3.1833 \(\int \frac{A+B x}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=400 \[ \frac{231 e^4 (3 a B e-13 A b e+10 b B d)}{128 \sqrt{d+e x} (b d-a e)^7}+\frac{77 e^4 (3 a B e-13 A b e+10 b B d)}{128 b (d+e x)^{3/2} (b d-a e)^6}+\frac{231 e^3 (3 a B e-13 A b e+10 b B d)}{640 b (a+b x) (d+e x)^{3/2} (b d-a e)^5}-\frac{33 e^2 (3 a B e-13 A b e+10 b B d)}{320 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac{231 \sqrt{b} e^4 (3 a B e-13 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}+\frac{11 e (3 a B e-13 A b e+10 b B d)}{240 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-13 A b e+10 b B d}{40 b (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \]

[Out]

(77*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*b*(b*d - a*e)^6*(d + e*x)^(3/2)) - (A*b - a*B)/(5*b*(b*d - a*e)*
(a + b*x)^5*(d + e*x)^(3/2)) - (10*b*B*d - 13*A*b*e + 3*a*B*e)/(40*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)
) + (11*e*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(240*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) - (33*e^2*(10*b*B
*d - 13*A*b*e + 3*a*B*e))/(320*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(3/2)) + (231*e^3*(10*b*B*d - 13*A*b*e +
3*a*B*e))/(640*b*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(3/2)) + (231*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*(b*
d - a*e)^7*Sqrt[d + e*x]) - (231*Sqrt[b]*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/S
qrt[b*d - a*e]])/(128*(b*d - a*e)^(15/2))

________________________________________________________________________________________

Rubi [A]  time = 0.447813, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {27, 78, 51, 63, 208} \[ \frac{231 e^4 (3 a B e-13 A b e+10 b B d)}{128 \sqrt{d+e x} (b d-a e)^7}+\frac{77 e^4 (3 a B e-13 A b e+10 b B d)}{128 b (d+e x)^{3/2} (b d-a e)^6}+\frac{231 e^3 (3 a B e-13 A b e+10 b B d)}{640 b (a+b x) (d+e x)^{3/2} (b d-a e)^5}-\frac{33 e^2 (3 a B e-13 A b e+10 b B d)}{320 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac{231 \sqrt{b} e^4 (3 a B e-13 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}+\frac{11 e (3 a B e-13 A b e+10 b B d)}{240 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-13 A b e+10 b B d}{40 b (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(77*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*b*(b*d - a*e)^6*(d + e*x)^(3/2)) - (A*b - a*B)/(5*b*(b*d - a*e)*
(a + b*x)^5*(d + e*x)^(3/2)) - (10*b*B*d - 13*A*b*e + 3*a*B*e)/(40*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)
) + (11*e*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(240*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) - (33*e^2*(10*b*B
*d - 13*A*b*e + 3*a*B*e))/(320*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(3/2)) + (231*e^3*(10*b*B*d - 13*A*b*e +
3*a*B*e))/(640*b*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(3/2)) + (231*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*(b*
d - a*e)^7*Sqrt[d + e*x]) - (231*Sqrt[b]*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/S
qrt[b*d - a*e]])/(128*(b*d - a*e)^(15/2))

