3.1829 \(\int \frac{(A+B x) (d+e x)^{3/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=313 \[ \frac{3 e^3 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{128 b^3 (a+b x) (b d-a e)^3}-\frac{e^2 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{64 b^3 (a+b x)^2 (b d-a e)^2}-\frac{3 e^4 (-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}-\frac{e \sqrt{d+e x} (-a B e-A b e+2 b B d)}{16 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{3/2} (-a B e-A b e+2 b B d)}{8 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[Out]

-(e*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(16*b^3*(b*d - a*e)*(a + b*x)^3) - (e^2*(2*b*B*d - A*b*e - a*B*e)
*Sqrt[d + e*x])/(64*b^3*(b*d - a*e)^2*(a + b*x)^2) + (3*e^3*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b^3*
(b*d - a*e)^3*(a + b*x)) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(3/2))/(8*b^2*(b*d - a*e)*(a + b*x)^4) - ((A*b
- a*B)*(d + e*x)^(5/2))/(5*b*(b*d - a*e)*(a + b*x)^5) - (3*e^4*(2*b*B*d - A*b*e - a*B*e)*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(7/2)*(b*d - a*e)^(7/2))

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Rubi [A]  time = 0.268544, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {27, 78, 47, 51, 63, 208} \[ \frac{3 e^3 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{128 b^3 (a+b x) (b d-a e)^3}-\frac{e^2 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{64 b^3 (a+b x)^2 (b d-a e)^2}-\frac{3 e^4 (-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}-\frac{e \sqrt{d+e x} (-a B e-A b e+2 b B d)}{16 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{3/2} (-a B e-A b e+2 b B d)}{8 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(16*b^3*(b*d - a*e)*(a + b*x)^3) - (e^2*(2*b*B*d - A*b*e - a*B*e)
*Sqrt[d + e*x])/(64*b^3*(b*d - a*e)^2*(a + b*x)^2) + (3*e^3*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b^3*
(b*d - a*e)^3*(a + b*x)) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(3/2))/(8*b^2*(b*d - a*e)*(a + b*x)^4) - ((A*b
- a*B)*(d + e*x)^(5/2))/(5*b*(b*d - a*e)*(a + b*x)^5) - (3*e^4*(2*b*B*d - A*b*e - a*B*e)*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(7/2)*(b*d - a*e)^(7/2))

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, 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 )^3} \, dx &=\int \frac{(A+B x) (d+e x)^{3/2}}{(a+b x)^6} \, dx\\ &=-\frac{(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac{(2 b B d-A b e-a B e) \int \frac{(d+e x)^{3/2}}{(a+b x)^5} \, dx}{2 b (b d-a e)}\\ &=-\frac{(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac{(3 e (2 b B d-A b e-a B e)) \int \frac{\sqrt{d+e x}}{(a+b x)^4} \, dx}{16 b^2 (b d-a e)}\\ &=-\frac{e (2 b B d-A b e-a B e) \sqrt{d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac{(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (e^2 (2 b B d-A b e-a B e)\right ) \int \frac{1}{(a+b x)^3 \sqrt{d+e x}} \, dx}{32 b^3 (b d-a e)}\\ &=-\frac{e (2 b B d-A b e-a B e) \sqrt{d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac{e^2 (2 b B d-A b e-a B e) \sqrt{d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}-\frac{(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}-\frac{\left (3 e^3 (2 b B d-A b e-a B e)\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{128 b^3 (b d-a e)^2}\\ &=-\frac{e (2 b B d-A b e-a B e) \sqrt{d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac{e^2 (2 b B d-A b e-a B e) \sqrt{d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac{3 e^3 (2 b B d-A b e-a B e) \sqrt{d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac{(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (3 e^4 (2 b B d-A b e-a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^3 (b d-a e)^3}\\ &=-\frac{e (2 b B d-A b e-a B e) \sqrt{d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac{e^2 (2 b B d-A b e-a B e) \sqrt{d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac{3 e^3 (2 b B d-A b e-a B e) \sqrt{d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac{(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (3 e^3 (2 b B d-A b e-a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{128 b^3 (b d-a e)^3}\\ &=-\frac{e (2 b B d-A b e-a B e) \sqrt{d+e x}}{16 b^3 (b d-a e) (a+b x)^3}-\frac{e^2 (2 b B d-A b e-a B e) \sqrt{d+e x}}{64 b^3 (b d-a e)^2 (a+b x)^2}+\frac{3 e^3 (2 b B d-A b e-a B e) \sqrt{d+e x}}{128 b^3 (b d-a e)^3 (a+b x)}-\frac{(2 b B d-A b e-a B e) (d+e x)^{3/2}}{8 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{5/2}}{5 b (b d-a e) (a+b x)^5}-\frac{3 e^4 (2 b B d-A b e-a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0732913, size = 98, normalized size = 0.31 \[ \frac{(d+e x)^{5/2} \left (\frac{5 e^4 (a B e+A b e-2 b B d) \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac{5 (a B-A b)}{(a+b x)^5}\right )}{25 b (b d-a e)} \]

