3.1822 \(\int \frac{A+B x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=250 \[ \frac{5 e^2 (a B e-7 A b e+6 b B d)}{8 b \sqrt{d+e x} (b d-a e)^4}-\frac{5 e^2 (a B e-7 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{9/2}}+\frac{5 e (a B e-7 A b e+6 b B d)}{24 b (a+b x) \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-7 A b e+6 b B d}{12 b (a+b x)^2 \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 \sqrt{d+e x} (b d-a e)} \]

[Out]

(5*e^2*(6*b*B*d - 7*A*b*e + a*B*e))/(8*b*(b*d - a*e)^4*Sqrt[d + e*x]) - (A*b - a*B)/(3*b*(b*d - a*e)*(a + b*x)
^3*Sqrt[d + e*x]) - (6*b*B*d - 7*A*b*e + a*B*e)/(12*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]) + (5*e*(6*b*B*d
- 7*A*b*e + a*B*e))/(24*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]) - (5*e^2*(6*b*B*d - 7*A*b*e + a*B*e)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*(b*d - a*e)^(9/2))

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Rubi [A]  time = 0.241646, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {27, 78, 51, 63, 208} \[ \frac{5 e^2 (a B e-7 A b e+6 b B d)}{8 b \sqrt{d+e x} (b d-a e)^4}-\frac{5 e^2 (a B e-7 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{9/2}}+\frac{5 e (a B e-7 A b e+6 b B d)}{24 b (a+b x) \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-7 A b e+6 b B d}{12 b (a+b x)^2 \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

(5*e^2*(6*b*B*d - 7*A*b*e + a*B*e))/(8*b*(b*d - a*e)^4*Sqrt[d + e*x]) - (A*b - a*B)/(3*b*(b*d - a*e)*(a + b*x)
^3*Sqrt[d + e*x]) - (6*b*B*d - 7*A*b*e + a*B*e)/(12*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]) + (5*e*(6*b*B*d
- 7*A*b*e + a*B*e))/(24*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]) - (5*e^2*(6*b*B*d - 7*A*b*e + a*B*e)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*Sqrt[b]*(b*d - a*e)^(9/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{A+B x}{(a+b x)^4 (d+e x)^{3/2}} \, dx\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt{d+e x}}+\frac{(6 b B d-7 A b e+a B e) \int \frac{1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{6 b (b d-a e)}\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt{d+e x}}-\frac{6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}-\frac{(5 e (6 b B d-7 A b e+a B e)) \int \frac{1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{24 b (b d-a e)^2}\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt{d+e x}}-\frac{6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}+\frac{5 e (6 b B d-7 A b e+a B e)}{24 b (b d-a e)^3 (a+b x) \sqrt{d+e x}}+\frac{\left (5 e^2 (6 b B d-7 A b e+a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 b (b d-a e)^3}\\ &=\frac{5 e^2 (6 b B d-7 A b e+a B e)}{8 b (b d-a e)^4 \sqrt{d+e x}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt{d+e x}}-\frac{6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}+\frac{5 e (6 b B d-7 A b e+a B e)}{24 b (b d-a e)^3 (a+b x) \sqrt{d+e x}}+\frac{\left (5 e^2 (6 b B d-7 A b e+a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 (b d-a e)^4}\\ &=\frac{5 e^2 (6 b B d-7 A b e+a B e)}{8 b (b d-a e)^4 \sqrt{d+e x}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt{d+e x}}-\frac{6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}+\frac{5 e (6 b B d-7 A b e+a B e)}{24 b (b d-a e)^3 (a+b x) \sqrt{d+e x}}+\frac{(5 e (6 b B d-7 A b e+a B e)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{8 (b d-a e)^4}\\ &=\frac{5 e^2 (6 b B d-7 A b e+a B e)}{8 b (b d-a e)^4 \sqrt{d+e x}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt{d+e x}}-\frac{6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x}}+\frac{5 e (6 b B d-7 A b e+a B e)}{24 b (b d-a e)^3 (a+b x) \sqrt{d+e x}}-\frac{5 e^2 (6 b B d-7 A b e+a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{b} (b d-a e)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0640666, size = 99, normalized size = 0.4 \[ \frac{\frac{a B-A b}{(a+b x)^3}-\frac{e^2 (-a B e+7 A b e-6 b B d) \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}}{3 b \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