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 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 )^3} \, dx &=\int \frac{A+B x}{(a+b x)^6 (d+e x)^{5/2}} \, dx\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac{(10 b B d-13 A b e+3 a B e) \int \frac{1}{(a+b x)^5 (d+e x)^{5/2}} \, dx}{10 b (b d-a e)}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac{10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac{(11 e (10 b B d-13 A b e+3 a B e)) \int \frac{1}{(a+b x)^4 (d+e x)^{5/2}} \, dx}{80 b (b d-a e)^2}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac{10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac{11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac{\left (33 e^2 (10 b B d-13 A b e+3 a B e)\right ) \int \frac{1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{160 b (b d-a e)^3}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac{10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac{11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac{33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac{\left (231 e^3 (10 b B d-13 A b e+3 a B e)\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{640 b (b d-a e)^4}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac{10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac{11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac{33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac{231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac{\left (231 e^4 (10 b B d-13 A b e+3 a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 b (b d-a e)^5}\\ &=\frac{77 e^4 (10 b B d-13 A b e+3 a B e)}{128 b (b d-a e)^6 (d+e x)^{3/2}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac{10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac{11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac{33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac{231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac{\left (231 e^4 (10 b B d-13 A b e+3 a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^6}\\ &=\frac{77 e^4 (10 b B d-13 A b e+3 a B e)}{128 b (b d-a e)^6 (d+e x)^{3/2}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac{10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac{11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac{33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac{231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac{231 e^4 (10 b B d-13 A b e+3 a B e)}{128 (b d-a e)^7 \sqrt{d+e x}}+\frac{\left (231 b e^4 (10 b B d-13 A b e+3 a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 (b d-a e)^7}\\ &=\frac{77 e^4 (10 b B d-13 A b e+3 a B e)}{128 b (b d-a e)^6 (d+e x)^{3/2}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac{10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac{11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac{33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac{231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac{231 e^4 (10 b B d-13 A b e+3 a B e)}{128 (b d-a e)^7 \sqrt{d+e x}}+\frac{\left (231 b e^3 (10 b B d-13 A b e+3 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 d-a e)^7}\\ &=\frac{77 e^4 (10 b B d-13 A b e+3 a B e)}{128 b (b d-a e)^6 (d+e x)^{3/2}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac{10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac{11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac{33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac{231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac{231 e^4 (10 b B d-13 A b e+3 a B e)}{128 (b d-a e)^7 \sqrt{d+e x}}-\frac{231 \sqrt{b} e^4 (10 b B d-13 A b e+3 a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}\\ \end{align*}

Mathematica [C]  time = 0.0978716, size = 99, normalized size = 0.25 \[ \frac{\frac{e^4 (3 a B e-13 A b e+10 b B d) \, _2F_1\left (-\frac{3}{2},5;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac{3 a B-3 A b}{(a+b x)^5}}{15 b (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*A*b + 3*a*B)/(a + b*x)^5 + (e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e)*Hypergeometric2F1[-3/2, 5, -1/2, (b*(d +
e*x))/(b*d - a*e)])/(b*d - a*e)^5)/(15*b*(b*d - a*e)*(d + e*x)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.042, size = 1653, normalized size = 4.1 \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)^3,x)

[Out]

1467/128*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A+3003/128*e^5/(a*e-b*d)^7*b^2/((a*e-b*d)*b)^(1/2)*ar
ctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A-10*e^4/(a*e-b*d)^7/(e*x+d)^(1/2)*B*b*d-2/3*e^5/(a*e-b*d)^6/(e*x+d)
^(3/2)*A+2/3*e^4/(a*e-b*d)^6/(e*x+d)^(3/2)*B*d+12*e^5/(a*e-b*d)^7/(e*x+d)^(1/2)*A*b-2*e^5/(a*e-b*d)^7/(e*x+d)^
(1/2)*a*B-843/128*e^9/(a*e-b*d)^7*b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5+921/64*e^8/(a*e-b*d)^7*b^2/(b*e*x+a*e)^5
*(e*x+d)^(1/2)*B*a^4*d+531/64*e^7/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^3*d^2-12131/64*e^7/(a*e-b*d)
^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^2*d+12131/64*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a*d^2+37
93/192*e^7/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*d+4169/64*e^6/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+
d)^(3/2)*B*a^2*d^2+1327/15*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*d^2-74/15*e^6/(a*e-b*d)^7*b^4/(
b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*d-2506/15*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a*d-363/8*e^6/(a*
e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*d^3+5277/128*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*
a*d^4-3767/192*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d-6823/64*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5
*(e*x+d)^(3/2)*B*a*d^3-2373/32*e^8/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*d+7119/64*e^7/(a*e-b*d)^7
*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^2*d^2-2373/32*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a*d^3-977
/64*e^6/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2-437/128*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(9
/2)*B*a+1253/15*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^2-172/3*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*
(e*x+d)^(5/2)*B*d^3+4075/96*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*d^4-765/64*e^4/(a*e-b*d)^7*b^6/(
b*e*x+a*e)^5*(e*x+d)^(1/2)*B*d^5+3349/96*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*d^2-1155/64*e^4/(a*
e-b*d)^7*b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*d+1253/15*e^5/(a*e-b*d)^7*b^6/(
b*e*x+a*e)^5*(e*x+d)^(5/2)*A*d^2+2373/128*e^9/(a*e-b*d)^7*b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^4+2373/128*e^5/(
a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*d^4+9629/192*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*a-
9629/192*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*d-515/64*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^
(9/2)*B*d-131/5*e^7/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3+12131/192*e^8/(a*e-b*d)^7*b^3/(b*e*x+a*e
)^5*(e*x+d)^(3/2)*A*a^3-12131/192*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*d^3-1327/64*e^8/(a*e-b*d)^
7*b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^4-693/128*e^5/(a*e-b*d)^7*b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/(
(a*e-b*d)*b)^(1/2))*a*B