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,x]

[Out]

((d + e*x)^(5/2)*((5*(-(A*b) + a*B))/(a + b*x)^5 + (5*e^4*(-2*b*B*d + A*b*e + a*B*e)*Hypergeometric2F1[5/2, 5,
7/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^5))/(25*b*(b*d - a*e))

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Maple [B]  time = 0.022, size = 871, normalized size = 2.8 \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)^3,x)

[Out]

3/128*e^5/(b*e*x+a*e)^5*b^2/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(9/2)*A+3/128*e^5/(b*e*x+a*e
)^5*b/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(9/2)*a*B-3/64*e^4/(b*e*x+a*e)^5*b^2/(a^3*e^3-3*a^
2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(9/2)*B*d+7/64*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^
(7/2)*A*b+7/64*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(7/2)*a*B-7/32*e^4/(b*e*x+a*e)^5/(a^2*e^2
-2*a*b*d*e+b^2*d^2)*(e*x+d)^(7/2)*B*b*d+1/5*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(5/2)*A-1/5*e^5/(b*e*x+a*e)^5/
(a*e-b*d)/b*(e*x+d)^(5/2)*a*B-7/64*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*A-7/64*e^5/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2
)*a*B+7/32*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*B*d-3/128*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*A*a+3/128*e^5/(b*e*
x+a*e)^5/b*(e*x+d)^(1/2)*A*d-3/128*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*a^2*B+9/128*e^5/(b*e*x+a*e)^5/b^2*(e*x+
d)^(1/2)*B*d*a-3/64*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*B*d^2+3/128*e^5/b^2/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e
-b^3*d^3)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A+3/128*e^5/b^3/(a^3*e^3-3*a^2*b*d*e
^2+3*a*b^2*d^2*e-b^3*d^3)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a*B-3/64*e^4/b^2/(a^
3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.72035, size = 4825, normalized size = 15.42 \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,x, algorithm="fricas")

[Out]

