3.1871 \(\int \frac{A+B x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=280 \[ -\frac{A b-a B}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{a B e-5 A b e+4 b B d}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{3 e (a+b x) (a B e-5 A b e+4 b B d)}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{3 e (a+b x) (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.255782, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {770, 78, 51, 63, 208} \[ -\frac{A b-a B}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}-\frac{a B e-5 A b e+4 b B d}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{3 e (a+b x) (a B e-5 A b e+4 b B d)}{4 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}+\frac{3 e (a+b x) (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4*b*B*d - 5*A*b*e + a*B*e)/(4*b*(b*d - a*e)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (A*b - a*B)/(2*
b*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*e*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b
*x))/(4*b*(b*d - a*e)^3*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b
*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*Sqrt[b]*(b*d - a*e)^(7/2)*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 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 )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{A+B x}{\left (a b+b^2 x\right )^3 (d+e x)^{3/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A b-a B}{2 b (b d-a e) (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left ((4 b B d-5 A b e+a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^2 (d+e x)^{3/2}} \, dx}{4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{2 b (b d-a e) (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (3 e (4 b B d-5 A b e+a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{8 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{2 b (b d-a e) (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (4 b B d-5 A b e+a B e) (a+b x)}{4 b (b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (3 e (4 b B d-5 A b e+a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{8 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{2 b (b d-a e) (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (4 b B d-5 A b e+a B e) (a+b x)}{4 b (b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (3 (4 b B d-5 A b e+a B e) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b-\frac{b^2 d}{e}+\frac{b^2 x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{4 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{2 b (b d-a e) (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (4 b B d-5 A b e+a B e) (a+b x)}{4 b (b d-a e)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (4 b B d-5 A b e+a B e) (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (b d-a e)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0803107, size = 110, normalized size = 0.39 \[ \frac{(a+b x) \left (\frac{e (a+b x)^2 (-a B e+5 A b e-4 b B d) \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+a B-A b\right )}{2 b \left ((a+b x)^2\right )^{3/2} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*(-(A*b) + a*B + (e*(-4*b*B*d + 5*A*b*e - a*B*e)*(a + b*x)^2*Hypergeometric2F1[-1/2, 2, 1/2, (b*(d +
e*x))/(b*d - a*e)])/(b*d - a*e)^2))/(2*b*(b*d - a*e)*((a + b*x)^2)^(3/2)*Sqrt[d + e*x])

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Maple [B]  time = 0.023, size = 681, normalized size = 2.4 \begin{align*} -{\frac{bx+a}{4\, \left ( ae-bd \right ) ^{3}} \left ( 15\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{x}^{2}{b}^{3}{e}^{2}-3\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{x}^{2}a{b}^{2}{e}^{2}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{x}^{2}{b}^{3}de+30\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}xa{b}^{2}{e}^{2}-6\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}x{a}^{2}b{e}^{2}-24\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}xa{b}^{2}de+15\,A\sqrt{ \left ( ae-bd \right ) b}{x}^{2}{b}^{2}{e}^{2}+15\,A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{a}^{2}b{e}^{2}-3\,B\sqrt{ \left ( ae-bd \right ) b}{x}^{2}ab{e}^{2}-12\,B\sqrt{ \left ( ae-bd \right ) b}{x}^{2}{b}^{2}de-3\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{a}^{3}{e}^{2}-12\,B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{a}^{2}bde+25\,A\sqrt{ \left ( ae-bd \right ) b}xab{e}^{2}+5\,A\sqrt{ \left ( ae-bd \right ) b}x{b}^{2}de-5\,B\sqrt{ \left ( ae-bd \right ) b}x{a}^{2}{e}^{2}-21\,B\sqrt{ \left ( ae-bd \right ) b}xabde-4\,B\sqrt{ \left ( ae-bd \right ) b}x{b}^{2}{d}^{2}+8\,A\sqrt{ \left ( ae-bd \right ) b}{a}^{2}{e}^{2}+9\,A\sqrt{ \left ( ae-bd \right ) b}abde-2\,A\sqrt{ \left ( ae-bd \right ) b}{b}^{2}{d}^{2}-13\,B\sqrt{ \left ( ae-bd \right ) b}{a}^{2}de-2\,B\sqrt{ \left ( ae-bd \right ) b}ab{d}^{2} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}{\frac{1}{\sqrt{ex+d}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

