3.1832 \(\int \frac{A+B x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)
Optimal. Leaf size=352 \[ \frac{63 e^4 (a B e-11 A b e+10 b B d)}{128 b \sqrt{d+e x} (b d-a e)^6}+\frac{21 e^3 (a B e-11 A b e+10 b B d)}{128 b (a+b x) \sqrt{d+e x} (b d-a e)^5}-\frac{21 e^2 (a B e-11 A b e+10 b B d)}{320 b (a+b x)^2 \sqrt{d+e x} (b d-a e)^4}-\frac{63 e^4 (a B e-11 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{13/2}}+\frac{3 e (a B e-11 A b e+10 b B d)}{80 b (a+b x)^3 \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-11 A b e+10 b B d}{40 b (a+b x)^4 \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 \sqrt{d+e x} (b d-a e)} \]
[Out]
(63*e^4*(10*b*B*d - 11*A*b*e + a*B*e))/(128*b*(b*d - a*e)^6*Sqrt[d + e*x]) - (A*b - a*B)/(5*b*(b*d - a*e)*(a +
b*x)^5*Sqrt[d + e*x]) - (10*b*B*d - 11*A*b*e + a*B*e)/(40*b*(b*d - a*e)^2*(a + b*x)^4*Sqrt[d + e*x]) + (3*e*(
10*b*B*d - 11*A*b*e + a*B*e))/(80*b*(b*d - a*e)^3*(a + b*x)^3*Sqrt[d + e*x]) - (21*e^2*(10*b*B*d - 11*A*b*e +
a*B*e))/(320*b*(b*d - a*e)^4*(a + b*x)^2*Sqrt[d + e*x]) + (21*e^3*(10*b*B*d - 11*A*b*e + a*B*e))/(128*b*(b*d -
a*e)^5*(a + b*x)*Sqrt[d + e*x]) - (63*e^4*(10*b*B*d - 11*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[
b*d - a*e]])/(128*Sqrt[b]*(b*d - a*e)^(13/2))
________________________________________________________________________________________
Rubi [A] time = 0.381629, antiderivative size = 352, normalized size of antiderivative = 1.,
number of steps used = 9, 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} \[ \frac{63 e^4 (a B e-11 A b e+10 b B d)}{128 b \sqrt{d+e x} (b d-a e)^6}+\frac{21 e^3 (a B e-11 A b e+10 b B d)}{128 b (a+b x) \sqrt{d+e x} (b d-a e)^5}-\frac{21 e^2 (a B e-11 A b e+10 b B d)}{320 b (a+b x)^2 \sqrt{d+e x} (b d-a e)^4}-\frac{63 e^4 (a B e-11 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{13/2}}+\frac{3 e (a B e-11 A b e+10 b B d)}{80 b (a+b x)^3 \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-11 A b e+10 b B d}{40 b (a+b x)^4 \sqrt{d+e x} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 \sqrt{d+e x} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
(63*e^4*(10*b*B*d - 11*A*b*e + a*B*e))/(128*b*(b*d - a*e)^6*Sqrt[d + e*x]) - (A*b - a*B)/(5*b*(b*d - a*e)*(a +
b*x)^5*Sqrt[d + e*x]) - (10*b*B*d - 11*A*b*e + a*B*e)/(40*b*(b*d - a*e)^2*(a + b*x)^4*Sqrt[d + e*x]) + (3*e*(
10*b*B*d - 11*A*b*e + a*B*e))/(80*b*(b*d - a*e)^3*(a + b*x)^3*Sqrt[d + e*x]) - (21*e^2*(10*b*B*d - 11*A*b*e +
a*B*e))/(320*b*(b*d - a*e)^4*(a + b*x)^2*Sqrt[d + e*x]) + (21*e^3*(10*b*B*d - 11*A*b*e + a*B*e))/(128*b*(b*d -
a*e)^5*(a + b*x)*Sqrt[d + e*x]) - (63*e^4*(10*b*B*d - 11*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[
b*d - a*e]])/(128*Sqrt[b]*(b*d - a*e)^(13/2))