[1/1280*(15*(2*B*a^5*b*d*e^4 - (B*a^6 + A*a^5*b)*e^5 + (2*B*b^6*d*e^4 - (B*a*b^5 + A*b^6)*e^5)*x^5 + 5*(2*B*a*
b^5*d*e^4 - (B*a^2*b^4 + A*a*b^5)*e^5)*x^4 + 10*(2*B*a^2*b^4*d*e^4 - (B*a^3*b^3 + A*a^2*b^4)*e^5)*x^3 + 10*(2*
B*a^3*b^3*d*e^4 - (B*a^4*b^2 + A*a^3*b^3)*e^5)*x^2 + 5*(2*B*a^4*b^2*d*e^4 - (B*a^5*b + A*a^4*b^2)*e^5)*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*(32*(B*a*b^6 +
4*A*b^7)*d^5 - 16*(6*B*a^2*b^5 + 29*A*a*b^6)*d^4*e + 4*(19*B*a^3*b^4 + 146*A*a^2*b^5)*d^3*e^2 + 2*(4*B*a^4*b^3
- 129*A*a^3*b^4)*d^2*e^3 - 5*(7*B*a^5*b^2 + A*a^4*b^3)*d*e^4 + 15*(B*a^6*b + A*a^5*b^2)*e^5 - 15*(2*B*b^7*d^2
*e^3 - (3*B*a*b^6 + A*b^7)*d*e^4 + (B*a^2*b^5 + A*a*b^6)*e^5)*x^4 + 10*(2*B*b^7*d^3*e^2 - (17*B*a*b^6 + A*b^7)
*d^2*e^3 + 2*(11*B*a^2*b^5 + 4*A*a*b^6)*d*e^4 - 7*(B*a^3*b^4 + A*a^2*b^5)*e^5)*x^3 + 2*(120*B*b^7*d^4*e - 2*(2
27*B*a*b^6 - 2*A*b^7)*d^3*e^2 + 27*(21*B*a^2*b^5 - A*a*b^6)*d^2*e^3 - 3*(99*B*a^3*b^4 - 29*A*a^2*b^5)*d*e^4 +
64*(B*a^4*b^3 - A*a^3*b^4)*e^5)*x^2 + 2*(80*B*b^7*d^5 - 8*(31*B*a*b^6 - 11*A*b^7)*d^4*e + 2*(107*B*a^2*b^5 - 1
72*A*a*b^6)*d^3*e^2 + (B*a^3*b^4 + 489*A*a^2*b^5)*d^2*e^3 - 2*(41*B*a^4*b^3 + 134*A*a^3*b^4)*d*e^4 + 35*(B*a^5
*b^2 + A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^4 - 4*a^6*b^7*d^3*e + 6*a^7*b^6*d^2*e^2 - 4*a^8*b^5*d*e^3
+ a^9*b^4*e^4 + (b^13*d^4 - 4*a*b^12*d^3*e + 6*a^2*b^11*d^2*e^2 - 4*a^3*b^10*d*e^3 + a^4*b^9*e^4)*x^5 + 5*(a*b
^12*d^4 - 4*a^2*b^11*d^3*e + 6*a^3*b^10*d^2*e^2 - 4*a^4*b^9*d*e^3 + a^5*b^8*e^4)*x^4 + 10*(a^2*b^11*d^4 - 4*a^
3*b^10*d^3*e + 6*a^4*b^9*d^2*e^2 - 4*a^5*b^8*d*e^3 + a^6*b^7*e^4)*x^3 + 10*(a^3*b^10*d^4 - 4*a^4*b^9*d^3*e + 6
*a^5*b^8*d^2*e^2 - 4*a^6*b^7*d*e^3 + a^7*b^6*e^4)*x^2 + 5*(a^4*b^9*d^4 - 4*a^5*b^8*d^3*e + 6*a^6*b^7*d^2*e^2 -
4*a^7*b^6*d*e^3 + a^8*b^5*e^4)*x), 1/640*(15*(2*B*a^5*b*d*e^4 - (B*a^6 + A*a^5*b)*e^5 + (2*B*b^6*d*e^4 - (B*a
*b^5 + A*b^6)*e^5)*x^5 + 5*(2*B*a*b^5*d*e^4 - (B*a^2*b^4 + A*a*b^5)*e^5)*x^4 + 10*(2*B*a^2*b^4*d*e^4 - (B*a^3*
b^3 + A*a^2*b^4)*e^5)*x^3 + 10*(2*B*a^3*b^3*d*e^4 - (B*a^4*b^2 + A*a^3*b^3)*e^5)*x^2 + 5*(2*B*a^4*b^2*d*e^4 -
(B*a^5*b + A*a^4*b^2)*e^5)*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)) -
(32*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(6*B*a^2*b^5 + 29*A*a*b^6)*d^4*e + 4*(19*B*a^3*b^4 + 146*A*a^2*b^5)*d^3*e^2 +
2*(4*B*a^4*b^3 - 129*A*a^3*b^4)*d^2*e^3 - 5*(7*B*a^5*b^2 + A*a^4*b^3)*d*e^4 + 15*(B*a^6*b + A*a^5*b^2)*e^5 -
15*(2*B*b^7*d^2*e^3 - (3*B*a*b^6 + A*b^7)*d*e^4 + (B*a^2*b^5 + A*a*b^6)*e^5)*x^4 + 10*(2*B*b^7*d^3*e^2 - (17*B
*a*b^6 + A*b^7)*d^2*e^3 + 2*(11*B*a^2*b^5 + 4*A*a*b^6)*d*e^4 - 7*(B*a^3*b^4 + A*a^2*b^5)*e^5)*x^3 + 2*(120*B*b
^7*d^4*e - 2*(227*B*a*b^6 - 2*A*b^7)*d^3*e^2 + 27*(21*B*a^2*b^5 - A*a*b^6)*d^2*e^3 - 3*(99*B*a^3*b^4 - 29*A*a^
2*b^5)*d*e^4 + 64*(B*a^4*b^3 - A*a^3*b^4)*e^5)*x^2 + 2*(80*B*b^7*d^5 - 8*(31*B*a*b^6 - 11*A*b^7)*d^4*e + 2*(10
7*B*a^2*b^5 - 172*A*a*b^6)*d^3*e^2 + (B*a^3*b^4 + 489*A*a^2*b^5)*d^2*e^3 - 2*(41*B*a^4*b^3 + 134*A*a^3*b^4)*d*
e^4 + 35*(B*a^5*b^2 + A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^4 - 4*a^6*b^7*d^3*e + 6*a^7*b^6*d^2*e^2 - 4
*a^8*b^5*d*e^3 + a^9*b^4*e^4 + (b^13*d^4 - 4*a*b^12*d^3*e + 6*a^2*b^11*d^2*e^2 - 4*a^3*b^10*d*e^3 + a^4*b^9*e^
4)*x^5 + 5*(a*b^12*d^4 - 4*a^2*b^11*d^3*e + 6*a^3*b^10*d^2*e^2 - 4*a^4*b^9*d*e^3 + a^5*b^8*e^4)*x^4 + 10*(a^2*
b^11*d^4 - 4*a^3*b^10*d^3*e + 6*a^4*b^9*d^2*e^2 - 4*a^5*b^8*d*e^3 + a^6*b^7*e^4)*x^3 + 10*(a^3*b^10*d^4 - 4*a^
4*b^9*d^3*e + 6*a^5*b^8*d^2*e^2 - 4*a^6*b^7*d*e^3 + a^7*b^6*e^4)*x^2 + 5*(a^4*b^9*d^4 - 4*a^5*b^8*d^3*e + 6*a^
6*b^7*d^2*e^2 - 4*a^7*b^6*d*e^3 + a^8*b^5*e^4)*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,x)

