3.2143 \(\int \frac{a+b x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=276 \[ -\frac{35 e^3 (a+b x)}{8 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}-\frac{35 e^2}{24 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{35 \sqrt{b} e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{7 e}{12 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

[Out]

(-35*e^2)/(24*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(3*(b*d - a*e)*(a + b*x)^2*Sqrt[d
+ e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e)/(12*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) - (35*e^3*(a + b*x))/(8*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*Sqrt[b]*e^3
*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])

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Rubi [A]  time = 0.179548, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {770, 21, 51, 63, 208} \[ -\frac{35 e^3 (a+b x)}{8 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}-\frac{35 e^2}{24 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{35 \sqrt{b} e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{7 e}{12 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-35*e^2)/(24*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(3*(b*d - a*e)*(a + b*x)^2*Sqrt[d
+ e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e)/(12*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) - (35*e^3*(a + b*x))/(8*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*Sqrt[b]*e^3
*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{a+b x}{\left (a b+b^2 x\right )^5 (d+e x)^{3/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (7 e \left (a b+b^2 x\right )\right ) \int \frac{1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{6 b (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e}{12 (b d-a e)^2 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 e^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{24 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^2}{24 (b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{3 (b d-a e) (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e}{12 (b d-a e)^2 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 e^3 \left (a b+b^2 x\right )\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 b (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^2}{24 (b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{3 (b d-a e) (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e}{12 (b d-a e)^2 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (a+b x)}{8 (b d-a e)^4 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 e^3 \left (a b+b^2 x\right )\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^2}{24 (b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{3 (b d-a e) (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e}{12 (b d-a e)^2 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (a+b x)}{8 (b d-a e)^4 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 e^2 \left (a b+b^2 x\right )\right ) \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 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^2}{24 (b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{3 (b d-a e) (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e}{12 (b d-a e)^2 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (a+b x)}{8 (b d-a e)^4 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 \sqrt{b} e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0288989, size = 66, normalized size = 0.24 \[ -\frac{2 e^3 (a+b x) \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{\sqrt{(a+b x)^2} \sqrt{d+e x} (a e-b d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e^3*(a + b*x)*Hypergeometric2F1[-1/2, 4, 1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/((-(b*d) + a*e)^4*Sqrt[(a
+ b*x)^2]*Sqrt[d + e*x])

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Maple [B]  time = 0.023, size = 431, normalized size = 1.6 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{2}}{24\, \left ( ae-bd \right ) ^{4}} \left ( 105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{x}^{3}{b}^{4}{e}^{3}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{x}^{2}a{b}^{3}{e}^{3}+105\,\sqrt{ \left ( ae-bd \right ) b}{x}^{3}{b}^{3}{e}^{3}+315\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}x{a}^{2}{b}^{2}{e}^{3}+280\,\sqrt{ \left ( ae-bd \right ) b}{x}^{2}a{b}^{2}{e}^{3}+35\,\sqrt{ \left ( ae-bd \right ) b}{x}^{2}{b}^{3}d{e}^{2}+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{a}^{3}b{e}^{3}+231\,\sqrt{ \left ( ae-bd \right ) b}x{a}^{2}b{e}^{3}+98\,\sqrt{ \left ( ae-bd \right ) b}xa{b}^{2}d{e}^{2}-14\,\sqrt{ \left ( ae-bd \right ) b}x{b}^{3}{d}^{2}e+48\,\sqrt{ \left ( ae-bd \right ) b}{a}^{3}{e}^{3}+87\,\sqrt{ \left ( ae-bd \right ) b}{a}^{2}bd{e}^{2}-38\,\sqrt{ \left ( ae-bd \right ) b}a{b}^{2}{d}^{2}e+8\,\sqrt{ \left ( ae-bd \right ) b}{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \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)^(5/2),x)

[Out]

-1/24*(105*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^3*b^4*e^3+315*arctan((e*x+d)^(1/2)*b/((
a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^2*a*b^3*e^3+105*((a*e-b*d)*b)^(1/2)*x^3*b^3*e^3+315*arctan((e*x+d)^(1/2)*b/
((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x*a^2*b^2*e^3+280*((a*e-b*d)*b)^(1/2)*x^2*a*b^2*e^3+35*((a*e-b*d)*b)^(1/2)*
x^2*b^3*d*e^2+105*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*a^3*b*e^3+231*((a*e-b*d)*b)^(1/2)*
x*a^2*b*e^3+98*((a*e-b*d)*b)^(1/2)*x*a*b^2*d*e^2-14*((a*e-b*d)*b)^(1/2)*x*b^3*d^2*e+48*((a*e-b*d)*b)^(1/2)*a^3
*e^3+87*((a*e-b*d)*b)^(1/2)*a^2*b*d*e^2-38*((a*e-b*d)*b)^(1/2)*a*b^2*d^2*e+8*((a*e-b*d)*b)^(1/2)*b^3*d^3)*(b*x
+a)^2/((a*e-b*d)*b)^(1/2)/(e*x+d)^(1/2)/(a*e-b*d)^4/((b*x+a)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \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)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(3/2)), x)

