3.1881 \(\int \frac{A+B x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=424 \[ -\frac{35 e^3 (a+b x) (a B e-9 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}-\frac{35 e^2 (a B e-9 A b e+8 b B d)}{192 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}+\frac{35 e^3 (a+b x) (a B e-9 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}+\frac{7 e (a B e-9 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-9 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

[Out]

(-35*e^2*(8*b*B*d - 9*A*b*e + a*B*e))/(192*b*(b*d - a*e)^4*Sqrt[d + e*x]*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*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 9*A*b*e + a*B*e
)/(24*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(8*b*B*d - 9*A*b*e + a*B
*e))/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b*B*d - 9*A*b*e +
a*B*e)*(a + b*x))/(64*b*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*e^3*(8*b*B*d - 9*A*b
*e + a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d - a*e)^(11/2)*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.436505, antiderivative size = 424, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{35 e^3 (a+b x) (a B e-9 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}-\frac{35 e^2 (a B e-9 A b e+8 b B d)}{192 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}+\frac{35 e^3 (a+b x) (a B e-9 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}+\frac{7 e (a B e-9 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-9 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-35*e^2*(8*b*B*d - 9*A*b*e + a*B*e))/(192*b*(b*d - a*e)^4*Sqrt[d + e*x]*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*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 9*A*b*e + a*B*e
)/(24*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(8*b*B*d - 9*A*b*e + a*B
*e))/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b*B*d - 9*A*b*e +
a*B*e)*(a + b*x))/(64*b*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*e^3*(8*b*B*d - 9*A*b
*e + a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(b*d - a*e)^(11/2)*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 )^{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)^{3/2}} \, 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 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^2 (8 b B d-9 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 (d+e x)^{3/2}} \, 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 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (7 b e (8 b B d-9 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 (d+e x)^{3/2}} \, dx}{48 (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 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 e^2 (8 b B d-9 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}{192 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt{d+e