3.1880 \(\int \frac{A+B x}{\sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=359 \[ -\frac{5 e^2 \sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{5 e^3 (a+b x) (-a B e-7 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{5 e \sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{\sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(-5*e^2*(8*b*B*d - 7*A*b*e - a*B*e)*Sqrt[d + e*x])/(64*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b
- a*B)*Sqrt[d + e*x])/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d - 7*A*b*e - a*B*
e)*Sqrt[d + e*x])/(24*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e*(8*b*B*d - 7*A*b*e - a
*B*e)*Sqrt[d + e*x])/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^3*(8*b*B*d - 7*A*b*e
- a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(3/2)*(b*d - a*e)^(9/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.348191, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{5 e^2 \sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{5 e^3 (a+b x) (-a B e-7 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}}+\frac{5 e \sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{\sqrt{d+e x} (-a B e-7 A b e+8 b B d)}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{\sqrt{d+e x} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*e^2*(8*b*B*d - 7*A*b*e - a*B*e)*Sqrt[d + e*x])/(64*b*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b
- a*B)*Sqrt[d + e*x])/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b*B*d - 7*A*b*e - a*B*
e)*Sqrt[d + e*x])/(24*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e*(8*b*B*d - 7*A*b*e - a
*B*e)*Sqrt[d + e*x])/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^3*(8*b*B*d - 7*A*b*e
- a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(3/2)*(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 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}{\sqrt{d+e x} \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 \sqrt{d+e x}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^2 (8 b B d-7 A b e-a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^4 \sqrt{d+e x}} \, dx}{8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d-7 A b e-a B e) \sqrt{d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (5 b e (8 b B d-7 A b e-a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^3 \sqrt{d+e x}} \, dx}{48 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d-7 A b e-a B e) \sqrt{d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (8 b B d-7 A b e-a B e) \sqrt{d+e x}}{96 b (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (5 e^2 (8 b B d-7 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 \sqrt{d+e x}} \, dx}{64 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{5 e^2 (8 b B d-7 A b e-a B e) \sqrt{d+e x}}{64 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) \sqrt{d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d-7 A b e-a B e) \sqrt{d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (8 b B d-7 A b e-a B e) \sqrt{d+e x}}{96 b (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (5 e^3 (8 b B d-7 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}{128 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{5 e^2 (8 b B d-7 A b e-a B e) \sqrt{d+e x}}{64 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) \sqrt{d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d-7 A b e-a B e) \sqrt{d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (8 b B d-7 A b e-a B e) \sqrt{d+e x}}{96 b (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (5 e^2 (8 b B d-7 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 )}{64 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{5 e^2 (8 b B d-7 A b e-a B e) \sqrt{d+e x}}{64 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) \sqrt{d+e x}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d-7 A b e-a B e) \sqrt{d+e x}}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e (8 b B d-7 A b e-a B e) \sqrt{d+e x}}{96 b (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e^3 (8 b B d-7 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 )}{64 b^{3/2} (b d-a e)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0996657, size = 114, normalized size = 0.32 \[ \frac{\sqrt{d+e x} \left (-\frac{e^3 (a+b x)^4 (a B e+7 A b e-8 b B d) \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+a B-A b\right )}{4 b (a+b x)^3 \sqrt{(a+b x)^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(-(A*b) + a*B - (e^3*(-8*b*B*d + 7*A*b*e + a*B*e)*(a + b*x)^4*Hypergeometric2F1[1/2, 4, 3/2, (b