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)^{3/2} \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)^{3/2}} \, dx\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt{d+e x}}+\frac{(10 b B d-11 A b e+a B e) \int \frac{1}{(a+b x)^5 (d+e x)^{3/2}} \, dx}{10 b (b d-a e)}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt{d+e x}}-\frac{10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}-\frac{(9 e (10 b B d-11 A b e+a B e)) \int \frac{1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{80 b (b d-a e)^2}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt{d+e x}}-\frac{10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}+\frac{3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}+\frac{\left (21 e^2 (10 b B d-11 A b e+a B e)\right ) \int \frac{1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{160 b (b d-a e)^3}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt{d+e x}}-\frac{10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}+\frac{3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}-\frac{\left (21 e^3 (10 b B d-11 A b e+a B e)\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{128 b (b d-a e)^4}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt{d+e x}}-\frac{10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}+\frac{3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}+\frac{21 e^3 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) \sqrt{d+e x}}+\frac{\left (63 e^4 (10 b B d-11 A b e+a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 b (b d-a e)^5}\\ &=\frac{63 e^4 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^6 \sqrt{d+e x}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt{d+e x}}-\frac{10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}+\frac{3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}+\frac{21 e^3 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) \sqrt{d+e x}}+\frac{\left (63 e^4 (10 b B d-11 A b e+a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 (b d-a e)^6}\\ &=\frac{63 e^4 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^6 \sqrt{d+e x}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt{d+e x}}-\frac{10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}+\frac{3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}+\frac{21 e^3 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) \sqrt{d+e x}}+\frac{\left (63 e^3 (10 b B d-11 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 d-a e)^6}\\ &=\frac{63 e^4 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^6 \sqrt{d+e x}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt{d+e x}}-\frac{10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt{d+e x}}+\frac{3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt{d+e x}}-\frac{21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt{d+e x}}+\frac{21 e^3 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) \sqrt{d+e x}}-\frac{63 e^4 (10 b B d-11 A b e+a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 \sqrt{b} (b d-a e)^{13/2}}\\ \end{align*}
Mathematica [C] time = 0.0820599, size = 97, normalized size = 0.28 \[ \frac{\frac{e^4 (a B e-11 A b e+10 b B d) \, _2F_1\left (-\frac{1}{2},5;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac{a B-A b}{(a+b x)^5}}{5 b \sqrt{d+e x} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