x} \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 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 e^3 (8 b B d-9 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}{128 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt{d+e x} \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 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 e^3 (8 b B d-9 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 d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt{d+e x} \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 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 e^2 (8 b B d-9 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 d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt{d+e x} \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 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^3 (8 b B d-9 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 \sqrt{b} (b d-a e)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.104483, size = 114, normalized size = 0.27 \[ \frac{\frac{e^3 (a+b x)^4 (-a B e+9 A b e-8 b B d) \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+a B-A b}{4 b (a+b x)^3 \sqrt{(a+b x)^2} \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(A*b) + a*B + (e^3*(-8*b*B*d + 9*A*b*e - a*B*e)*(a + b*x)^4*Hypergeometric2F1[-1/2, 4, 1/2, (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]*Sqrt[d + e*x])

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Maple [B]  time = 0.032, size = 1493, normalized size = 3.5 \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)^(5/2),x)

[Out]

-1/192*(-280*B*((a*e-b*d)*b)^(1/2)*x^3*b^4*d^2*e^2-3115*B*((a*e-b*d)*b)^(1/2)*x^3*a*b^3*d*e^3+945*A*arctan((e*
x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^4*b^5*e^4-105*B*((a*e-b*d)*b)^(1/2)*x^4*a*b^3*e^4-840*B*((a*
e-b*d)*b)^(1/2)*x^4*b^4*d*e^3+3465*A*((a*e-b*d)*b)^(1/2)*x^3*a*b^3*e^4+72*A*((a*e-b*d)*b)^(1/2)*x*b^4*d^3*e+31
5*A*((a*e-b*d)*b)^(1/2)*x^3*b^4*d*e^3-385*B*((a*e-b*d)*b)^(1/2)*x^3*a^2*b^2*e^4+4599*A*((a*e-b*d)*b)^(1/2)*x^2
*a^2*b^2*e^4-126*A*((a*e-b*d)*b)^(1/2)*x^2*b^4*d^2*e^2-511*B*((a*e-b*d)*b)^(1/2)*x^2*a^3*b*e^4+112*B*((a*e-b*d
)*b)^(1/2)*x^2*b^4*d^3*e+2511*A*((a*e-b*d)*b)^(1/2)*x*a^3*b*e^4+264*A*((a*e-b*d)*b)^(1/2)*a*b^3*d^3*e-370*B*((
a*e-b*d)*b)^(1/2)*a^3*b*d^2*e^2+104*B*((a*e-b*d)*b)^(1/2)*a^2*b^2*d^3*e+975*A*((a*e-b*d)*b)^(1/2)*a^3*b*d*e^3-
630*A*((a*e-b*d)*b)^(1/2)*a^2*b^2*d^2*e^2+384*A*((a*e-b*d)*b)^(1/2)*a^4*e^4-48*A*((a*e-b*d)*b)^(1/2)*b^4*d^4+9
45*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*a^4*b*e^4-64*B*((a*e-b*d)*b)^(1/2)*x*b^4*d^4+94
5*A*((a*e-b*d)*b)^(1/2)*x^4*b^4*e^4-663*B*((a*e-b*d)*b)^(1/2)*a^4*d*e^3-16*B*((a*e-b*d)*b)^(1/2)*a*b^3*d^4-105
*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*a^5*e^4+1197*A*((a*e-b*d)*b)^(1/2)*x^2*a*b^3*d*e^
3-4221*B*((a*e-b*d)*b)^(1/2)*x^2*a^2*b^2*d*e^3-1050*B*((a*e-b*d)*b)^(1/2)*x^2*a*b^3*d^2*e^2+1665*A*((a*e-b*d)*
