9 + a^9*d*e^11)*x), -1/6*(1
5*(6*a^4*c^2*d^4*e^4 - a^5*c*d^2*e^6 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*e^8)*x^6 + 2*(6*a^2*c^4*d^3*e^5 - a^3*c^3*
d*e^7)*x^5 + (6*a^2*c^4*d^4*e^4 + 11*a^3*c^3*d^2*e^6 - 2*a^4*c^2*e^8)*x^4 + 4*(6*a^3*c^3*d^3*e^5 - a^4*c^2*d*e
^7)*x^3 + (12*a^3*c^3*d^4*e^4 + 4*a^4*c^2*d^2*e^6 - a^5*c*e^8)*x^2 + 2*(6*a^4*c^2*d^3*e^5 - a^5*c*d*e^7)*x)*sq
rt(-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)) - (6*a^2*c^4*d^8*e + 70*a^3*c^3*d^6*e^3 + 14*a^4*c^2*d^4*e^5 - 53*a^5*c*d^2*e^7 - 3*a^6*e^9 + (4
*c^6*d^7*e^2 + 32*a*c^5*d^5*e^4 - 53*a^2*c^4*d^3*e^6 - 81*a^3*c^3*d*e^8)*x^5 + (8*c^6*d^8*e + 64*a*c^5*d^6*e^3
- 16*a^2*c^4*d^4*e^5 - 87*a^3*c^3*d^2*e^7 - 15*a^4*c^2*e^9)*x^4 + 2*(2*c^6*d^9 + 19*a*c^5*d^7*e^2 + 65*a^2*c^
4*d^5*e^4 - 24*a^3*c^3*d^3*e^6 - 72*a^4*c^2*d*e^8)*x^3 + 2*(6*a*c^5*d^8*e + 63*a^2*c^4*d^6*e^3 - 7*a^3*c^3*d^4
*e^5 - 74*a^4*c^2*d^2*e^7 - 10*a^5*c*e^9)*x^2 + (6*a*c^5*d^9 + 42*a^2*c^4*d^7*e^2 + 110*a^3*c^3*d^5*e^4 + 13*a
^4*c^2*d^3*e^6 - 61*a^5*c*d*e^8)*x)*sqrt(c*x^2 + a))/(a^4*c^5*d^12 + 5*a^5*c^4*d^10*e^2 + 10*a^6*c^3*d^8*e^4 +
10*a^7*c^2*d^6*e^6 + 5*a^8*c*d^4*e^8 + a^9*d^2*e^10 + (a^2*c^7*d^10*e^2 + 5*a^3*c^6*d^8*e^4 + 10*a^4*c^5*d^6*
e^6 + 10*a^5*c^4*d^4*e^8 + 5*a^6*c^3*d^2*e^10 + a^7*c^2*e^12)*x^6 + 2*(a^2*c^7*d^11*e + 5*a^3*c^6*d^9*e^3 + 10
*a^4*c^5*d^7*e^5 + 10*a^5*c^4*d^5*e^7 + 5*a^6*c^3*d^3*e^9 + a^7*c^2*d*e^11)*x^5 + (a^2*c^7*d^12 + 7*a^3*c^6*d^
10*e^2 + 20*a^4*c^5*d^8*e^4 + 30*a^5*c^4*d^6*e^6 + 25*a^6*c^3*d^4*e^8 + 11*a^7*c^2*d^2*e^10 + 2*a^8*c*e^12)*x^
4 + 4*(a^3*c^6*d^11*e + 5*a^4*c^5*d^9*e^3 + 10*a^5*c^4*d^7*e^5 + 10*a^6*c^3*d^5*e^7 + 5*a^7*c^2*d^3*e^9 + a^8*
c*d*e^11)*x^3 + (2*a^3*c^6*d^12 + 11*a^4*c^5*d^10*e^2 + 25*a^5*c^4*d^8*e^4 + 30*a^6*c^3*d^6*e^6 + 20*a^7*c^2*d
^4*e^8 + 7*a^8*c*d^2*e^10 + a^9*e^12)*x^2 + 2*(a^4*c^5*d^11*e + 5*a^5*c^4*d^9*e^3 + 10*a^6*c^3*d^7*e^5 + 10*a^
7*c^2*d^5*e^7 + 5*a^8*c*d^3*e^9 + a^9*d*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{5}{2}} \left (d + e x\right )^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(c*x**2+a)**(5/2),x)

[Out]

Integral(1/((a + c*x**2)**(5/2)*(d + e*x)**3), x)

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Giac [B]  time = 2.57672, size = 2759, normalized size = 8.44 \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)^3/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