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Fricas [B]  time = 1.19649, size = 2441, normalized size = 8.84 \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)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(b^3*e^4*x^4 + a^3*d*e^3 + (b^3*d*e^3 + 3*a*b^2*e^4)*x^3 + 3*(a*b^2*d*e^3 + a^2*b*e^4)*x^2 + (3*a^2
*b*d*e^3 + a^3*e^4)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d
- a*e)))/(b*x + a)) - 2*(105*b^3*e^3*x^3 + 8*b^3*d^3 - 38*a*b^2*d^2*e + 87*a^2*b*d*e^2 + 48*a^3*e^3 + 35*(b^3*
d*e^2 + 8*a*b^2*e^3)*x^2 - 7*(2*b^3*d^2*e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3)*x)*sqrt(e*x + d))/(a^3*b^4*d^5 - 4*
a^4*b^3*d^4*e + 6*a^5*b^2*d^3*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 + 6*a^2*b^5*d^2
*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e^5)*x^4 + (b^7*d^5 - a*b^6*d^4*e - 6*a^2*b^5*d^3*e^2 + 14*a^3*b^4*d^2*e^3 -
11*a^4*b^3*d*e^4 + 3*a^5*b^2*e^5)*x^3 + 3*(a*b^6*d^5 - 3*a^2*b^5*d^4*e + 2*a^3*b^4*d^3*e^2 + 2*a^4*b^3*d^2*e^3
- 3*a^5*b^2*d*e^4 + a^6*b*e^5)*x^2 + (3*a^2*b^5*d^5 - 11*a^3*b^4*d^4*e + 14*a^4*b^3*d^3*e^2 - 6*a^5*b^2*d^2*e
^3 - a^6*b*d*e^4 + a^7*e^5)*x), 1/24*(105*(b^3*e^4*x^4 + a^3*d*e^3 + (b^3*d*e^3 + 3*a*b^2*e^4)*x^3 + 3*(a*b^2*
d*e^3 + a^2*b*e^4)*x^2 + (3*a^2*b*d*e^3 + a^3*e^4)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*s
qrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (105*b^3*e^3*x^3 + 8*b^3*d^3 - 38*a*b^2*d^2*e + 87*a^2*b*d*e^2 + 48*a^3*e
^3 + 35*(b^3*d*e^2 + 8*a*b^2*e^3)*x^2 - 7*(2*b^3*d^2*e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3)*x)*sqrt(e*x + d))/(a^3
*b^4*d^5 - 4*a^4*b^3*d^4*e + 6*a^5*b^2*d^3*e^2 - 4*a^6*b*d^2*e^3 + a^7*d*e^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 +
6*a^2*b^5*d^2*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e^5)*x^4 + (b^7*d^5 - a*b^6*d^4*e - 6*a^2*b^5*d^3*e^2 + 14*a^3*b
^4*d^2*e^3 - 11*a^4*b^3*d*e^4 + 3*a^5*b^2*e^5)*x^3 + 3*(a*b^6*d^5 - 3*a^2*b^5*d^4*e + 2*a^3*b^4*d^3*e^2 + 2*a^
4*b^3*d^2*e^3 - 3*a^5*b^2*d*e^4 + a^6*b*e^5)*x^2 + (3*a^2*b^5*d^5 - 11*a^3*b^4*d^4*e + 14*a^4*b^3*d^3*e^2 - 6*
a^5*b^2*d^2*e^3 - a^6*b*d*e^4 + a^7*e^5)*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)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.30494, size = 883, normalized size = 3.2 \begin{align*} -\frac{35 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{4} d^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \, e^{3}}{{\left (b^{4} d^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt{x e + d}} - \frac{57 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{3} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{3} + 87 \, \sqrt{x e + d} b^{3} d^{2} e^{3} + 136 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} e^{4} - 174 \, \sqrt{x e + d} a b^{2} d e^{4} + 87 \, \sqrt{x e + d} a^{2} b e^{5}}{24 \,{\left (b^{4} d^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{4} e^{4} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\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)^(5/2),x, algorithm="giac")

[Out]

-35/8*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^4*d^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a*b^
3*d^3*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 6*a^2*b^2*d^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a^3*b*d*
e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^4*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) - 2
*e^3/((b^4*d^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a*b^3*d^3*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 6*a^2*b
^2*d^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a^3*b*d*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^4*e^4*sgn
((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(x*e + d)) - 1/24*(57*(x*e + d)^(5/2)*b^3*e^3 - 136*(x*e + d)^(3/2)*b^3*d
*e^3 + 87*sqrt(x*e + d)*b^3*d^2*e^3 + 136*(x*e + d)^(3/2)*a*b^2*e^4 - 174*sqrt(x*e + d)*a*b^2*d*e^4 + 87*sqrt(
x*e + d)*a^2*b*e^5)/((b^4*d^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a*b^3*d^3*e*sgn((x*e + d)*b*e - b*d*e + a
*e^2) + 6*a^2*b^2*d^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a^3*b*d*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2
) + a^4*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^3)