b)^(1/2)*x*a^2*b^2*d*e^3-468*A*((a*e-b*d)*b)^(1/2)*x*a*b^3*d^2*e^2-2417*B*((a*e-b*d)*b)^(1/2)*x*a^3*b*d*e^3-14
28*B*((a*e-b*d)*b)^(1/2)*x*a^2*b^2*d^2*e^2+408*B*((a*e-b*d)*b)^(1/2)*x*a*b^3*d^3*e-105*B*arctan((e*x+d)^(1/2)*
b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^4*a*b^4*e^4-840*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(
1/2)*x^4*b^5*d*e^3+3780*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^3*a*b^4*e^4-420*B*arctan
((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^3*a^2*b^3*e^4+5670*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b
)^(1/2))*(e*x+d)^(1/2)*x^2*a^2*b^3*e^4-630*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^2*a^3
*b^2*e^4+3780*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x*a^3*b^2*e^4-420*B*arctan((e*x+d)^(
1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x*a^4*b*e^4-840*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d
)^(1/2)*a^4*b*d*e^3-3360*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x*a^3*b^2*d*e^3-3360*B*ar
ctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)*x^3*a*b^4*d*e^3-5040*B*arctan((e*x+d)^(1/2)*b/((a*e-b*
d)*b)^(1/2))*(e*x+d)^(1/2)*x^2*a^2*b^3*d*e^3-279*B*((a*e-b*d)*b)^(1/2)*x*a^4*e^4)*(b*x+a)/((a*e-b*d)*b)^(1/2)/
(e*x+d)^(1/2)/(a*e-b*d)^5/((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{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)^(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)^(3/2)), x)

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Fricas [B]  time = 2.19571, size = 6215, normalized size = 14.66 \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)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(105*(8*B*a^4*b*d^2*e^3 + (B*a^5 - 9*A*a^4*b)*d*e^4 + (8*B*b^5*d*e^4 + (B*a*b^4 - 9*A*b^5)*e^5)*x^5 + (
8*B*b^5*d^2*e^3 + 3*(11*B*a*b^4 - 3*A*b^5)*d*e^4 + 4*(B*a^2*b^3 - 9*A*a*b^4)*e^5)*x^4 + 2*(16*B*a*b^4*d^2*e^3
+ 2*(13*B*a^2*b^3 - 9*A*a*b^4)*d*e^4 + 3*(B*a^3*b^2 - 9*A*a^2*b^3)*e^5)*x^3 + 2*(24*B*a^2*b^3*d^2*e^3 + (19*B*
a^3*b^2 - 27*A*a^2*b^3)*d*e^4 + 2*(B*a^4*b - 9*A*a^3*b^2)*e^5)*x^2 + (32*B*a^3*b^2*d^2*e^3 + 12*(B*a^4*b - 3*A
*a^3*b^2)*d*e^4 + (B*a^5 - 9*A*a^4*b)*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*(384*A*a^5*b*e^5 + 16*(B*a*b^5 + 3*A*b^6)*d^5 - 24*(5*B*a^2*b^4 + 13*A*a*b^
5)*d^4*e + 6*(79*B*a^3*b^3 + 149*A*a^2*b^4)*d^3*e^2 + (293*B*a^4*b^2 - 1605*A*a^3*b^3)*d^2*e^3 - 3*(221*B*a^5*
b - 197*A*a^4*b^2)*d*e^4 + 105*(8*B*b^6*d^2*e^3 - (7*B*a*b^5 + 9*A*b^6)*d*e^4 - (B*a^2*b^4 - 9*A*a*b^5)*e^5)*x
^4 + 35*(8*B*b^6*d^3*e^2 + 9*(9*B*a*b^5 - A*b^6)*d^2*e^3 - 6*(13*B*a^2*b^4 + 15*A*a*b^5)*d*e^4 - 11*(B*a^3*b^3
- 9*A*a^2*b^4)*e^5)*x^3 - 7*(16*B*b^6*d^4*e - 2*(83*B*a*b^5 + 9*A*b^6)*d^3*e^2 - 3*(151*B*a^2*b^4 - 63*A*a*b^
5)*d^2*e^3 + 2*(265*B*a^3*b^3 + 243*A*a^2*b^4)*d*e^4 + 73*(B*a^4*b^2 - 9*A*a^3*b^3)*e^5)*x^2 + (64*B*b^6*d^5 -
