3.1882 \(\int \frac{A+B x}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=496 \[ -\frac{105 e^3 (a+b x) (3 a B e-11 A b e+8 b B d)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}-\frac{35 e^3 (a+b x) (3 a B e-11 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}-\frac{21 e^2 (3 a B e-11 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac{105 \sqrt{b} e^3 (a+b x) (3 a B e-11 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}}+\frac{3 e (3 a B e-11 A b e+8 b B d)}{32 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-11 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)} \]

[Out]

(-21*e^2*(8*b*B*d - 11*A*b*e + 3*a*B*e))/(64*b*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 11*A*b*e
+ 3*a*B*e)/(24*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e*(8*b*B*d - 11
*A*b*e + 3*a*B*e))/(32*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b
*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(64*b*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10
5*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(64*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (105*Sqrt[b]*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])
/(64*(b*d - a*e)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.545861, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{105 e^3 (a+b x) (3 a B e-11 A b e+8 b B d)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}-\frac{35 e^3 (a+b x) (3 a B e-11 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}-\frac{21 e^2 (3 a B e-11 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac{105 \sqrt{b} e^3 (a+b x) (3 a B e-11 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}}+\frac{3 e (3 a B e-11 A b e+8 b B d)}{32 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-11 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*e^2*(8*b*B*d - 11*A*b*e + 3*a*B*e))/(64*b*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 11*A*b*e
+ 3*a*B*e)/(24*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e*(8*b*B*d - 11
*A*b*e + 3*a*B*e))/(32*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b
*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(64*b*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10
5*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(64*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (105*Sqrt[b]*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])
