### 3.583 $$\int \frac{1}{(d+e x)^2 (a+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=244 $\frac{e \sqrt{a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )}-\frac{5 c d e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}}$

[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)

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Rubi [A]  time = 0.233254, antiderivative size = 244, normalized size of antiderivative = 1., 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 = {741, 823, 807, 725, 206} $\frac{e \sqrt{a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )}-\frac{5 c d e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}}$

Antiderivative was successfully veriﬁed.

[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 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 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)^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{\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 ) \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 )}{\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]  time = 0.298793, size = 237, normalized size = 0.97 $\frac{a^2 c^2 e \left (21 d^2 e^2 x^2+11 d^3 e x+2 d^4+7 d e^3 x^3-8 e^4 x^4\right )+2 a^3 c e^3 \left (7 d^2+4 d e x-6 e^2 x^2\right )-3 a^4 e^5+3 a c^3 d^2 x \left (d^2 e x+d^3+3 d e^2 x^2+3 e^3 x^3\right )+2 c^4 d^4 x^3 (d+e x)}{3 a^2 \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )^3}-\frac{5 c d e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}}$

Antiderivative was successfully veriﬁed.

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

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Maple [B]  time = 0.2, size = 667, normalized size = 2.7 \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)^2/(c*x^2+a)^(5/2),x)

[Out]

-1/(a*e^2+c*d^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)+5/3*e*c*d/(a*e^2+c*d^2)^2/(c*(d
/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)+5/3*c^2*d^2/(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+10/3*c^2*d^2/(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*e^3*c*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)+5*e^2*c^2*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-5*e^3*c*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)*c/a/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x-8/3/(a
*e^2+c*d^2)*c/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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 6.18523, size = 3310, normalized size = 13.57 \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)^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)]

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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 )^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out