*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^4))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.027, size = 1296, normalized size = 3.6 \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)/(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x)

[Out]

1/192*(279*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b*e^4+15*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a*b^3*e+385*A*
((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^3*e^2-385*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^4*d*e+15*B*arctan((e*x+d
)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^4*a*b^4*e^5-120*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^4*b^5*d*e^4+4
20*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^3*a*b^4*e^5+60*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2)
)*x^3*a^2*b^3*e^5+60*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^4*b*e^5+511*A*((a*e-b*d)*b)^(1/2)*(e*x+
d)^(3/2)*a^2*b^2*e^3+511*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^4*d^2*e+73*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*
a^3*b*e^3-120*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^4*b*d*e^4+55*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)
*a^2*b^2*e^2+420*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^3*b^2*e^5-279*A*((a*e-b*d)*b)^(1/2)*(e*x+d)
^(1/2)*b^4*d^3*e+630*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^2*b^3*e^5+90*B*arctan((e*x+d)^(1/2)*b
/((a*e-b*d)*b)^(1/2))*x^2*a^3*b^2*e^5+15*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^5*e^5-120*B*((a*e-b*d
)*b)^(1/2)*(e*x+d)^(7/2)*b^4*d+440*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^4*d^2+105*A*arctan((e*x+d)^(1/2)*b/((
a*e-b*d)*b)^(1/2))*x^4*b^5*e^5+105*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^4*e-15*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^
(1/2)*a^4*e^4+264*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b^4*d^4+105*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2)
)*a^4*b*e^5-584*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^4*d^3-1022*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^3*d*e
^2-730*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^2*d*e^2-219*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b*d*e^3-7
20*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^2*b^3*d*e^4-495*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b
^3*d*e-480*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^3*a*b^4*d*e^4+747*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/
2)*a^2*b^2*d^2*e^2-777*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e+1241*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2
)*a*b^3*d^2*e-837*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3+837*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*
b^3*d^2*e^2-480*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^3*b^2*d*e^4)/e*(b*x+a)/((a*e-b*d)*b)^(1/2)/b
/(a*e-b*d)^2/(a^2*e^2-2*a*b*d*e+b^2*d^2)/((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}} \sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d)), x)

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Fricas [B]  time = 1.84903, size = 4076, normalized size = 11.35 \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)/(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[-1/384*(15*(8*B*a^4*b*d*e^3 - (B*a^5 + 7*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (B*a*b^4 + 7*A*b^5)*e^4)*x^4 + 4*(8*
B*a*b^4*d*e^3 - (B*a^2*b^3 + 7*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (B*a^3*b^2 + 7*A*a^2*b^3)*e^4)*x^2 +
4*(8*B*a^3*b^2*d*e^3 - (B*a^4*b + 7*A*a^3*b^2)*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*(16*(B*a*b^5 + 3*A*b^6)*d^4 - 8*(11*B*a^2*b^4 + 31*A*a*b^5)*d^3*e
+ 2*(109*B*a^3*b^3 + 263*A*a^2*b^4)*d^2*e^2 - (131*B*a^4*b^2 + 605*A*a^3*b^3)*d*e^3 - 3*(5*B*a^5*b - 93*A*a^4
*b^2)*e^4 + 15*(8*B*b^6*d^2*e^2 - (9*B*a*b^5 + 7*A*b^6)*d*e^3 + (B*a^2*b^4 + 7*A*a*b^5)*e^4)*x^3 - 5*(16*B*b^6