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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)^(5/2)/(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 = 2.31753, size = 10005, normalized size = 25.01 \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)^3,x, algorithm="fricas")

[Out]

[1/3840*(3465*(10*B*a^5*b*d^3*e^4 + (3*B*a^6 - 13*A*a^5*b)*d^2*e^5 + (10*B*b^6*d*e^6 + (3*B*a*b^5 - 13*A*b^6)*
e^7)*x^7 + (20*B*b^6*d^2*e^5 + 2*(28*B*a*b^5 - 13*A*b^6)*d*e^6 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*e^7)*x^6 + (10*B
*b^6*d^3*e^4 + (103*B*a*b^5 - 13*A*b^6)*d^2*e^5 + 130*(B*a^2*b^4 - A*a*b^5)*d*e^6 + 10*(3*B*a^3*b^3 - 13*A*a^2
*b^4)*e^7)*x^5 + 5*(10*B*a*b^5*d^3*e^4 + (43*B*a^2*b^4 - 13*A*a*b^5)*d^2*e^5 + 4*(8*B*a^3*b^3 - 13*A*a^2*b^4)*
d*e^6 + 2*(3*B*a^4*b^2 - 13*A*a^3*b^3)*e^7)*x^4 + 5*(20*B*a^2*b^4*d^3*e^4 + 2*(23*B*a^3*b^3 - 13*A*a^2*b^4)*d^
2*e^5 + 2*(11*B*a^4*b^2 - 26*A*a^3*b^3)*d*e^6 + (3*B*a^5*b - 13*A*a^4*b^2)*e^7)*x^3 + (100*B*a^3*b^3*d^3*e^4 +
130*(B*a^4*b^2 - A*a^3*b^3)*d^2*e^5 + 10*(4*B*a^5*b - 13*A*a^4*b^2)*d*e^6 + (3*B*a^6 - 13*A*a^5*b)*e^7)*x^2 +
(50*B*a^4*b^2*d^3*e^4 + 5*(7*B*a^5*b - 13*A*a^4*b^2)*d^2*e^5 + 2*(3*B*a^6 - 13*A*a^5*b)*d*e^6)*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*(1280*A*a^
6*e^6 - 96*(B*a*b^5 + 4*A*b^6)*d^6 + 16*(52*B*a^2*b^4 + 183*A*a*b^5)*d^5*e - 28*(127*B*a^3*b^3 + 358*A*a^2*b^4
)*d^4*e^2 + 70*(174*B*a^4*b^2 + 301*A*a^3*b^3)*d^3*e^3 + 5*(6625*B*a^5*b - 7119*A*a^4*b^2)*d^2*e^4 + 1280*(2*B
*a^6 - 19*A*a^5*b)*d*e^5 + 3465*(10*B*b^6*d*e^5 + (3*B*a*b^5 - 13*A*b^6)*e^6)*x^6 + 2310*(20*B*b^6*d^2*e^4 + 2
*(38*B*a*b^5 - 13*A*b^6)*d*e^5 + 7*(3*B*a^2*b^4 - 13*A*a*b^5)*e^6)*x^5 + 231*(30*B*b^6*d^3*e^3 + 13*(73*B*a*b^
5 - 3*A*b^6)*d^2*e^4 + 2*(781*B*a^2*b^4 - 611*A*a*b^5)*d*e^5 + 128*(3*B*a^3*b^3 - 13*A*a^2*b^4)*e^6)*x^4 - 66*
(30*B*b^6*d^4*e^2 - 3*(167*B*a*b^5 + 13*A*b^6)*d^3*e^3 - (6223*B*a^2*b^4 - 663*A*a*b^5)*d^2*e^4 - (5771*B*a^3*
b^3 - 7891*A*a^2*b^4)*d*e^5 - 395*(3*B*a^4*b^2 - 13*A*a^3*b^3)*e^6)*x^3 + 11*(80*B*b^6*d^5*e - 4*(209*B*a*b^5
+ 26*A*b^6)*d^4*e^2 + 2*(2811*B*a^2*b^4 + 559*A*a*b^5)*d^3*e^3 + 4*(8566*B*a^3*b^3 - 1911*A*a^2*b^4)*d^2*e^4 +
50*(388*B*a^4*b^2 - 845*A*a^3*b^3)*d*e^5 + 965*(3*B*a^5*b - 13*A*a^4*b^2)*e^6)*x^2 - 2*(240*B*b^6*d^6 - 8*(25
1*B*a*b^5 + 39*A*b^6)*d^5*e + 2*(4133*B*a^2*b^4 + 1352*A*a*b^5)*d^4*e^2 - 7*(3969*B*a^3*b^3 + 1651*A*a^2*b^4)*
d^3*e^3 - 5*(16657*B*a^4*b^2 - 7917*A*a^3*b^3)*d^2*e^4 - 5*(5729*B*a^5*b - 19279*A*a^4*b^2)*d*e^5 - 640*(3*B*a
^6 - 13*A*a^5*b)*e^6)*x)*sqrt(e*x + d))/(a^5*b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e^2 - 35*a^8*b^4*d^6*e
^3 + 35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5 + 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d^7*e^2 - 7*a*b^11*d^6