((-(A*b) + a*B)/(a + b*x)^5 + (e^4*(10*b*B*d - 11*A*b*e + a*B*e)*Hypergeometric2F1[-1/2, 5, 1/2, (b*(d + e*x))
/(b*d - a*e)])/(b*d - a*e)^5)/(5*b*(b*d - a*e)*Sqrt[d + e*x])
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Maple [B] time = 0.039, size = 1568, normalized size = 4.5 \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)^3,x)
[Out]
-693/128*e^5/(a*e-b*d)^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*b-437/128*e^5/(a*e-
b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A*b^5+193/128*e^9/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5+63/128*e^5/
(a*e-b*d)^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a*B-2*e^5/(a*e-b*d)^6/(e*x+d)^(1/2
)*A+2*e^4/(a*e-b*d)^6/(e*x+d)^(1/2)*B*d-199/5*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*b^4*d^2-721/64*e
^7/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^3*b^2*d^2-1327/64*e^8/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A
*a^3*b^2+1327/64*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*b^5*d^3+22*e^4/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d
)^(5/2)*B*b^5*d^3-545/32*e^4/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*b^5*d^4+325/64*e^4/(a*e-b*d)^6/(b*e*x+a
*e)^5*(e*x+d)^(1/2)*B*b^5*d^5+3033/64*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a*b^4*d^3+843/32*e^8/(a*e-
b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*b^2*d-2529/64*e^7/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^2*b^3*d
^2+68/5*e^6/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*b^3*d-2407/128*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)
^(1/2)*B*a*b^4*d^4+683/64*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*b^4*d+262/5*e^6/(a*e-b*d)^6/(b*e*x+a
*e)^5*(e*x+d)^(5/2)*A*a*b^4*d+3981/64*e^7/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^2*b^3*d-3981/64*e^6/(a*e
-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a*b^4*d^2+379/64*e^7/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*b^2*d
+391/16*e^6/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*b^3*d^3+843/32*e^6/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)
^(1/2)*A*a*b^4*d^3-61/64*e^8/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^4*b*d-2559/64*e^6/(a*e-b*d)^6/(b*e*x+
a*e)^5*(e*x+d)^(3/2)*B*a^2*b^3*d^2-415/32*e^4/(a*e-b*d)^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*b^5*d^2+315/64*e^4/(a*
e-b*d)^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*b*d+147/64*e^6/(a*e-b*d)^6/(b*e*x+a
*e)^5*B*(e*x+d)^(7/2)*a^2*b^3+63/128*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*a*b^4-131/5*e^7/(a*e-b*d)^6
/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^2*b^3-131/5*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*b^5*d^2+21/5*e^7/(a
*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3*b^2+237/64*e^8/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^4*b-843