/(64*(b*d - a*e)^(13/2)*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)^{5/2} \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)^{5/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^2 (8 b B d-11 A b e+3 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^4 (d+e x)^{5/2}} \, dx}{8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (3 b e (8 b B d-11 A b e+3 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^3 (d+e x)^{5/2}} \, dx}{16 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (21 e^2 (8 b B d-11 A b e+3 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)^{5/2}} \, dx}{64 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (105 e^3 (8 b B d-11 A b e+3 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{128 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 b (b d-a e)^5 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (105 e^3 (8 b B d-11 A b e+3 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}{128 (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 b (b d-a e)^5 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 (b d-a e)^6 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (105 b e^3 (8 b B d-11 A b e+3 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 d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 b (b d-a e)^5 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 (b d-a e)^6 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (105 b e^2 (8 b B d-11 A b e+3 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 d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 b (b d-a e)^5 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 (b d-a e)^6 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 \sqrt{b} e^3 (8 b B d-11 A b e+3 a B e) (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 (b d-a e)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.10832, size = 115, normalized size = 0.23 \[ \frac{\frac{e^3 (a+b x)^4 (-3 a B e+11 A b e-8 b B d) \, _2F_1\left (-\frac{3}{2},4;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+3 a B-3 A b}{12 b (a+b x)^3 \sqrt{(a+b x)^2} (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*A*b + 3*a*B + (e^3*(-8*b*B*d + 11*A*b*e - 3*a*B*e)*(a + b*x)^4*Hypergeometric2F1[-3/2, 4, -1/2, (b*(d + e*
x))/(b*d - a*e)])/(b*d - a*e)^4)/(12*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2))

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Maple [B]  time = 0.039, size = 1860, normalized size = 3.8 \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(-504*B*((a*e-b*d)*b)^(1/2)*x^3*b^5*d^3*e^2-945*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3
/2)*a^5*b*e^4+9207*A*((a*e-b*d)*b)^(1/2)*x^2*a^3*b^2*e^5-198*A*((a*e-b*d)*b)^(1/2)*x^2*b^5*d^3*e^2-2511*B*((a*
e-b*d)*b)^(1/2)*x^2*a^4*b*e^5+144*B*((a*e-b*d)*b)^(1/2)*x^2*b^5*d^4*e+1408*A*((a*e-b*d)*b)^(1/2)*x*a^4*b*e^5+8
8*A*((a*e-b*d)*b)^(1/2)*x*b^5*d^4*e+3465*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*x^4*b^6*e
^4-945*B*((a*e-b*d)*b)^(1/2)*x^5*a*b^4*e^5-2520*B*((a*e-b*d)*b)^(1/2)*x^5*b^5*d*e^4+12705*A*((a*e-b*d)*b)^(1/2
)*x^4*a*b^4*e^5+4620*A*((a*e-b*d)*b)^(1/2)*x^4*b^5*d*e^4-3465*B*((a*e-b*d)*b)^(1/2)*x^4*a^2*b^3*e^5-3360*B*((a