-1/4*(15*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^2*b^3*e^2-3*B*arctan((e*x+d)^(1/2)*b/((
a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^2*a*b^2*e^2-12*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*
x^2*b^3*d*e+30*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x*a*b^2*e^2-6*B*arctan((e*x+d)^(1/2
)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x*a^2*b*e^2-24*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1
/2)*x*a*b^2*d*e+15*A*((a*e-b*d)*b)^(1/2)*x^2*b^2*e^2+15*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^
(1/2)*a^2*b*e^2-3*B*((a*e-b*d)*b)^(1/2)*x^2*a*b*e^2-12*B*((a*e-b*d)*b)^(1/2)*x^2*b^2*d*e-3*B*arctan((e*x+d)^(1
/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*a^3*e^2-12*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2
)*a^2*b*d*e+25*A*((a*e-b*d)*b)^(1/2)*x*a*b*e^2+5*A*((a*e-b*d)*b)^(1/2)*x*b^2*d*e-5*B*((a*e-b*d)*b)^(1/2)*x*a^2
*e^2-21*B*((a*e-b*d)*b)^(1/2)*x*a*b*d*e-4*B*((a*e-b*d)*b)^(1/2)*x*b^2*d^2+8*A*((a*e-b*d)*b)^(1/2)*a^2*e^2+9*A*
((a*e-b*d)*b)^(1/2)*a*b*d*e-2*A*((a*e-b*d)*b)^(1/2)*b^2*d^2-13*B*((a*e-b*d)*b)^(1/2)*a^2*d*e-2*B*((a*e-b*d)*b)
^(1/2)*a*b*d^2)*(b*x+a)/((a*e-b*d)*b)^(1/2)/(e*x+d)^(1/2)/(a*e-b*d)^3/((b*x+a)^2)^(3/2)

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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{3}{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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.86079, size = 2884, normalized size = 10.3 \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)^(3/2),x, algorithm="fricas")

[Out]