/128*e^9/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^4*b-843/128*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A
*b^5*d^4-977/64*e^6/(a*e-b*d)^6/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*a*b^4+977/64*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*A*(e*
x+d)^(7/2)*b^5*d+187/64*e^4/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*b^5*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 = 2.18568, size = 8244, normalized size = 23.42 \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*(315*(10*B*a^5*b*d^2*e^4 + (B*a^6 - 11*A*a^5*b)*d*e^5 + (10*B*b^6*d*e^5 + (B*a*b^5 - 11*A*b^6)*e^6)*x
^6 + (10*B*b^6*d^2*e^4 + (51*B*a*b^5 - 11*A*b^6)*d*e^5 + 5*(B*a^2*b^4 - 11*A*a*b^5)*e^6)*x^5 + 5*(10*B*a*b^5*d
^2*e^4 + (21*B*a^2*b^4 - 11*A*a*b^5)*d*e^5 + 2*(B*a^3*b^3 - 11*A*a^2*b^4)*e^6)*x^4 + 10*(10*B*a^2*b^4*d^2*e^4
+ 11*(B*a^3*b^3 - A*a^2*b^4)*d*e^5 + (B*a^4*b^2 - 11*A*a^3*b^3)*e^6)*x^3 + 5*(20*B*a^3*b^3*d^2*e^4 + 2*(6*B*a^
4*b^2 - 11*A*a^3*b^3)*d*e^5 + (B*a^5*b - 11*A*a^4*b^2)*e^6)*x^2 + (50*B*a^4*b^2*d^2*e^4 + 5*(3*B*a^5*b - 11*A*
a^4*b^2)*d*e^5 + (B*a^6 - 11*A*a^5*b)*e^6)*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*(1280*A*a^6*b*e^6 - 32*(B*a*b^6 + 4*A*b^7)*d^6 + 16*(16*B*a^2*b^5 + 59*A*a*
b^6)*d^5*e - 4*(239*B*a^3*b^4 + 766*A*a^2*b^5)*d^4*e^2 + 14*(178*B*a^4*b^3 + 417*A*a^3*b^4)*d^3*e^3 + 5*(97*B*
a^5*b^2 - 1561*A*a^4*b^3)*d^2*e^4 - 5*(449*B*a^6*b - 587*A*a^5*b^2)*d*e^5 + 315*(10*B*b^7*d^2*e^4 - (9*B*a*b^6
+ 11*A*b^7)*d*e^5 - (B*a^2*b^5 - 11*A*a*b^6)*e^6)*x^5 + 105*(10*B*b^7*d^3*e^3 + (131*B*a*b^6 - 11*A*b^7)*d^2*
e^4 - (127*B*a^2*b^5 + 143*A*a*b^6)*d*e^5 - 14*(B*a^3*b^4 - 11*A*a^2*b^5)*e^6)*x^4 - 42*(10*B*b^7*d^4*e^2 - (1
29*B*a*b^6 + 11*A*b^7)*d^3*e^3 - 13*(41*B*a^2*b^5 - 11*A*a*b^6)*d^2*e^4 + 4*(147*B*a^3*b^4 + 143*A*a^2*b^5)*d*
e^5 + 64*(B*a^4*b^3 - 11*A*a^3*b^4)*e^6)*x^3 + 6*(40*B*b^7*d^5*e - 2*(183*B*a*b^6 + 22*A*b^7)*d^4*e^2 + (1883*
B*a^2*b^5 + 407*A*a*b^6)*d^3*e^3 + 88*(29*B*a^3*b^4 - 24*A*a^2*b^5)*d^2*e^4 - 2*(1857*B*a^4*b^3 + 1298*A*a^3*b
^4)*d*e^5 - 395*(B*a^5*b^2 - 11*A*a^4*b^3)*e^6)*x^2 - (160*B*b^7*d^6 - 16*(79*B*a*b^6 + 11*A*b^7)*d^5*e + 4*(1
163*B*a^2*b^5 + 352*A*a*b^6)*d^4*e^2 - 2*(5991*B*a^3*b^4 + 2629*A*a^2*b^5)*d^3*e^3 - 2*(1048*B*a^4*b^3 - 6853*
A*a^3*b^4)*d^2*e^4 + 5*(1913*B*a^5*b^2 + 187*A*a^4*b^3)*d*e^5 + 965*(B*a^6*b - 11*A*a^5*b^2)*e^6)*x)*sqrt(e*x
+ d))/(a^5*b^8*d^8 - 7*a^6*b^7*d^7*e + 21*a^7*b^6*d^6*e^2 - 35*a^8*b^5*d^5*e^3 + 35*a^9*b^4*d^4*e^4 - 21*a^10*
b^3*d^3*e^5 + 7*a^11*b^2*d^2*e^6 - a^12*b*d*e^7 + (b^13*d^7*e - 7*a*b^12*d^6*e^2 + 21*a^2*b^11*d^5*e^3 - 35*a^
3*b^10*d^4*e^4 + 35*a^4*b^9*d^3*e^5 - 21*a^5*b^8*d^2*e^6 + 7*a^6*b^7*d*e^7 - a^7*b^6*e^8)*x^6 + (b^13*d^8 - 2*
a*b^12*d^7*e - 14*a^2*b^11*d^6*e^2 + 70*a^3*b^10*d^5*e^3 - 140*a^4*b^9*d^4*e^4 + 154*a^5*b^8*d^3*e^5 - 98*a^6*
b^7*d^2*e^6 + 34*a^7*b^6*d*e^7 - 5*a^8*b^5*e^8)*x^5 + 5*(a*b^12*d^8 - 5*a^2*b^11*d^7*e + 7*a^3*b^10*d^6*e^2 +