8*(59*B*a*b^5 + 9*A*b^6)*d^4*e + 108*(17*B*a^2*b^4 + 5*A*a*b^5)*d^3*e^2 + (989*B*a^3*b^3 - 2133*A*a^2*b^4)*d^
2*e^3 - 2*(1069*B*a^4*b^2 + 423*A*a^3*b^3)*d*e^4 - 279*(B*a^5*b - 9*A*a^4*b^2)*e^5)*x)*sqrt(e*x + d))/(a^4*b^7
*d^7 - 6*a^5*b^6*d^6*e + 15*a^6*b^5*d^5*e^2 - 20*a^7*b^4*d^4*e^3 + 15*a^8*b^3*d^3*e^4 - 6*a^9*b^2*d^2*e^5 + a^
10*b*d*e^6 + (b^11*d^6*e - 6*a*b^10*d^5*e^2 + 15*a^2*b^9*d^4*e^3 - 20*a^3*b^8*d^3*e^4 + 15*a^4*b^7*d^2*e^5 - 6
*a^5*b^6*d*e^6 + a^6*b^5*e^7)*x^5 + (b^11*d^7 - 2*a*b^10*d^6*e - 9*a^2*b^9*d^5*e^2 + 40*a^3*b^8*d^4*e^3 - 65*a
^4*b^7*d^3*e^4 + 54*a^5*b^6*d^2*e^5 - 23*a^6*b^5*d*e^6 + 4*a^7*b^4*e^7)*x^4 + 2*(2*a*b^10*d^7 - 9*a^2*b^9*d^6*
e + 12*a^3*b^8*d^5*e^2 + 5*a^4*b^7*d^4*e^3 - 30*a^5*b^6*d^3*e^4 + 33*a^6*b^5*d^2*e^5 - 16*a^7*b^4*d*e^6 + 3*a^
8*b^3*e^7)*x^3 + 2*(3*a^2*b^9*d^7 - 16*a^3*b^8*d^6*e + 33*a^4*b^7*d^5*e^2 - 30*a^5*b^6*d^4*e^3 + 5*a^6*b^5*d^3
*e^4 + 12*a^7*b^4*d^2*e^5 - 9*a^8*b^3*d*e^6 + 2*a^9*b^2*e^7)*x^2 + (4*a^3*b^8*d^7 - 23*a^4*b^7*d^6*e + 54*a^5*
b^6*d^5*e^2 - 65*a^6*b^5*d^4*e^3 + 40*a^7*b^4*d^3*e^4 - 9*a^8*b^3*d^2*e^5 - 2*a^9*b^2*d*e^6 + a^10*b*e^7)*x),
-1/192*(105*(8*B*a^4*b*d^2*e^3 + (B*a^5 - 9*A*a^4*b)*d*e^4 + (8*B*b^5*d*e^4 + (B*a*b^4 - 9*A*b^5)*e^5)*x^5 + (
8*B*b^5*d^2*e^3 + 3*(11*B*a*b^4 - 3*A*b^5)*d*e^4 + 4*(B*a^2*b^3 - 9*A*a*b^4)*e^5)*x^4 + 2*(16*B*a*b^4*d^2*e^3
+ 2*(13*B*a^2*b^3 - 9*A*a*b^4)*d*e^4 + 3*(B*a^3*b^2 - 9*A*a^2*b^3)*e^5)*x^3 + 2*(24*B*a^2*b^3*d^2*e^3 + (19*B*
a^3*b^2 - 27*A*a^2*b^3)*d*e^4 + 2*(B*a^4*b - 9*A*a^3*b^2)*e^5)*x^2 + (32*B*a^3*b^2*d^2*e^3 + 12*(B*a^4*b - 3*A
*a^3*b^2)*d*e^4 + (B*a^5 - 9*A*a^4*b)*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)) + (384*A*a^5*b*e^5 + 16*(B*a*b^5 + 3*A*b^6)*d^5 - 24*(5*B*a^2*b^4 + 13*A*a*b^5)*d^4*e + 6*(79*B*
a^3*b^3 + 149*A*a^2*b^4)*d^3*e^2 + (293*B*a^4*b^2 - 1605*A*a^3*b^3)*d^2*e^3 - 3*(221*B*a^5*b - 197*A*a^4*b^2)*
d*e^4 + 105*(8*B*b^6*d^2*e^3 - (7*B*a*b^5 + 9*A*b^6)*d*e^4 - (B*a^2*b^4 - 9*A*a*b^5)*e^5)*x^4 + 35*(8*B*b^6*d^
3*e^2 + 9*(9*B*a*b^5 - A*b^6)*d^2*e^3 - 6*(13*B*a^2*b^4 + 15*A*a*b^5)*d*e^4 - 11*(B*a^3*b^3 - 9*A*a^2*b^4)*e^5
)*x^3 - 7*(16*B*b^6*d^4*e - 2*(83*B*a*b^5 + 9*A*b^6)*d^3*e^2 - 3*(151*B*a^2*b^4 - 63*A*a*b^5)*d^2*e^3 + 2*(265
*B*a^3*b^3 + 243*A*a^2*b^4)*d*e^4 + 73*(B*a^4*b^2 - 9*A*a^3*b^3)*e^5)*x^2 + (64*B*b^6*d^5 - 8*(59*B*a*b^5 + 9*
A*b^6)*d^4*e + 108*(17*B*a^2*b^4 + 5*A*a*b^5)*d^3*e^2 + (989*B*a^3*b^3 - 2133*A*a^2*b^4)*d^2*e^3 - 2*(1069*B*a
^4*b^2 + 423*A*a^3*b^3)*d*e^4 - 279*(B*a^5*b - 9*A*a^4*b^2)*e^5)*x)*sqrt(e*x + d))/(a^4*b^7*d^7 - 6*a^5*b^6*d^
6*e + 15*a^6*b^5*d^5*e^2 - 20*a^7*b^4*d^4*e^3 + 15*a^8*b^3*d^3*e^4 - 6*a^9*b^2*d^2*e^5 + a^10*b*d*e^6 + (b^11*
d^6*e - 6*a*b^10*d^5*e^2 + 15*a^2*b^9*d^4*e^3 - 20*a^3*b^8*d^3*e^4 + 15*a^4*b^7*d^2*e^5 - 6*a^5*b^6*d*e^6 + a^
6*b^5*e^7)*x^5 + (b^11*d^7 - 2*a*b^10*d^6*e - 9*a^2*b^9*d^5*e^2 + 40*a^3*b^8*d^4*e^3 - 65*a^4*b^7*d^3*e^4 + 54