[1/8*(3*(4*B*a^2*b*d^2*e + (B*a^3 - 5*A*a^2*b)*d*e^2 + (4*B*b^3*d*e^2 + (B*a*b^2 - 5*A*b^3)*e^3)*x^3 + (4*B*b^
3*d^2*e + (9*B*a*b^2 - 5*A*b^3)*d*e^2 + 2*(B*a^2*b - 5*A*a*b^2)*e^3)*x^2 + (8*B*a*b^2*d^2*e + 2*(3*B*a^2*b - 5
*A*a*b^2)*d*e^2 + (B*a^3 - 5*A*a^2*b)*e^3)*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*(8*A*a^3*b*e^3 + 2*(B*a*b^3 + A*b^4)*d^3 + 11*(B*a^2*b^2 - A*a*b^3)*d^2*e -
(13*B*a^3*b - A*a^2*b^2)*d*e^2 + 3*(4*B*b^4*d^2*e - (3*B*a*b^3 + 5*A*b^4)*d*e^2 - (B*a^2*b^2 - 5*A*a*b^3)*e^3
)*x^2 + (4*B*b^4*d^3 + (17*B*a*b^3 - 5*A*b^4)*d^2*e - 4*(4*B*a^2*b^2 + 5*A*a*b^3)*d*e^2 - 5*(B*a^3*b - 5*A*a^2
*b^2)*e^3)*x)*sqrt(e*x + d))/(a^2*b^5*d^5 - 4*a^3*b^4*d^4*e + 6*a^4*b^3*d^3*e^2 - 4*a^5*b^2*d^2*e^3 + a^6*b*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^3 + (b^7*d^5 - 2*a*b
^6*d^4*e - 2*a^2*b^5*d^3*e^2 + 8*a^3*b^4*d^2*e^3 - 7*a^4*b^3*d*e^4 + 2*a^5*b^2*e^5)*x^2 + (2*a*b^6*d^5 - 7*a^2
*b^5*d^4*e + 8*a^3*b^4*d^3*e^2 - 2*a^4*b^3*d^2*e^3 - 2*a^5*b^2*d*e^4 + a^6*b*e^5)*x), -1/4*(3*(4*B*a^2*b*d^2*e
+ (B*a^3 - 5*A*a^2*b)*d*e^2 + (4*B*b^3*d*e^2 + (B*a*b^2 - 5*A*b^3)*e^3)*x^3 + (4*B*b^3*d^2*e + (9*B*a*b^2 - 5
*A*b^3)*d*e^2 + 2*(B*a^2*b - 5*A*a*b^2)*e^3)*x^2 + (8*B*a*b^2*d^2*e + 2*(3*B*a^2*b - 5*A*a*b^2)*d*e^2 + (B*a^3
- 5*A*a^2*b)*e^3)*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)) + (8*A*a^3
*b*e^3 + 2*(B*a*b^3 + A*b^4)*d^3 + 11*(B*a^2*b^2 - A*a*b^3)*d^2*e - (13*B*a^3*b - A*a^2*b^2)*d*e^2 + 3*(4*B*b^
4*d^2*e - (3*B*a*b^3 + 5*A*b^4)*d*e^2 - (B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + (4*B*b^4*d^3 + (17*B*a*b^3 - 5*A*b^
4)*d^2*e - 4*(4*B*a^2*b^2 + 5*A*a*b^3)*d*e^2 - 5*(B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(a^2*b^5*d^5 -
4*a^3*b^4*d^4*e + 6*a^4*b^3*d^3*e^2 - 4*a^5*b^2*d^2*e^3 + a^6*b*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^3 + (b^7*d^5 - 2*a*b^6*d^4*e - 2*a^2*b^5*d^3*e^2 + 8*a^3*b^4*d^
2*e^3 - 7*a^4*b^3*d*e^4 + 2*a^5*b^2*e^5)*x^2 + (2*a*b^6*d^5 - 7*a^2*b^5*d^4*e + 8*a^3*b^4*d^3*e^2 - 2*a^4*b^3*
d^2*e^3 - 2*a^5*b^2*d*e^4 + a^6*b*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)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.30776, size = 832, normalized size = 2.97 \begin{align*} -\frac{3 \,{\left (4 \, B b d e^{2} + B a e^{3} - 5 \, A b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 3 \, a^{2} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{3} 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 \,{\left (B d e^{2} - A e^{3}\right )}}{{\left (b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 3 \, a^{2} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{3} 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{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{2} - 4 \, \sqrt{x e + d} B b^{2} d^{2} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{3} - 7 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{3} - \sqrt{x e + d} B a b d e^{3} + 9 \, \sqrt{x e + d} A b^{2} d e^{3} + 5 \, \sqrt{x e + d} B a^{2} e^{4} - 9 \, \sqrt{x e + d} A a b e^{4}}{4 \,{\left (b^{3} d^{3} e \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 3 \, a b^{2} d^{2} e^{2} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 3 \, a^{2} b d e^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{3} 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 )}^{2}} \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)^(3/2),x, algorithm="giac")

[Out]

-3/4*(4*B*b*d*e^2 + B*a*e^3 - 5*A*b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^3*d^3*e*sgn((x*e + d
)*b*e - b*d*e + a*e^2) - 3*a*b^2*d^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 3*a^2*b*d*e^3*sgn((x*e + d)*b*e
- b*d*e + a*e^2) - a^3*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) - 2*(B*d*e^2 - A*e^3)/((b
^3*d^3*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 3*a*b^2*d^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 3*a^2*b*d*e
^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) - a^3*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(x*e + d)) - 1/4*(4*(x
*e + d)^(3/2)*B*b^2*d*e^2 - 4*sqrt(x*e + d)*B*b^2*d^2*e^2 + 3*(x*e + d)^(3/2)*B*a*b*e^3 - 7*(x*e + d)^(3/2)*A*
b^2*e^3 - sqrt(x*e + d)*B*a*b*d*e^3 + 9*sqrt(x*e + d)*A*b^2*d*e^3 + 5*sqrt(x*e + d)*B*a^2*e^4 - 9*sqrt(x*e + d
)*A*a*b*e^4)/((b^3*d^3*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 3*a*b^2*d^2*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^
2) + 3*a^2*b*d*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) - a^3*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2))*((x*e + d)
*b - b*d + a*e)^2)