7*a^4*b^9*d^5*e^3 - 35*a^5*b^8*d^4*e^4 + 49*a^6*b^7*d^3*e^5 - 35*a^7*b^6*d^2*e^6 + 13*a^8*b^5*d*e^7 - 2*a^9*b^
4*e^8)*x^4 + 10*(a^2*b^11*d^8 - 6*a^3*b^10*d^7*e + 14*a^4*b^9*d^6*e^2 - 14*a^5*b^8*d^5*e^3 + 14*a^7*b^6*d^3*e^
5 - 14*a^8*b^5*d^2*e^6 + 6*a^9*b^4*d*e^7 - a^10*b^3*e^8)*x^3 + 5*(2*a^3*b^10*d^8 - 13*a^4*b^9*d^7*e + 35*a^5*b
^8*d^6*e^2 - 49*a^6*b^7*d^5*e^3 + 35*a^7*b^6*d^4*e^4 - 7*a^8*b^5*d^3*e^5 - 7*a^9*b^4*d^2*e^6 + 5*a^10*b^3*d*e^
7 - a^11*b^2*e^8)*x^2 + (5*a^4*b^9*d^8 - 34*a^5*b^8*d^7*e + 98*a^6*b^7*d^6*e^2 - 154*a^7*b^6*d^5*e^3 + 140*a^8
*b^5*d^4*e^4 - 70*a^9*b^4*d^3*e^5 + 14*a^10*b^3*d^2*e^6 + 2*a^11*b^2*d*e^7 - a^12*b*e^8)*x), 1/640*(315*(10*B*
a^5*b*d^2*e^4 + (B*a^6 - 11*A*a^5*b)*d*e^5 + (10*B*b^6*d*e^5 + (B*a*b^5 - 11*A*b^6)*e^6)*x^6 + (10*B*b^6*d^2*e
^4 + (51*B*a*b^5 - 11*A*b^6)*d*e^5 + 5*(B*a^2*b^4 - 11*A*a*b^5)*e^6)*x^5 + 5*(10*B*a*b^5*d^2*e^4 + (21*B*a^2*b
^4 - 11*A*a*b^5)*d*e^5 + 2*(B*a^3*b^3 - 11*A*a^2*b^4)*e^6)*x^4 + 10*(10*B*a^2*b^4*d^2*e^4 + 11*(B*a^3*b^3 - A*
a^2*b^4)*d*e^5 + (B*a^4*b^2 - 11*A*a^3*b^3)*e^6)*x^3 + 5*(20*B*a^3*b^3*d^2*e^4 + 2*(6*B*a^4*b^2 - 11*A*a^3*b^3
)*d*e^5 + (B*a^5*b - 11*A*a^4*b^2)*e^6)*x^2 + (50*B*a^4*b^2*d^2*e^4 + 5*(3*B*a^5*b - 11*A*a^4*b^2)*d*e^5 + (B*
a^6 - 11*A*a^5*b)*e^6)*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)) + (128
0*A*a^6*b*e^6 - 32*(B*a*b^6 + 4*A*b^7)*d^6 + 16*(16*B*a^2*b^5 + 59*A*a*b^6)*d^5*e - 4*(239*B*a^3*b^4 + 766*A*a
^2*b^5)*d^4*e^2 + 14*(178*B*a^4*b^3 + 417*A*a^3*b^4)*d^3*e^3 + 5*(97*B*a^5*b^2 - 1561*A*a^4*b^3)*d^2*e^4 - 5*(
449*B*a^6*b - 587*A*a^5*b^2)*d*e^5 + 315*(10*B*b^7*d^2*e^4 - (9*B*a*b^6 + 11*A*b^7)*d*e^5 - (B*a^2*b^5 - 11*A*
a*b^6)*e^6)*x^5 + 105*(10*B*b^7*d^3*e^3 + (131*B*a*b^6 - 11*A*b^7)*d^2*e^4 - (127*B*a^2*b^5 + 143*A*a*b^6)*d*e
^5 - 14*(B*a^3*b^4 - 11*A*a^2*b^5)*e^6)*x^4 - 42*(10*B*b^7*d^4*e^2 - (129*B*a*b^6 + 11*A*b^7)*d^3*e^3 - 13*(41
*B*a^2*b^5 - 11*A*a*b^6)*d^2*e^4 + 4*(147*B*a^3*b^4 + 143*A*a^2*b^5)*d*e^5 + 64*(B*a^4*b^3 - 11*A*a^3*b^4)*e^6
)*x^3 + 6*(40*B*b^7*d^5*e - 2*(183*B*a*b^6 + 22*A*b^7)*d^4*e^2 + (1883*B*a^2*b^5 + 407*A*a*b^6)*d^3*e^3 + 88*(
29*B*a^3*b^4 - 24*A*a^2*b^5)*d^2*e^4 - 2*(1857*B*a^4*b^3 + 1298*A*a^3*b^4)*d*e^5 - 395*(B*a^5*b^2 - 11*A*a^4*b
^3)*e^6)*x^2 - (160*B*b^7*d^6 - 16*(79*B*a*b^6 + 11*A*b^7)*d^5*e + 4*(1163*B*a^2*b^5 + 352*A*a*b^6)*d^4*e^2 -
2*(5991*B*a^3*b^4 + 2629*A*a^2*b^5)*d^3*e^3 - 2*(1048*B*a^4*b^3 - 6853*A*a^3*b^4)*d^2*e^4 + 5*(1913*B*a^5*b^2
+ 187*A*a^4*b^3)*d*e^5 + 965*(B*a^6*b - 11*A*a^5*b^2)*e^6)*x)*sqrt(e*x + d))/(a^5*b^8*d^8 - 7*a^6*b^7*d^7*e +
21*a^7*b^6*d^6*e^2 - 35*a^8*b^5*d^5*e^3 + 35*a^9*b^4*d^4*e^4 - 21*a^10*b^3*d^3*e^5 + 7*a^11*b^2*d^2*e^6 - a^12
*b*d*e^7 + (b^13*d^7*e - 7*a*b^12*d^6*e^2 + 21*a^2*b^11*d^5*e^3 - 35*a^3*b^10*d^4*e^4 + 35*a^4*b^9*d^3*e^5 - 2
1*a^5*b^8*d^2*e^6 + 7*a^6*b^7*d*e^7 - a^7*b^6*e^8)*x^6 + (b^13*d^8 - 2*a*b^12*d^7*e - 14*a^2*b^11*d^6*e^2 + 70
*a^3*b^10*d^5*e^3 - 140*a^4*b^9*d^4*e^4 + 154*a^5*b^8*d^3*e^5 - 98*a^6*b^7*d^2*e^6 + 34*a^7*b^6*d*e^7 - 5*a^8*