*d^3*e - 2*(53*B*a*b^5 + 7*A*b^6)*d^2*e^2 + (101*B*a^2*b^4 + 91*A*a*b^5)*d*e^3 - 11*(B*a^3*b^3 + 7*A*a^2*b^4)*
e^4)*x^2 + (64*B*b^6*d^4 - 8*(45*B*a*b^5 + 7*A*b^6)*d^3*e + 4*(229*B*a^2*b^4 + 77*A*a*b^5)*d^2*e^2 - 7*(99*B*a
^3*b^3 + 109*A*a^2*b^4)*d*e^3 + 73*(B*a^4*b^2 + 7*A*a^3*b^3)*e^4)*x)*sqrt(e*x + d))/(a^4*b^7*d^5 - 5*a^5*b^6*d
^4*e + 10*a^6*b^5*d^3*e^2 - 10*a^7*b^4*d^2*e^3 + 5*a^8*b^3*d*e^4 - a^9*b^2*e^5 + (b^11*d^5 - 5*a*b^10*d^4*e +
10*a^2*b^9*d^3*e^2 - 10*a^3*b^8*d^2*e^3 + 5*a^4*b^7*d*e^4 - a^5*b^6*e^5)*x^4 + 4*(a*b^10*d^5 - 5*a^2*b^9*d^4*e
+ 10*a^3*b^8*d^3*e^2 - 10*a^4*b^7*d^2*e^3 + 5*a^5*b^6*d*e^4 - a^6*b^5*e^5)*x^3 + 6*(a^2*b^9*d^5 - 5*a^3*b^8*d
^4*e + 10*a^4*b^7*d^3*e^2 - 10*a^5*b^6*d^2*e^3 + 5*a^6*b^5*d*e^4 - a^7*b^4*e^5)*x^2 + 4*(a^3*b^8*d^5 - 5*a^4*b
^7*d^4*e + 10*a^5*b^6*d^3*e^2 - 10*a^6*b^5*d^2*e^3 + 5*a^7*b^4*d*e^4 - a^8*b^3*e^5)*x), -1/192*(15*(8*B*a^4*b*
d*e^3 - (B*a^5 + 7*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (B*a*b^4 + 7*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (B*a^2*
b^3 + 7*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (B*a^3*b^2 + 7*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 -
(B*a^4*b + 7*A*a^3*b^2)*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))
+ (16*(B*a*b^5 + 3*A*b^6)*d^4 - 8*(11*B*a^2*b^4 + 31*A*a*b^5)*d^3*e + 2*(109*B*a^3*b^3 + 263*A*a^2*b^4)*d^2*e
^2 - (131*B*a^4*b^2 + 605*A*a^3*b^3)*d*e^3 - 3*(5*B*a^5*b - 93*A*a^4*b^2)*e^4 + 15*(8*B*b^6*d^2*e^2 - (9*B*a*b
^5 + 7*A*b^6)*d*e^3 + (B*a^2*b^4 + 7*A*a*b^5)*e^4)*x^3 - 5*(16*B*b^6*d^3*e - 2*(53*B*a*b^5 + 7*A*b^6)*d^2*e^2
+ (101*B*a^2*b^4 + 91*A*a*b^5)*d*e^3 - 11*(B*a^3*b^3 + 7*A*a^2*b^4)*e^4)*x^2 + (64*B*b^6*d^4 - 8*(45*B*a*b^5 +
7*A*b^6)*d^3*e + 4*(229*B*a^2*b^4 + 77*A*a*b^5)*d^2*e^2 - 7*(99*B*a^3*b^3 + 109*A*a^2*b^4)*d*e^3 + 73*(B*a^4*
b^2 + 7*A*a^3*b^3)*e^4)*x)*sqrt(e*x + d))/(a^4*b^7*d^5 - 5*a^5*b^6*d^4*e + 10*a^6*b^5*d^3*e^2 - 10*a^7*b^4*d^2
*e^3 + 5*a^8*b^3*d*e^4 - a^9*b^2*e^5 + (b^11*d^5 - 5*a*b^10*d^4*e + 10*a^2*b^9*d^3*e^2 - 10*a^3*b^8*d^2*e^3 +
5*a^4*b^7*d*e^4 - a^5*b^6*e^5)*x^4 + 4*(a*b^10*d^5 - 5*a^2*b^9*d^4*e + 10*a^3*b^8*d^3*e^2 - 10*a^4*b^7*d^2*e^3
+ 5*a^5*b^6*d*e^4 - a^6*b^5*e^5)*x^3 + 6*(a^2*b^9*d^5 - 5*a^3*b^8*d^4*e + 10*a^4*b^7*d^3*e^2 - 10*a^5*b^6*d^2
*e^3 + 5*a^6*b^5*d*e^4 - a^7*b^4*e^5)*x^2 + 4*(a^3*b^8*d^5 - 5*a^4*b^7*d^4*e + 10*a^5*b^6*d^3*e^2 - 10*a^6*b^5
*d^2*e^3 + 5*a^7*b^4*d*e^4 - a^8*b^3*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)/(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.41297, size = 1149, normalized size = 3.2 \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)/(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-5/64*(8*B*b*d*e^3 - B*a*e^4 - 7*A*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^4*sgn((x*e + d)
*b*e - b*d*e + a*e^2) - 4*a*b^4*d^3*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 6*a^2*b^3*d^2*e^2*sgn((x*e + d)*b*e
- b*d*e + a*e^2) - 4*a^3*b^2*d*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^4*b*e^4*sgn((x*e + d)*b*e - b*d*e +
a*e^2))*sqrt(-b^2*d + a*b*e)) - 1/192*(120*(x*e + d)^(7/2)*B*b^4*d*e^3 - 440*(x*e + d)^(5/2)*B*b^4*d^2*e^3 +
584*(x*e + d)^(3/2)*B*b^4*d^3*e^3 - 264*sqrt(x*e + d)*B*b^4*d^4*e^3 - 15*(x*e + d)^(7/2)*B*a*b^3*e^4 - 105*(x*
e + d)^(7/2)*A*b^4*e^4 + 495*(x*e + d)^(5/2)*B*a*b^3*d*e^4 + 385*(x*e + d)^(5/2)*A*b^4*d*e^4 - 1241*(x*e + d)^
(3/2)*B*a*b^3*d^2*e^4 - 511*(x*e + d)^(3/2)*A*b^4*d^2*e^4 + 777*sqrt(x*e + d)*B*a*b^3*d^3*e^4 + 279*sqrt(x*e +
d)*A*b^4*d^3*e^4 - 55*(x*e + d)^(5/2)*B*a^2*b^2*e^5 - 385*(x*e + d)^(5/2)*A*a*b^3*e^5 + 730*(x*e + d)^(3/2)*B
*a^2*b^2*d*e^5 + 1022*(x*e + d)^(3/2)*A*a*b^3*d*e^5 - 747*sqrt(x*e + d)*B*a^2*b^2*d^2*e^5 - 837*sqrt(x*e + d)*
A*a*b^3*d^2*e^5 - 73*(x*e + d)^(3/2)*B*a^3*b*e^6 - 511*(x*e + d)^(3/2)*A*a^2*b^2*e^6 + 219*sqrt(x*e + d)*B*a^3
*b*d*e^6 + 837*sqrt(x*e + d)*A*a^2*b^2*d*e^6 + 15*sqrt(x*e + d)*B*a^4*e^7 - 279*sqrt(x*e + d)*A*a^3*b*e^7)/((b
^5*d^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a*b^4*d^3*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 6*a^2*b^3*d^2*e
^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 4*a^3*b^2*d*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^4*b*e^4*sgn((x*
e + d)*b*e - b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)