[Out]

Timed out

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Giac [B]  time = 1.21789, size = 1099, normalized size = 3.51 \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,x, algorithm="giac")

[Out]

3/128*(2*B*b*d*e^4 - B*a*e^5 - A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^3 - 3*a*b^5*d^2*e
+ 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*sqrt(-b^2*d + a*b*e)) + 1/640*(30*(x*e + d)^(9/2)*B*b^5*d*e^4 - 140*(x*e + d
)^(7/2)*B*b^5*d^2*e^4 + 140*(x*e + d)^(3/2)*B*b^5*d^4*e^4 - 30*sqrt(x*e + d)*B*b^5*d^5*e^4 - 15*(x*e + d)^(9/2
)*B*a*b^4*e^5 - 15*(x*e + d)^(9/2)*A*b^5*e^5 + 210*(x*e + d)^(7/2)*B*a*b^4*d*e^5 + 70*(x*e + d)^(7/2)*A*b^5*d*
e^5 + 128*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5 - 128*(x*e + d)^(5/2)*A*b^5*d^2*e^5 - 490*(x*e + d)^(3/2)*B*a*b^4*d^
3*e^5 - 70*(x*e + d)^(3/2)*A*b^5*d^3*e^5 + 135*sqrt(x*e + d)*B*a*b^4*d^4*e^5 + 15*sqrt(x*e + d)*A*b^5*d^4*e^5
- 70*(x*e + d)^(7/2)*B*a^2*b^3*e^6 - 70*(x*e + d)^(7/2)*A*a*b^4*e^6 - 256*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 + 25
6*(x*e + d)^(5/2)*A*a*b^4*d*e^6 + 630*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^6 + 210*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6
- 240*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6 - 60*sqrt(x*e + d)*A*a*b^4*d^3*e^6 + 128*(x*e + d)^(5/2)*B*a^3*b^2*e^7 -
128*(x*e + d)^(5/2)*A*a^2*b^3*e^7 - 350*(x*e + d)^(3/2)*B*a^3*b^2*d*e^7 - 210*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7
+ 210*sqrt(x*e + d)*B*a^3*b^2*d^2*e^7 + 90*sqrt(x*e + d)*A*a^2*b^3*d^2*e^7 + 70*(x*e + d)^(3/2)*B*a^4*b*e^8 +
70*(x*e + d)^(3/2)*A*a^3*b^2*e^8 - 90*sqrt(x*e + d)*B*a^4*b*d*e^8 - 60*sqrt(x*e + d)*A*a^3*b^2*d*e^8 + 15*sqr
t(x*e + d)*B*a^5*e^9 + 15*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3
)*((x*e + d)*b - b*d + a*e)^5)