3.1828 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)
Optimal. Leaf size=313 \[ -\frac{e^3 \sqrt{d+e x} (-7 a B e-3 A b e+10 b B d)}{128 b^4 (a+b x) (b d-a e)^2}-\frac{e^2 \sqrt{d+e x} (-7 a B e-3 A b e+10 b B d)}{64 b^4 (a+b x)^2 (b d-a e)}+\frac{e^4 (-7 a B e-3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}-\frac{e (d+e x)^{3/2} (-7 a B e-3 A b e+10 b B d)}{48 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e-3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
[Out]
-(e^2*(10*b*B*d - 3*A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(64*b^4*(b*d - a*e)*(a + b*x)^2) - (e^3*(10*b*B*d - 3*A*b*
e - 7*a*B*e)*Sqrt[d + e*x])/(128*b^4*(b*d - a*e)^2*(a + b*x)) - (e*(10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(3
/2))/(48*b^3*(b*d - a*e)*(a + b*x)^3) - ((10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(40*b^2*(b*d - a*e)*(
a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(7/2))/(5*b*(b*d - a*e)*(a + b*x)^5) + (e^4*(10*b*B*d - 3*A*b*e - 7*a*B*e
)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(5/2))
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Rubi [A] time = 0.273428, 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} \[ -\frac{e^3 \sqrt{d+e x} (-7 a B e-3 A b e+10 b B d)}{128 b^4 (a+b x) (b d-a e)^2}-\frac{e^2 \sqrt{d+e x} (-7 a B e-3 A b e+10 b B d)}{64 b^4 (a+b x)^2 (b d-a e)}+\frac{e^4 (-7 a B e-3 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}-\frac{e (d+e x)^{3/2} (-7 a B e-3 A b e+10 b B d)}{48 b^3 (a+b x)^3 (b d-a e)}-\frac{(d+e x)^{5/2} (-7 a B e-3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac{(d+e x)^{7/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
-(e^2*(10*b*B*d - 3*A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(64*b^4*(b*d - a*e)*(a + b*x)^2) - (e^3*(10*b*B*d - 3*A*b*
e - 7*a*B*e)*Sqrt[d + e*x])/(128*b^4*(b*d - a*e)^2*(a + b*x)) - (e*(10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(3
/2))/(48*b^3*(b*d - a*e)*(a + b*x)^3) - ((10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(40*b^2*(b*d - a*e)*(
a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(7/2))/(5*b*(b*d - a*e)*(a + b*x)^5) + (e^4*(10*b*B*d - 3*A*b*e - 7*a*B*e
)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(5/2))
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)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{(A+B x) (d+e x)^{5/2}}{(a+b x)^6} \, dx\\ &=-\frac{(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac{(10 b B d-3 A b e-7 a B e) \int \frac{(d+e x)^{5/2}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac{(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac{(e (10 b B d-3 A b e-7 a B e)) \int \frac{(d+e x)^{3/2}}{(a+b x)^4} \, dx}{16 b^2 (b d-a e)}\\ &=-\frac{e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (e^2 (10 b B d-3 A b e-7 a B e)\right ) \int \frac{\sqrt{d+e x}}{(a+b x)^3} \, dx}{32 b^3 (b d-a e)}\\ &=-\frac{e^2 (10 b B d-3 A b e-7 a B e) \sqrt{d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac{e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac{\left (e^3 (10 b B d-3 A b e-7 a B e)\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{128 b^4 (b d-a e)}\\ &=-\frac{e^2 (10 b B d-3 A b e-7 a B e) \sqrt{d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac{e^3 (10 b B d-3 A b e-7 a B e) \sqrt{d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac{e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}-\frac{\left (e^4 (10 b B d-3 A b e-7 a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^4 (b d-a e)^2}\\ &=-\frac{e^2 (10 b B d-3 A b e-7 a B e) \sqrt{d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac{e^3 (10 b B d-3 A b e-7 a B e) \sqrt{d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac{e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}-\frac{\left (e^3 (10 b B d-3 A b e-7 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^4 (b d-a e)^2}\\ &=-\frac{e^2 (10 b B d-3 A b e-7 a B e) \sqrt{d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac{e^3 (10 b B d-3 A b e-7 a B e) \sqrt{d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac{e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac{(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac{(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac{e^4 (10 b B d-3 A b e-7 a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0738982, size = 99, normalized size = 0.32 \[ \frac{(d+e x)^{7/2} \left (\frac{e^4 (7 a B e+3 A b e-10 b B d) \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac{7 (a B-A b)}{(a+b x)^5}\right )}{35 b (b d-a e)} \]