*e^3 + 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7 + 7*a^6*b^6*d*e^8 -
a^7*b^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5*e^4 - 105*a^4*b^8*d^
4*e^5 + 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6 + (b^12*d^9 + 3*a*b^1
1*d^8*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^4*e^5 + 147*a^6*b^6*d
^3*e^6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3*a^2*b^10*d^8*e - 5*a
^3*b^9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5*d^3*e^6 - 15*a^8*b^4
*d^2*e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e + 15*a^4*b^8*d^7*e^2
+ 7*a^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^9*b^3*d^2*e^7 + 3*a^1
0*b^2*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2 - 147*a^6*b^6*d^6*e^3
+ 21*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 - 105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 - 3*a^11*b*d*e^8 - a^1
2*e^9)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3 + 105*a^8*b^4*d^5*e^
4 - 35*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 + 9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*x), -1/1920*(3465*(10*B*a^5*b*d
^3*e^4 + (3*B*a^6 - 13*A*a^5*b)*d^2*e^5 + (10*B*b^6*d*e^6 + (3*B*a*b^5 - 13*A*b^6)*e^7)*x^7 + (20*B*b^6*d^2*e^
5 + 2*(28*B*a*b^5 - 13*A*b^6)*d*e^6 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*e^7)*x^6 + (10*B*b^6*d^3*e^4 + (103*B*a*b^5
- 13*A*b^6)*d^2*e^5 + 130*(B*a^2*b^4 - A*a*b^5)*d*e^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*e^7)*x^5 + 5*(10*B*a*
b^5*d^3*e^4 + (43*B*a^2*b^4 - 13*A*a*b^5)*d^2*e^5 + 4*(8*B*a^3*b^3 - 13*A*a^2*b^4)*d*e^6 + 2*(3*B*a^4*b^2 - 13
*A*a^3*b^3)*e^7)*x^4 + 5*(20*B*a^2*b^4*d^3*e^4 + 2*(23*B*a^3*b^3 - 13*A*a^2*b^4)*d^2*e^5 + 2*(11*B*a^4*b^2 - 2
6*A*a^3*b^3)*d*e^6 + (3*B*a^5*b - 13*A*a^4*b^2)*e^7)*x^3 + (100*B*a^3*b^3*d^3*e^4 + 130*(B*a^4*b^2 - A*a^3*b^3
)*d^2*e^5 + 10*(4*B*a^5*b - 13*A*a^4*b^2)*d*e^6 + (3*B*a^6 - 13*A*a^5*b)*e^7)*x^2 + (50*B*a^4*b^2*d^3*e^4 + 5*
(7*B*a^5*b - 13*A*a^4*b^2)*d^2*e^5 + 2*(3*B*a^6 - 13*A*a^5*b)*d*e^6)*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)) - (1280*A*a^6*e^6 - 96*(B*a*b^5 + 4*A*b^6)*d^6 + 16*(52*B
*a^2*b^4 + 183*A*a*b^5)*d^5*e - 28*(127*B*a^3*b^3 + 358*A*a^2*b^4)*d^4*e^2 + 70*(174*B*a^4*b^2 + 301*A*a^3*b^3
)*d^3*e^3 + 5*(6625*B*a^5*b - 7119*A*a^4*b^2)*d^2*e^4 + 1280*(2*B*a^6 - 19*A*a^5*b)*d*e^5 + 3465*(10*B*b^6*d*e
^5 + (3*B*a*b^5 - 13*A*b^6)*e^6)*x^6 + 2310*(20*B*b^6*d^2*e^4 + 2*(38*B*a*b^5 - 13*A*b^6)*d*e^5 + 7*(3*B*a^2*b
^4 - 13*A*a*b^5)*e^6)*x^5 + 231*(30*B*b^6*d^3*e^3 + 13*(73*B*a*b^5 - 3*A*b^6)*d^2*e^4 + 2*(781*B*a^2*b^4 - 611
*A*a*b^5)*d*e^5 + 128*(3*B*a^3*b^3 - 13*A*a^2*b^4)*e^6)*x^4 - 66*(30*B*b^6*d^4*e^2 - 3*(167*B*a*b^5 + 13*A*b^6