((-(A*b) + a*B)/(a + b*x)^3 - (e^2*(-6*b*B*d + 7*A*b*e - a*B*e)*Hypergeometric2F1[-1/2, 3, 1/2, (b*(d + e*x))/
(b*d - a*e)])/(b*d - a*e)^3)/(3*b*(b*d - a*e)*Sqrt[d + e*x])

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Maple [B]  time = 0.032, size = 768, 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((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-2*e^3/(a*e-b*d)^4/(e*x+d)^(1/2)*A+2*e^2/(a*e-b*d)^4/(e*x+d)^(1/2)*B*d-19/8*e^3/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x
+d)^(5/2)*A*b^3+7/4*e^2/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*b^3*d+5/8*e^3/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x
+d)^(5/2)*B*a*b^2-17/3*e^4/(a*e-b*d)^4/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a*b^2+17/3*e^3/(a*e-b*d)^4/(b*e*x+a*e)^3*
A*(e*x+d)^(3/2)*b^3*d+5/3*e^4/(a*e-b*d)^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^2*b+7/3*e^3/(a*e-b*d)^4/(b*e*x+a*e)^
3*B*(e*x+d)^(3/2)*a*b^2*d-4*e^2/(a*e-b*d)^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*b^3*d^2-29/8*e^5/(a*e-b*d)^4/(b*e*x+
a*e)^3*(e*x+d)^(1/2)*A*a^2*b+29/4*e^4/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*b^2*d-29/8*e^3/(a*e-b*d)^4/(
b*e*x+a*e)^3*(e*x+d)^(1/2)*A*b^3*d^2+11/8*e^5/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3-1/2*e^4/(a*e-b*d)^
4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^2*b*d-25/8*e^3/(a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*b^2*d^2+9/4*e^2/(
a*e-b*d)^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*b^3*d^3-35/8*e^3/(a*e-b*d)^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)
*b/((a*e-b*d)*b)^(1/2))*A*b+5/8*e^3/(a*e-b*d)^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2)
)*a*B+15/4*e^2/(a*e-b*d)^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*b*d

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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((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.6424, size = 4342, normalized size = 17.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