*e-b*d)*b)^(1/2)*x^4*b^5*d^2*e^3+16863*A*((a*e-b*d)*b)^(1/2)*x^3*a^2*b^3*e^5+693*A*((a*e-b*d)*b)^(1/2)*x^3*b^5
*d^2*e^3+3465*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*a^4*b^2*e^4+2048*A*((a*e-b*d)*b)^(1/
2)*a^4*b*d*e^4+2295*A*((a*e-b*d)*b)^(1/2)*a^3*b^2*d^2*e^3-1030*A*((a*e-b*d)*b)^(1/2)*a^2*b^3*d^3*e^2+328*A*((a
*e-b*d)*b)^(1/2)*a*b^4*d^4*e-2639*B*((a*e-b*d)*b)^(1/2)*a^4*b*d^2*e^3-690*B*((a*e-b*d)*b)^(1/2)*a^3*b^2*d^3*e^
2+136*B*((a*e-b*d)*b)^(1/2)*a^2*b^3*d^4*e-128*A*((a*e-b*d)*b)^(1/2)*a^5*e^5-48*A*((a*e-b*d)*b)^(1/2)*b^5*d^5+2
0790*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*x^2*a^2*b^4*e^4-5670*B*arctan((e*x+d)^(1/2)*b
/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*x^2*a^3*b^3*e^4+13860*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d
)^(3/2)*x*a^3*b^3*e^4-10500*B*((a*e-b*d)*b)^(1/2)*x^4*a*b^4*d*e^4-3780*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^
(1/2))*(e*x+d)^(3/2)*x*a^4*b^2*e^4+17094*A*((a*e-b*d)*b)^(1/2)*x^3*a*b^4*d*e^4-16926*B*((a*e-b*d)*b)^(1/2)*x^3
*a^2*b^3*d*e^4-12621*B*((a*e-b*d)*b)^(1/2)*x^3*a*b^4*d^2*e^3-2520*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2)
)*(e*x+d)^(3/2)*a^4*b^2*d*e^3+22968*A*((a*e-b*d)*b)^(1/2)*x^2*a^2*b^3*d*e^4+2673*A*((a*e-b*d)*b)^(1/2)*x^2*a*b
^4*d^2*e^3-12960*B*((a*e-b*d)*b)^(1/2)*x^2*a^3*b^2*d*e^4-17433*B*((a*e-b*d)*b)^(1/2)*x^2*a^2*b^3*d^2*e^3-1890*
B*((a*e-b*d)*b)^(1/2)*x^2*a*b^4*d^3*e^2+12782*A*((a*e-b*d)*b)^(1/2)*x*a^3*b^2*d*e^4+3795*A*((a*e-b*d)*b)^(1/2)
*x*a^2*b^3*d^2*e^3-748*A*((a*e-b*d)*b)^(1/2)*x*a*b^4*d^3*e^2-4510*B*((a*e-b*d)*b)^(1/2)*x*a^4*b*d*e^4-945*B*ar
ctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*x^4*a*b^5*e^4-2520*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)
*b)^(1/2))*(e*x+d)^(3/2)*x^4*b^6*d*e^3+13860*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*x^3*a
*b^5*e^4-3780*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*x^3*a^2*b^4*e^4-10331*B*((a*e-b*d)*b
)^(1/2)*x*a^3*b^2*d^2*e^3-2556*B*((a*e-b*d)*b)^(1/2)*x*a^2*b^3*d^3*e^2+520*B*((a*e-b*d)*b)^(1/2)*x*a*b^4*d^4*e
-4599*B*((a*e-b*d)*b)^(1/2)*x^3*a^3*b^2*e^5-10080*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*
x*a^3*b^3*d*e^3+3465*A*((a*e-b*d)*b)^(1/2)*x^5*b^5*e^5-384*B*((a*e-b*d)*b)^(1/2)*x*a^5*e^5-64*B*((a*e-b*d)*b)^
(1/2)*x*b^5*d^5-256*B*((a*e-b*d)*b)^(1/2)*a^5*d*e^4-16*B*((a*e-b*d)*b)^(1/2)*a*b^4*d^5-10080*B*arctan((e*x+d)^
(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*x^3*a*b^5*d*e^3-15120*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))
*(e*x+d)^(3/2)*x^2*a^2*b^4*d*e^3)*(b*x+a)/((a*e-b*d)*b)^(1/2)/(e*x+d)^(3/2)/(a*e-b*d)^6/((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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B]  time = 2.36088, size = 7596, normalized size = 15.31 \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(315*(8*B*a^4*b*d^3*e^3 + (3*B*a^5 - 11*A*a^4*b)*d^2*e^4 + (8*B*b^5*d*e^5 + (3*B*a*b^4 - 11*A*b^5)*e^6