)*d^3*e^3 - (6223*B*a^2*b^4 - 663*A*a*b^5)*d^2*e^4 - (5771*B*a^3*b^3 - 7891*A*a^2*b^4)*d*e^5 - 395*(3*B*a^4*b^
2 - 13*A*a^3*b^3)*e^6)*x^3 + 11*(80*B*b^6*d^5*e - 4*(209*B*a*b^5 + 26*A*b^6)*d^4*e^2 + 2*(2811*B*a^2*b^4 + 559
*A*a*b^5)*d^3*e^3 + 4*(8566*B*a^3*b^3 - 1911*A*a^2*b^4)*d^2*e^4 + 50*(388*B*a^4*b^2 - 845*A*a^3*b^3)*d*e^5 + 9
65*(3*B*a^5*b - 13*A*a^4*b^2)*e^6)*x^2 - 2*(240*B*b^6*d^6 - 8*(251*B*a*b^5 + 39*A*b^6)*d^5*e + 2*(4133*B*a^2*b
^4 + 1352*A*a*b^5)*d^4*e^2 - 7*(3969*B*a^3*b^3 + 1651*A*a^2*b^4)*d^3*e^3 - 5*(16657*B*a^4*b^2 - 7917*A*a^3*b^3
)*d^2*e^4 - 5*(5729*B*a^5*b - 19279*A*a^4*b^2)*d*e^5 - 640*(3*B*a^6 - 13*A*a^5*b)*e^6)*x)*sqrt(e*x + d))/(a^5*
b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e^2 - 35*a^8*b^4*d^6*e^3 + 35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5
+ 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d^7*e^2 - 7*a*b^11*d^6*e^3 + 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e
^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7 + 7*a^6*b^6*d*e^8 - a^7*b^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d
^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5*e^4 - 105*a^4*b^8*d^4*e^5 + 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*
e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6 + (b^12*d^9 + 3*a*b^11*d^8*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d
^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^4*e^5 + 147*a^6*b^6*d^3*e^6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e
^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3*a^2*b^10*d^8*e - 5*a^3*b^9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*
b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5*d^3*e^6 - 15*a^8*b^4*d^2*e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)
*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e + 15*a^4*b^8*d^7*e^2 + 7*a^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63
*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^9*b^3*d^2*e^7 + 3*a^10*b^2*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d
^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2 - 147*a^6*b^6*d^6*e^3 + 21*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 -
105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 - 3*a^11*b*d*e^8 - a^12*e^9)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*
e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3 + 105*a^8*b^4*d^5*e^4 - 35*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 +
9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.34682, size = 1489, normalized size = 3.72 \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)^3,x, algorithm="giac")