[-1/48*(15*(6*B*a^3*b*d^2*e^2 + (B*a^4 - 7*A*a^3*b)*d*e^3 + (6*B*b^4*d*e^3 + (B*a*b^3 - 7*A*b^4)*e^4)*x^4 + (6
*B*b^4*d^2*e^2 + (19*B*a*b^3 - 7*A*b^4)*d*e^3 + 3*(B*a^2*b^2 - 7*A*a*b^3)*e^4)*x^3 + 3*(6*B*a*b^3*d^2*e^2 + 7*
(B*a^2*b^2 - A*a*b^3)*d*e^3 + (B*a^3*b - 7*A*a^2*b^2)*e^4)*x^2 + (18*B*a^2*b^2*d^2*e^2 + 3*(3*B*a^3*b - 7*A*a^
2*b^2)*d*e^3 + (B*a^4 - 7*A*a^3*b)*e^4)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e + 2*sqrt(b^2*d - a*b*e
)*sqrt(e*x + d))/(b*x + a)) - 2*(48*A*a^4*b*e^4 - 4*(B*a*b^4 + 2*A*b^5)*d^4 + 2*(16*B*a^2*b^3 + 23*A*a*b^4)*d^
3*e + (53*B*a^3*b^2 - 125*A*a^2*b^3)*d^2*e^2 - 3*(27*B*a^4*b - 13*A*a^3*b^2)*d*e^3 + 15*(6*B*b^5*d^2*e^2 - (5*
B*a*b^4 + 7*A*b^5)*d*e^3 - (B*a^2*b^3 - 7*A*a*b^4)*e^4)*x^3 + 5*(6*B*b^5*d^3*e + (43*B*a*b^4 - 7*A*b^5)*d^2*e^
2 - (41*B*a^2*b^3 + 49*A*a*b^4)*d*e^3 - 8*(B*a^3*b^2 - 7*A*a^2*b^3)*e^4)*x^2 - (12*B*b^5*d^4 - 2*(47*B*a*b^4 +
7*A*b^5)*d^3*e - 2*(65*B*a^2*b^3 - 56*A*a*b^4)*d^2*e^2 + (179*B*a^3*b^2 + 133*A*a^2*b^3)*d*e^3 + 33*(B*a^4*b
- 7*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(a^3*b^6*d^6 - 5*a^4*b^5*d^5*e + 10*a^5*b^4*d^4*e^2 - 10*a^6*b^3*d^3*e^3
+ 5*a^7*b^2*d^2*e^4 - a^8*b*d*e^5 + (b^9*d^5*e - 5*a*b^8*d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10*a^3*b^6*d^2*e^4 +
5*a^4*b^5*d*e^5 - a^5*b^4*e^6)*x^4 + (b^9*d^6 - 2*a*b^8*d^5*e - 5*a^2*b^7*d^4*e^2 + 20*a^3*b^6*d^3*e^3 - 25*a^
4*b^5*d^2*e^4 + 14*a^5*b^4*d*e^5 - 3*a^6*b^3*e^6)*x^3 + 3*(a*b^8*d^6 - 4*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 - 5
*a^5*b^4*d^2*e^4 + 4*a^6*b^3*d*e^5 - a^7*b^2*e^6)*x^2 + (3*a^2*b^7*d^6 - 14*a^3*b^6*d^5*e + 25*a^4*b^5*d^4*e^2
- 20*a^5*b^4*d^3*e^3 + 5*a^6*b^3*d^2*e^4 + 2*a^7*b^2*d*e^5 - a^8*b*e^6)*x), 1/24*(15*(6*B*a^3*b*d^2*e^2 + (B*
a^4 - 7*A*a^3*b)*d*e^3 + (6*B*b^4*d*e^3 + (B*a*b^3 - 7*A*b^4)*e^4)*x^4 + (6*B*b^4*d^2*e^2 + (19*B*a*b^3 - 7*A*
b^4)*d*e^3 + 3*(B*a^2*b^2 - 7*A*a*b^3)*e^4)*x^3 + 3*(6*B*a*b^3*d^2*e^2 + 7*(B*a^2*b^2 - A*a*b^3)*d*e^3 + (B*a^
3*b - 7*A*a^2*b^2)*e^4)*x^2 + (18*B*a^2*b^2*d^2*e^2 + 3*(3*B*a^3*b - 7*A*a^2*b^2)*d*e^3 + (B*a^4 - 7*A*a^3*b)*
e^4)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (48*A*a^4*b*e^4 - 4*(B
*a*b^4 + 2*A*b^5)*d^4 + 2*(16*B*a^2*b^3 + 23*A*a*b^4)*d^3*e + (53*B*a^3*b^2 - 125*A*a^2*b^3)*d^2*e^2 - 3*(27*B
*a^4*b - 13*A*a^3*b^2)*d*e^3 + 15*(6*B*b^5*d^2*e^2 - (5*B*a*b^4 + 7*A*b^5)*d*e^3 - (B*a^2*b^3 - 7*A*a*b^4)*e^4
)*x^3 + 5*(6*B*b^5*d^3*e + (43*B*a*b^4 - 7*A*b^5)*d^2*e^2 - (41*B*a^2*b^3 + 49*A*a*b^4)*d*e^3 - 8*(B*a^3*b^2 -
7*A*a^2*b^3)*e^4)*x^2 - (12*B*b^5*d^4 - 2*(47*B*a*b^4 + 7*A*b^5)*d^3*e - 2*(65*B*a^2*b^3 - 56*A*a*b^4)*d^2*e^
2 + (179*B*a^3*b^2 + 133*A*a^2*b^3)*d*e^3 + 33*(B*a^4*b - 7*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(a^3*b^6*d^6 - 5
*a^4*b^5*d^5*e + 10*a^5*b^4*d^4*e^2 - 10*a^6*b^3*d^3*e^3 + 5*a^7*b^2*d^2*e^4 - a^8*b*d*e^5 + (b^9*d^5*e - 5*a*
b^8*d^4*e^2 + 10*a^2*b^7*d^3*e^3 - 10*a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*e^5 - a^5*b^4*e^6)*x^4 + (b^9*d^6 - 2*a*b^
8*d^5*e - 5*a^2*b^7*d^4*e^2 + 20*a^3*b^6*d^3*e^3 - 25*a^4*b^5*d^2*e^4 + 14*a^5*b^4*d*e^5 - 3*a^6*b^3*e^6)*x^3
+ 3*(a*b^8*d^6 - 4*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 - 5*a^5*b^4*d^2*e^4 + 4*a^6*b^3*d*e^5 - a^7*b^2*e^6)*x^2
+ (3*a^2*b^7*d^6 - 14*a^3*b^6*d^5*e + 25*a^4*b^5*d^4*e^2 - 20*a^5*b^4*d^3*e^3 + 5*a^6*b^3*d^2*e^4 + 2*a^7*b^2*
d*e^5 - a^8*b*e^6)*x)]