)*x^6 + 2*(8*B*b^5*d^2*e^4 + (19*B*a*b^4 - 11*A*b^5)*d*e^5 + 2*(3*B*a^2*b^3 - 11*A*a*b^4)*e^6)*x^5 + (8*B*b^5*
d^3*e^3 + (67*B*a*b^4 - 11*A*b^5)*d^2*e^4 + 8*(9*B*a^2*b^3 - 11*A*a*b^4)*d*e^5 + 6*(3*B*a^3*b^2 - 11*A*a^2*b^3
)*e^6)*x^4 + 4*(8*B*a*b^4*d^3*e^3 + (27*B*a^2*b^3 - 11*A*a*b^4)*d^2*e^4 + (17*B*a^3*b^2 - 33*A*a^2*b^3)*d*e^5
+ (3*B*a^4*b - 11*A*a^3*b^2)*e^6)*x^3 + (48*B*a^2*b^3*d^3*e^3 + 2*(41*B*a^3*b^2 - 33*A*a^2*b^3)*d^2*e^4 + 8*(4
*B*a^4*b - 11*A*a^3*b^2)*d*e^5 + (3*B*a^5 - 11*A*a^4*b)*e^6)*x^2 + 2*(16*B*a^3*b^2*d^3*e^3 + 2*(7*B*a^4*b - 11
*A*a^3*b^2)*d^2*e^4 + (3*B*a^5 - 11*A*a^4*b)*d*e^5)*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*(128*A*a^5*e^5 + 16*(B*a*b^4 + 3*A*b^5)*d^5 - 8*(17*B*
a^2*b^3 + 41*A*a*b^4)*d^4*e + 10*(69*B*a^3*b^2 + 103*A*a^2*b^3)*d^3*e^2 + (2639*B*a^4*b - 2295*A*a^3*b^2)*d^2*
e^3 + 256*(B*a^5 - 8*A*a^4*b)*d*e^4 + 315*(8*B*b^5*d*e^4 + (3*B*a*b^4 - 11*A*b^5)*e^5)*x^5 + 105*(32*B*b^5*d^2
*e^3 + 4*(25*B*a*b^4 - 11*A*b^5)*d*e^4 + 11*(3*B*a^2*b^3 - 11*A*a*b^4)*e^5)*x^4 + 21*(24*B*b^5*d^3*e^2 + (601*
B*a*b^4 - 33*A*b^5)*d^2*e^3 + 2*(403*B*a^2*b^3 - 407*A*a*b^4)*d*e^4 + 73*(3*B*a^3*b^2 - 11*A*a^2*b^3)*e^5)*x^3
- 9*(16*B*b^5*d^4*e - 2*(105*B*a*b^4 + 11*A*b^5)*d^3*e^2 - (1937*B*a^2*b^3 - 297*A*a*b^4)*d^2*e^3 - 8*(180*B*
a^3*b^2 - 319*A*a^2*b^3)*d*e^4 - 93*(3*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + (64*B*b^5*d^5 - 8*(65*B*a*b^4 + 11*A
*b^5)*d^4*e + 4*(639*B*a^2*b^3 + 187*A*a*b^4)*d^3*e^2 + (10331*B*a^3*b^2 - 3795*A*a^2*b^3)*d^2*e^3 + 22*(205*B
*a^4*b - 581*A*a^3*b^2)*d*e^4 + 128*(3*B*a^5 - 11*A*a^4*b)*e^5)*x)*sqrt(e*x + d))/(a^4*b^6*d^8 - 6*a^5*b^5*d^7
*e + 15*a^6*b^4*d^6*e^2 - 20*a^7*b^3*d^5*e^3 + 15*a^8*b^2*d^4*e^4 - 6*a^9*b*d^3*e^5 + a^10*d^2*e^6 + (b^10*d^6
*e^2 - 6*a*b^9*d^5*e^3 + 15*a^2*b^8*d^4*e^4 - 20*a^3*b^7*d^3*e^5 + 15*a^4*b^6*d^2*e^6 - 6*a^5*b^5*d*e^7 + a^6*
b^4*e^8)*x^6 + 2*(b^10*d^7*e - 4*a*b^9*d^6*e^2 + 3*a^2*b^8*d^5*e^3 + 10*a^3*b^7*d^4*e^4 - 25*a^4*b^6*d^3*e^5 +
24*a^5*b^5*d^2*e^6 - 11*a^6*b^4*d*e^7 + 2*a^7*b^3*e^8)*x^5 + (b^10*d^8 + 2*a*b^9*d^7*e - 27*a^2*b^8*d^6*e^2 +
64*a^3*b^7*d^5*e^3 - 55*a^4*b^6*d^4*e^4 - 6*a^5*b^5*d^3*e^5 + 43*a^6*b^4*d^2*e^6 - 28*a^7*b^3*d*e^7 + 6*a^8*b
^2*e^8)*x^4 + 4*(a*b^9*d^8 - 3*a^2*b^8*d^7*e - 2*a^3*b^7*d^6*e^2 + 19*a^4*b^6*d^5*e^3 - 30*a^5*b^5*d^4*e^4 + 1
9*a^6*b^4*d^3*e^5 - 2*a^7*b^3*d^2*e^6 - 3*a^8*b^2*d*e^7 + a^9*b*e^8)*x^3 + (6*a^2*b^8*d^8 - 28*a^3*b^7*d^7*e +
43*a^4*b^6*d^6*e^2 - 6*a^5*b^5*d^5*e^3 - 55*a^6*b^4*d^4*e^4 + 64*a^7*b^3*d^3*e^5 - 27*a^8*b^2*d^2*e^6 + 2*a^9
*b*d*e^7 + a^10*e^8)*x^2 + 2*(2*a^3*b^7*d^8 - 11*a^4*b^6*d^7*e + 24*a^5*b^5*d^6*e^2 - 25*a^6*b^4*d^5*e^3 + 10*