[Out]

231/128*(10*B*b^2*d*e^4 + 3*B*a*b*e^5 - 13*A*b^2*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^7*d^7 -
7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d
*e^6 - a^7*e^7)*sqrt(-b^2*d + a*b*e)) + 2/3*(15*(x*e + d)*B*b*d*e^4 + B*b*d^2*e^4 + 3*(x*e + d)*B*a*e^5 - 18*(
x*e + d)*A*b*e^5 - B*a*d*e^5 - A*b*d*e^5 + A*a*e^6)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^
4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(x*e + d)^(3/2)) + 1/1920*(1545
0*(x*e + d)^(9/2)*B*b^6*d*e^4 - 66980*(x*e + d)^(7/2)*B*b^6*d^2*e^4 + 110080*(x*e + d)^(5/2)*B*b^6*d^3*e^4 - 8
1500*(x*e + d)^(3/2)*B*b^6*d^4*e^4 + 22950*sqrt(x*e + d)*B*b^6*d^5*e^4 + 6555*(x*e + d)^(9/2)*B*a*b^5*e^5 - 22
005*(x*e + d)^(9/2)*A*b^6*e^5 + 37670*(x*e + d)^(7/2)*B*a*b^5*d*e^5 + 96290*(x*e + d)^(7/2)*A*b^6*d*e^5 - 1698
56*(x*e + d)^(5/2)*B*a*b^5*d^2*e^5 - 160384*(x*e + d)^(5/2)*A*b^6*d^2*e^5 + 204690*(x*e + d)^(3/2)*B*a*b^5*d^3
*e^5 + 121310*(x*e + d)^(3/2)*A*b^6*d^3*e^5 - 79155*sqrt(x*e + d)*B*a*b^5*d^4*e^5 - 35595*sqrt(x*e + d)*A*b^6*
d^4*e^5 + 29310*(x*e + d)^(7/2)*B*a^2*b^4*e^6 - 96290*(x*e + d)^(7/2)*A*a*b^5*e^6 + 9472*(x*e + d)^(5/2)*B*a^2
*b^4*d*e^6 + 320768*(x*e + d)^(5/2)*A*a*b^5*d*e^6 - 125070*(x*e + d)^(3/2)*B*a^2*b^4*d^2*e^6 - 363930*(x*e + d
)^(3/2)*A*a*b^5*d^2*e^6 + 87120*sqrt(x*e + d)*B*a^2*b^4*d^3*e^6 + 142380*sqrt(x*e + d)*A*a*b^5*d^3*e^6 + 50304
*(x*e + d)^(5/2)*B*a^3*b^3*e^7 - 160384*(x*e + d)^(5/2)*A*a^2*b^4*e^7 - 37930*(x*e + d)^(3/2)*B*a^3*b^3*d*e^7
+ 363930*(x*e + d)^(3/2)*A*a^2*b^4*d*e^7 - 15930*sqrt(x*e + d)*B*a^3*b^3*d^2*e^7 - 213570*sqrt(x*e + d)*A*a^2*
b^4*d^2*e^7 + 39810*(x*e + d)^(3/2)*B*a^4*b^2*e^8 - 121310*(x*e + d)^(3/2)*A*a^3*b^3*e^8 - 27630*sqrt(x*e + d)
*B*a^4*b^2*d*e^8 + 142380*sqrt(x*e + d)*A*a^3*b^3*d*e^8 + 12645*sqrt(x*e + d)*B*a^5*b*e^9 - 35595*sqrt(x*e + d
)*A*a^4*b^2*e^9)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21
*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*((x*e + d)*b - b*d + a*e)^5)