b^5*e^8)*x^5 + 5*(a*b^12*d^8 - 5*a^2*b^11*d^7*e + 7*a^3*b^10*d^6*e^2 + 7*a^4*b^9*d^5*e^3 - 35*a^5*b^8*d^4*e^4
+ 49*a^6*b^7*d^3*e^5 - 35*a^7*b^6*d^2*e^6 + 13*a^8*b^5*d*e^7 - 2*a^9*b^4*e^8)*x^4 + 10*(a^2*b^11*d^8 - 6*a^3*b
^10*d^7*e + 14*a^4*b^9*d^6*e^2 - 14*a^5*b^8*d^5*e^3 + 14*a^7*b^6*d^3*e^5 - 14*a^8*b^5*d^2*e^6 + 6*a^9*b^4*d*e^
7 - a^10*b^3*e^8)*x^3 + 5*(2*a^3*b^10*d^8 - 13*a^4*b^9*d^7*e + 35*a^5*b^8*d^6*e^2 - 49*a^6*b^7*d^5*e^3 + 35*a^
7*b^6*d^4*e^4 - 7*a^8*b^5*d^3*e^5 - 7*a^9*b^4*d^2*e^6 + 5*a^10*b^3*d*e^7 - a^11*b^2*e^8)*x^2 + (5*a^4*b^9*d^8
- 34*a^5*b^8*d^7*e + 98*a^6*b^7*d^6*e^2 - 154*a^7*b^6*d^5*e^3 + 140*a^8*b^5*d^4*e^4 - 70*a^9*b^4*d^3*e^5 + 14*
a^10*b^3*d^2*e^6 + 2*a^11*b^2*d*e^7 - a^12*b*e^8)*x)]
________________________________________________________________________________________
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
________________________________________________________________________________________
Giac [B] time = 1.3177, size = 1342, normalized size = 3.81 \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]
63/128*(10*B*b*d*e^4 + B*a*e^5 - 11*A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6 - 6*a*b^5*
d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(-b^2*d +
a*b*e)) + 2*(B*d*e^4 - A*e^5)/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2
*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(x*e + d)) + 1/640*(1870*(x*e + d)^(9/2)*B*b^5*d*e^4 - 8300*(x*e + d)^
(7/2)*B*b^5*d^2*e^4 + 14080*(x*e + d)^(5/2)*B*b^5*d^3*e^4 - 10900*(x*e + d)^(3/2)*B*b^5*d^4*e^4 + 3250*sqrt(x*
e + d)*B*b^5*d^5*e^4 + 315*(x*e + d)^(9/2)*B*a*b^4*e^5 - 2185*(x*e + d)^(9/2)*A*b^5*e^5 + 6830*(x*e + d)^(7/2)
*B*a*b^4*d*e^5 + 9770*(x*e + d)^(7/2)*A*b^5*d*e^5 - 25472*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5 - 16768*(x*e + d)^(5
/2)*A*b^5*d^2*e^5 + 30330*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 + 13270*(x*e + d)^(3/2)*A*b^5*d^3*e^5 - 12035*sqrt(x
*e + d)*B*a*b^4*d^4*e^5 - 4215*sqrt(x*e + d)*A*b^5*d^4*e^5 + 1470*(x*e + d)^(7/2)*B*a^2*b^3*e^6 - 9770*(x*e +
d)^(7/2)*A*a*b^4*e^6 + 8704*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 + 33536*(x*e + d)^(5/2)*A*a*b^4*d*e^6 - 25590*(x*e
+ d)^(3/2)*B*a^2*b^3*d^2*e^6 - 39810*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6 + 15640*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6
+ 16860*sqrt(x*e + d)*A*a*b^4*d^3*e^6 + 2688*(x*e + d)^(5/2)*B*a^3*b^2*e^7 - 16768*(x*e + d)^(5/2)*A*a^2*b^3*e
^7 + 3790*(x*e + d)^(3/2)*B*a^3*b^2*d*e^7 + 39810*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7 - 7210*sqrt(x*e + d)*B*a^3*b
^2*d^2*e^7 - 25290*sqrt(x*e + d)*A*a^2*b^3*d^2*e^7 + 2370*(x*e + d)^(3/2)*B*a^4*b*e^8 - 13270*(x*e + d)^(3/2)*
A*a^3*b^2*e^8 - 610*sqrt(x*e + d)*B*a^4*b*d*e^8 + 16860*sqrt(x*e + d)*A*a^3*b^2*d*e^8 + 965*sqrt(x*e + d)*B*a^
5*e^9 - 4215*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 +
15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*((x*e + d)*b - b*d + a*e)^5)