a^7*b^3*d^4*e^4 + 3*a^8*b^2*d^3*e^5 - 4*a^9*b*d^2*e^6 + a^10*d*e^7)*x), 1/192*(315*(8*B*a^4*b*d^3*e^3 + (3*B*a
^5 - 11*A*a^4*b)*d^2*e^4 + (8*B*b^5*d*e^5 + (3*B*a*b^4 - 11*A*b^5)*e^6)*x^6 + 2*(8*B*b^5*d^2*e^4 + (19*B*a*b^4
- 11*A*b^5)*d*e^5 + 2*(3*B*a^2*b^3 - 11*A*a*b^4)*e^6)*x^5 + (8*B*b^5*d^3*e^3 + (67*B*a*b^4 - 11*A*b^5)*d^2*e^
4 + 8*(9*B*a^2*b^3 - 11*A*a*b^4)*d*e^5 + 6*(3*B*a^3*b^2 - 11*A*a^2*b^3)*e^6)*x^4 + 4*(8*B*a*b^4*d^3*e^3 + (27*
B*a^2*b^3 - 11*A*a*b^4)*d^2*e^4 + (17*B*a^3*b^2 - 33*A*a^2*b^3)*d*e^5 + (3*B*a^4*b - 11*A*a^3*b^2)*e^6)*x^3 +
(48*B*a^2*b^3*d^3*e^3 + 2*(41*B*a^3*b^2 - 33*A*a^2*b^3)*d^2*e^4 + 8*(4*B*a^4*b - 11*A*a^3*b^2)*d*e^5 + (3*B*a^
5 - 11*A*a^4*b)*e^6)*x^2 + 2*(16*B*a^3*b^2*d^3*e^3 + 2*(7*B*a^4*b - 11*A*a^3*b^2)*d^2*e^4 + (3*B*a^5 - 11*A*a^
4*b)*d*e^5)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (1
28*A*a^5*e^5 + 16*(B*a*b^4 + 3*A*b^5)*d^5 - 8*(17*B*a^2*b^3 + 41*A*a*b^4)*d^4*e + 10*(69*B*a^3*b^2 + 103*A*a^2
*b^3)*d^3*e^2 + (2639*B*a^4*b - 2295*A*a^3*b^2)*d^2*e^3 + 256*(B*a^5 - 8*A*a^4*b)*d*e^4 + 315*(8*B*b^5*d*e^4 +
(3*B*a*b^4 - 11*A*b^5)*e^5)*x^5 + 105*(32*B*b^5*d^2*e^3 + 4*(25*B*a*b^4 - 11*A*b^5)*d*e^4 + 11*(3*B*a^2*b^3 -
11*A*a*b^4)*e^5)*x^4 + 21*(24*B*b^5*d^3*e^2 + (601*B*a*b^4 - 33*A*b^5)*d^2*e^3 + 2*(403*B*a^2*b^3 - 407*A*a*b
^4)*d*e^4 + 73*(3*B*a^3*b^2 - 11*A*a^2*b^3)*e^5)*x^3 - 9*(16*B*b^5*d^4*e - 2*(105*B*a*b^4 + 11*A*b^5)*d^3*e^2
- (1937*B*a^2*b^3 - 297*A*a*b^4)*d^2*e^3 - 8*(180*B*a^3*b^2 - 319*A*a^2*b^3)*d*e^4 - 93*(3*B*a^4*b - 11*A*a^3*
b^2)*e^5)*x^2 + (64*B*b^5*d^5 - 8*(65*B*a*b^4 + 11*A*b^5)*d^4*e + 4*(639*B*a^2*b^3 + 187*A*a*b^4)*d^3*e^2 + (1
0331*B*a^3*b^2 - 3795*A*a^2*b^3)*d^2*e^3 + 22*(205*B*a^4*b - 581*A*a^3*b^2)*d*e^4 + 128*(3*B*a^5 - 11*A*a^4*b)
*e^5)*x)*sqrt(e*x + d))/(a^4*b^6*d^8 - 6*a^5*b^5*d^7*e + 15*a^6*b^4*d^6*e^2 - 20*a^7*b^3*d^5*e^3 + 15*a^8*b^2*
d^4*e^4 - 6*a^9*b*d^3*e^5 + a^10*d^2*e^6 + (b^10*d^6*e^2 - 6*a*b^9*d^5*e^3 + 15*a^2*b^8*d^4*e^4 - 20*a^3*b^7*d
^3*e^5 + 15*a^4*b^6*d^2*e^6 - 6*a^5*b^5*d*e^7 + a^6*b^4*e^8)*x^6 + 2*(b^10*d^7*e - 4*a*b^9*d^6*e^2 + 3*a^2*b^8
*d^5*e^3 + 10*a^3*b^7*d^4*e^4 - 25*a^4*b^6*d^3*e^5 + 24*a^5*b^5*d^2*e^6 - 11*a^6*b^4*d*e^7 + 2*a^7*b^3*e^8)*x^
5 + (b^10*d^8 + 2*a*b^9*d^7*e - 27*a^2*b^8*d^6*e^2 + 64*a^3*b^7*d^5*e^3 - 55*a^4*b^6*d^4*e^4 - 6*a^5*b^5*d^3*e
^5 + 43*a^6*b^4*d^2*e^6 - 28*a^7*b^3*d*e^7 + 6*a^8*b^2*e^8)*x^4 + 4*(a*b^9*d^8 - 3*a^2*b^8*d^7*e - 2*a^3*b^7*d
^6*e^2 + 19*a^4*b^6*d^5*e^3 - 30*a^5*b^5*d^4*e^4 + 19*a^6*b^4*d^3*e^5 - 2*a^7*b^3*d^2*e^6 - 3*a^8*b^2*d*e^7 +
a^9*b*e^8)*x^3 + (6*a^2*b^8*d^8 - 28*a^3*b^7*d^7*e + 43*a^4*b^6*d^6*e^2 - 6*a^5*b^5*d^5*e^3 - 55*a^6*b^4*d^4*e
^4 + 64*a^7*b^3*d^3*e^5 - 27*a^8*b^2*d^2*e^6 + 2*a^9*b*d*e^7 + a^10*e^8)*x^2 + 2*(2*a^3*b^7*d^8 - 11*a^4*b^6*d