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Sympy [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((B*x+A)/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.19191, size = 697, normalized size = 2.79 \begin{align*} \frac{5 \,{\left (6 \, B b d e^{2} + B a e^{3} - 7 \, A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} + \frac{2 \,{\left (B d e^{2} - A e^{3}\right )}}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{x e + d}} + \frac{42 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{3} d e^{2} - 96 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d^{2} e^{2} + 54 \, \sqrt{x e + d} B b^{3} d^{3} e^{2} + 15 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{2} e^{3} - 57 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{3} e^{3} + 56 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} d e^{3} + 136 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} d e^{3} - 75 \, \sqrt{x e + d} B a b^{2} d^{2} e^{3} - 87 \, \sqrt{x e + d} A b^{3} d^{2} e^{3} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b e^{4} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{2} e^{4} - 12 \, \sqrt{x e + d} B a^{2} b d e^{4} + 174 \, \sqrt{x e + d} A a b^{2} d e^{4} + 33 \, \sqrt{x e + d} B a^{3} e^{5} - 87 \, \sqrt{x e + d} A a^{2} b e^{5}}{24 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

5/8*(6*B*b*d*e^2 + B*a*e^3 - 7*A*b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d^4 - 4*a*b^3*d^3*e
+ 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(-b^2*d + a*b*e)) + 2*(B*d*e^2 - A*e^3)/((b^4*d^4 - 4*a*b^
3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(x*e + d)) + 1/24*(42*(x*e + d)^(5/2)*B*b^3*d*e^2 -
96*(x*e + d)^(3/2)*B*b^3*d^2*e^2 + 54*sqrt(x*e + d)*B*b^3*d^3*e^2 + 15*(x*e + d)^(5/2)*B*a*b^2*e^3 - 57*(x*e
+ d)^(5/2)*A*b^3*e^3 + 56*(x*e + d)^(3/2)*B*a*b^2*d*e^3 + 136*(x*e + d)^(3/2)*A*b^3*d*e^3 - 75*sqrt(x*e + d)*B
*a*b^2*d^2*e^3 - 87*sqrt(x*e + d)*A*b^3*d^2*e^3 + 40*(x*e + d)^(3/2)*B*a^2*b*e^4 - 136*(x*e + d)^(3/2)*A*a*b^2
*e^4 - 12*sqrt(x*e + d)*B*a^2*b*d*e^4 + 174*sqrt(x*e + d)*A*a*b^2*d*e^4 + 33*sqrt(x*e + d)*B*a^3*e^5 - 87*sqrt
(x*e + d)*A*a^2*b*e^5)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*((x*e + d)*b -
b*d + a*e)^3)