5*(6*c^2*d^2*e^4 - a*c*e^6)*arctan(-((sqrt(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*((2*((c^18*
d^29 + 19*a*c^17*d^27*e^2 + 138*a^2*c^16*d^25*e^4 + 538*a^3*c^15*d^23*e^6 + 1243*a^4*c^14*d^21*e^8 + 1617*a^5*
c^13*d^19*e^10 + 528*a^6*c^12*d^17*e^12 - 2244*a^7*c^11*d^15*e^14 - 5049*a^8*c^10*d^13*e^16 - 5819*a^9*c^9*d^1
1*e^18 - 4334*a^10*c^8*d^9*e^20 - 2166*a^11*c^7*d^7*e^22 - 707*a^12*c^6*d^5*e^24 - 137*a^13*c^5*d^3*e^26 - 12*
a^14*c^4*d*e^28)*x/(a^2*c^17*d^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 182
0*a^6*c^13*d^24*e^8 + 4368*a^7*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^1
0*c^9*d^16*e^16 + 11440*a^11*c^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5
*d^8*e^24 + 560*a^15*c^4*d^6*e^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32) + 3*(5*a^2*c^1
6*d^26*e^3 + 59*a^3*c^15*d^24*e^5 + 318*a^4*c^14*d^22*e^7 + 1034*a^5*c^13*d^20*e^9 + 2255*a^6*c^12*d^18*e^11 +
3465*a^7*c^11*d^16*e^13 + 3828*a^8*c^10*d^14*e^15 + 3036*a^9*c^9*d^12*e^17 + 1683*a^10*c^8*d^10*e^19 + 605*a^
11*c^7*d^8*e^21 + 110*a^12*c^6*d^6*e^23 - 6*a^13*c^5*d^4*e^25 - 7*a^14*c^4*d^2*e^27 - a^15*c^3*e^29)/(a^2*c^17
*d^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 1820*a^6*c^13*d^24*e^8 + 4368*a
^7*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^10*c^9*d^16*e^16 + 11440*a^11
*c^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5*d^8*e^24 + 560*a^15*c^4*d^6
*e^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32))*x + 3*(a*c^17*d^29 + 16*a^2*c^16*d^27*e^2
+ 105*a^3*c^15*d^25*e^4 + 376*a^4*c^14*d^23*e^6 + 781*a^5*c^13*d^21*e^8 + 792*a^6*c^12*d^19*e^10 - 363*a^7*c^
11*d^17*e^12 - 2640*a^8*c^10*d^15*e^14 - 4653*a^9*c^9*d^13*e^16 - 4928*a^10*c^8*d^11*e^18 - 3509*a^11*c^7*d^9*
e^20 - 1704*a^12*c^6*d^7*e^22 - 545*a^13*c^5*d^5*e^24 - 104*a^14*c^4*d^3*e^26 - 9*a^15*c^3*d*e^28)/(a^2*c^17*d
^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 1820*a^6*c^13*d^24*e^8 + 4368*a^7
*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^10*c^9*d^16*e^16 + 11440*a^11*c
^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5*d^8*e^24 + 560*a^15*c^4*d^6*e
^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32))*x + (3*a^2*c^16*d^28*e + 68*a^3*c^15*d^26*e
^3 + 575*a^4*c^14*d^24*e^5 + 2688*a^5*c^13*d^22*e^7 + 8063*a^6*c^12*d^20*e^9 + 16676*a^7*c^11*d^18*e^11 + 2465
1*a^8*c^10*d^16*e^13 + 26400*a^9*c^9*d^14*e^15 + 20361*a^10*c^8*d^12*e^17 + 10956*a^11*c^7*d^10*e^19 + 3773*a^
12*c^6*d^8*e^21 + 608*a^13*c^5*d^6*e^23 - 75*a^14*c^4*d^4*e^25 - 52*a^15*c^3*d^2*e^27 - 7*a^16*c^2*e^29)/(a^2*
c^17*d^32 + 16*a^3*c^16*d^30*e^2 + 120*a^4*c^15*d^28*e^4 + 560*a^5*c^14*d^26*e^6 + 1820*a^6*c^13*d^24*e^8 + 43
68*a^7*c^12*d^22*e^10 + 8008*a^8*c^11*d^20*e^12 + 11440*a^9*c^10*d^18*e^14 + 12870*a^10*c^9*d^16*e^16 + 11440*
a^11*c^8*d^14*e^18 + 8008*a^12*c^7*d^12*e^20 + 4368*a^13*c^6*d^10*e^22 + 1820*a^14*c^5*d^8*e^24 + 560*a^15*c^4
*d^6*e^26 + 120*a^16*c^3*d^4*e^28 + 16*a^17*c^2*d^2*e^30 + a^18*c*e^32))/(c*x^2 + a)^(3/2) - (22*(sqrt(c)*x -
sqrt(c*x^2 + a))^2*c^(5/2)*d^3*e^4 + 10*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^2*d^2*e^5 - 34*(sqrt(c)*x - sqrt(c*x
^2 + a))*a*c^2*d^2*e^5 - 11*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^(3/2)*d*e^6 - (sqrt(c)*x - sqrt(c*x^2 + a))^3*
a*c*e^7 + 11*a^2*c^(3/2)*d*e^6 - (sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c*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)^2)