^7*e + 24*a^5*b^5*d^6*e^2 - 25*a^6*b^4*d^5*e^3 + 10*a^7*b^3*d^4*e^4 + 3*a^8*b^2*d^3*e^5 - 4*a^9*b*d^2*e^6 + a^
10*d*e^7)*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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.52042, size = 1756, normalized size = 3.54 \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

-105/64*(8*B*b^2*d*e^3 + 3*B*a*b*e^4 - 11*A*b^2*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6*sg
n((x*e + d)*b*e - b*d*e + a*e^2) - 6*a*b^5*d^5*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^2*b^4*d^4*e^2*sgn((
x*e + d)*b*e - b*d*e + a*e^2) - 20*a^3*b^3*d^3*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^4*b^2*d^2*e^4*sgn
((x*e + d)*b*e - b*d*e + a*e^2) - 6*a^5*b*d*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^6*e^6*sgn((x*e + d)*b*e
- b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) - 2/3*(12*(x*e + d)*B*b*d*e^3 + B*b*d^2*e^3 + 3*(x*e + d)*B*a*e^4 - 1
5*(x*e + d)*A*b*e^4 - B*a*d*e^4 - A*b*d*e^4 + A*a*e^5)/((b^6*d^6*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a*b^5*
d^5*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^2*b^4*d^4*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 20*a^3*b^3*
d^3*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^4*b^2*d^2*e^4*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a^5*b*d
*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^6*e^6*sgn((x*e + d)*b*e - b*d*e + a*e^2))*(x*e + d)^(3/2)) - 1/192
*(984*(x*e + d)^(7/2)*B*b^5*d*e^3 - 3224*(x*e + d)^(5/2)*B*b^5*d^2*e^3 + 3560*(x*e + d)^(3/2)*B*b^5*d^3*e^3 -
1320*sqrt(x*e + d)*B*b^5*d^4*e^3 + 561*(x*e + d)^(7/2)*B*a*b^4*e^4 - 1545*(x*e + d)^(7/2)*A*b^5*e^4 + 1295*(x*
e + d)^(5/2)*B*a*b^4*d*e^4 + 5153*(x*e + d)^(5/2)*A*b^5*d*e^4 - 4825*(x*e + d)^(3/2)*B*a*b^4*d^2*e^4 - 5855*(x
*e + d)^(3/2)*A*b^5*d^2*e^4 + 2985*sqrt(x*e + d)*B*a*b^4*d^3*e^4 + 2295*sqrt(x*e + d)*A*b^5*d^3*e^4 + 1929*(x*
e + d)^(5/2)*B*a^2*b^3*e^5 - 5153*(x*e + d)^(5/2)*A*a*b^4*e^5 - 1030*(x*e + d)^(3/2)*B*a^2*b^3*d*e^5 + 11710*(
x*e + d)^(3/2)*A*a*b^4*d*e^5 - 1035*sqrt(x*e + d)*B*a^2*b^3*d^2*e^5 - 6885*sqrt(x*e + d)*A*a*b^4*d^2*e^5 + 229
5*(x*e + d)^(3/2)*B*a^3*b^2*e^6 - 5855*(x*e + d)^(3/2)*A*a^2*b^3*e^6 - 1605*sqrt(x*e + d)*B*a^3*b^2*d*e^6 + 68
85*sqrt(x*e + d)*A*a^2*b^3*d*e^6 + 975*sqrt(x*e + d)*B*a^4*b*e^7 - 2295*sqrt(x*e + d)*A*a^3*b^2*e^7)/((b^6*d^6
*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a*b^5*d^5*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^2*b^4*d^4*e^2*sg
n((x*e + d)*b*e - b*d*e + a*e^2) - 20*a^3*b^3*d^3*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 15*a^4*b^2*d^2*e^4*
sgn((x*e + d)*b*e - b*d*e + a*e^2) - 6*a^5*b*d*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^6*e^6*sgn((x*e + d)*
b*e - b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)