*a^5*b^6*d^2*e^5 - 23*a^6*b^5*d*e^6 + 4*a^7*b^4*e^7)*x^4 + 2*(2*a*b^10*d^7 - 9*a^2*b^9*d^6*e + 12*a^3*b^8*d^5*
e^2 + 5*a^4*b^7*d^4*e^3 - 30*a^5*b^6*d^3*e^4 + 33*a^6*b^5*d^2*e^5 - 16*a^7*b^4*d*e^6 + 3*a^8*b^3*e^7)*x^3 + 2*
(3*a^2*b^9*d^7 - 16*a^3*b^8*d^6*e + 33*a^4*b^7*d^5*e^2 - 30*a^5*b^6*d^4*e^3 + 5*a^6*b^5*d^3*e^4 + 12*a^7*b^4*d
^2*e^5 - 9*a^8*b^3*d*e^6 + 2*a^9*b^2*e^7)*x^2 + (4*a^3*b^8*d^7 - 23*a^4*b^7*d^6*e + 54*a^5*b^6*d^5*e^2 - 65*a^
6*b^5*d^4*e^3 + 40*a^7*b^4*d^3*e^4 - 9*a^8*b^3*d^2*e^5 - 2*a^9*b^2*d*e^6 + a^10*b*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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.44934, size = 1528, 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]

integrate((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

-35/64*(8*B*b*d*e^3 + B*a*e^4 - 9*A*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^5*sgn((x*e + d
)*b*e - b*d*e + a*e^2) - 5*a*b^4*d^4*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 10*a^2*b^3*d^3*e^2*sgn((x*e + d)*b
*e - b*d*e + a*e^2) - 10*a^3*b^2*d^2*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 5*a^4*b*d*e^4*sgn((x*e + d)*b*e
- b*d*e + a*e^2) - a^5*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) - 2*(B*d*e^3 - A*e^4)/((b
^5*d^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 5*a*b^4*d^4*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 10*a^2*b^3*d^3*
e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 10*a^3*b^2*d^2*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 5*a^4*b*d*e^4
*sgn((x*e + d)*b*e - b*d*e + a*e^2) - a^5*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(x*e + d)) - 1/192*(456*
(x*e + d)^(7/2)*B*b^4*d*e^3 - 1544*(x*e + d)^(5/2)*B*b^4*d^2*e^3 + 1784*(x*e + d)^(3/2)*B*b^4*d^3*e^3 - 696*sq
rt(x*e + d)*B*b^4*d^4*e^3 + 105*(x*e + d)^(7/2)*B*a*b^3*e^4 - 561*(x*e + d)^(7/2)*A*b^4*e^4 + 1159*(x*e + d)^(
5/2)*B*a*b^3*d*e^4 + 1929*(x*e + d)^(5/2)*A*b^4*d*e^4 - 3057*(x*e + d)^(3/2)*B*a*b^3*d^2*e^4 - 2295*(x*e + d)^
(3/2)*A*b^4*d^2*e^4 + 1809*sqrt(x*e + d)*B*a*b^3*d^3*e^4 + 975*sqrt(x*e + d)*A*b^4*d^3*e^4 + 385*(x*e + d)^(5/
2)*B*a^2*b^2*e^5 - 1929*(x*e + d)^(5/2)*A*a*b^3*e^5 + 762*(x*e + d)^(3/2)*B*a^2*b^2*d*e^5 + 4590*(x*e + d)^(3/
2)*A*a*b^3*d*e^5 - 1251*sqrt(x*e + d)*B*a^2*b^2*d^2*e^5 - 2925*sqrt(x*e + d)*A*a*b^3*d^2*e^5 + 511*(x*e + d)^(
3/2)*B*a^3*b*e^6 - 2295*(x*e + d)^(3/2)*A*a^2*b^2*e^6 - 141*sqrt(x*e + d)*B*a^3*b*d*e^6 + 2925*sqrt(x*e + d)*A
*a^2*b^2*d*e^6 + 279*sqrt(x*e + d)*B*a^4*e^7 - 975*sqrt(x*e + d)*A*a^3*b*e^7)/((b^5*d^5*sgn((x*e + d)*b*e - b*
d*e + a*e^2) - 5*a*b^4*d^4*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 10*a^2*b^3*d^3*e^2*sgn((x*e + d)*b*e - b*d*e
+ a*e^2) - 10*a^3*b^2*d^2*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 5*a^4*b*d*e^4*sgn((x*e + d)*b*e - b*d*e +
a*e^2) - a^5*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)