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]
((d + e*x)^(7/2)*((7*(-(A*b) + a*B))/(a + b*x)^5 + (e^4*(-10*b*B*d + 3*A*b*e + 7*a*B*e)*Hypergeometric2F1[7/2,
5, 9/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^5))/(35*b*(b*d - a*e))
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Maple [B] time = 0.023, size = 872, 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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
3/128*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(9/2)*A*b+7/128*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d
*e+b^2*d^2)*(e*x+d)^(9/2)*a*B-5/64*e^4/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(9/2)*B*b*d+7/64*e^5/
(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(7/2)*A-79/192*e^5/(b*e*x+a*e)^5/b/(a*e-b*d)*(e*x+d)^(7/2)*a*B+29/96*e^4/(b*e*
x+a*e)^5/(a*e-b*d)*(e*x+d)^(7/2)*B*d-1/5*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*A-7/15*e^5/(b*e*x+a*e)^5/b^2*(e*x+d
)^(5/2)*a*B+2/3*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*B*d-7/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*A*a+7/64*e^5/(b
*e*x+a*e)^5/b*(e*x+d)^(3/2)*A*d-49/192*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(3/2)*a^2*B+119/192*e^5/(b*e*x+a*e)^5/b^2
*(e*x+d)^(3/2)*B*d*a-35/96*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*B*d^2-3/128*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*A
*a^2+3/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*A*a*d-3/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*A*d^2-7/128*e^7/(b
*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*B*a^3+3/16*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*B*a^2*d-27/128*e^5/(b*e*x+a*e)^5/
b^2*(e*x+d)^(1/2)*B*a*d^2+5/64*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*B*d^3+3/128*e^5/b^3/(a^2*e^2-2*a*b*d*e+b^2*d^
2)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A+7/128*e^5/b^4/(a^2*e^2-2*a*b*d*e+b^2*d^2)
/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a*B-5/64*e^4/b^3/(a^2*e^2-2*a*b*d*e+b^2*d^2)/
((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)^(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 = 1.6514, size = 4670, normalized size = 14.92 \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*(15*(10*B*a^5*b*d*e^4 - (7*B*a^6 + 3*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (7*B*a*b^5 + 3*A*b^6)*e^5)*x^5
+ 5*(10*B*a*b^5*d*e^4 - (7*B*a^2*b^4 + 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (7*B*a^3*b^3 + 3*A*a^2*b
^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (7*B*a^4*b^2 + 3*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (7*B*a^
5*b + 3*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*(96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(8*B*a^2*b^5 + 57*A*a*b^6)*d^4*e - 12*(B*a^3*b^4 - 46*A*a^2*b
^5)*d^3*e^2 - 6*(6*B*a^4*b^3 - A*a^3*b^4)*d^2*e^3 + 5*(37*B*a^5*b^2 + 3*A*a^4*b^3)*d*e^4 - 15*(7*B*a^6*b + 3*A
*a^5*b^2)*e^5 + 15*(10*B*b^7*d^2*e^3 - (17*B*a*b^6 + 3*A*b^7)*d*e^4 + (7*B*a^2*b^5 + 3*A*a*b^6)*e^5)*x^4 + 10*
(118*B*b^7*d^3*e^2 - 3*(99*B*a*b^6 - A*b^7)*d^2*e^3 + 6*(43*B*a^2*b^5 - 4*A*a*b^6)*d*e^4 - (79*B*a^3*b^4 - 21*
A*a^2*b^5)*e^5)*x^3 + 2*(680*B*b^7*d^4*e - 2*(661*B*a*b^6 - 186*A*b^7)*d^3*e^2 + 3*(97*B*a^2*b^5 - 357*A*a*b^6
)*d^2*e^3 + (799*B*a^3*b^4 + 891*A*a^2*b^5)*d*e^4 - 64*(7*B*a^4*b^3 + 3*A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^5
- 8*(43*B*a*b^6 - 63*A*b^7)*d^4*e + 2*(B*a^2*b^5 - 636*A*a*b^6)*d^3*e^2 - 3*(29*B*a^3*b^4 - 279*A*a^2*b^5)*d^
2*e^3 + 2*(217*B*a^4*b^3 + 18*A*a^3*b^4)*d*e^4 - 35*(7*B*a^5*b^2 + 3*A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^
8*d^3 - 3*a^6*b^7*d^2*e + 3*a^7*b^6*d*e^2 - a^8*b^5*e^3 + (b^13*d^3 - 3*a*b^12*d^2*e + 3*a^2*b^11*d*e^2 - a^3*
b^10*e^3)*x^5 + 5*(a*b^12*d^3 - 3*a^2*b^11*d^2*e + 3*a^3*b^10*d*e^2 - a^4*b^9*e^3)*x^4 + 10*(a^2*b^11*d^3 - 3*
a^3*b^10*d^2*e + 3*a^4*b^9*d*e^2 - a^5*b^8*e^3)*x^3 + 10*(a^3*b^10*d^3 - 3*a^4*b^9*d^2*e + 3*a^5*b^8*d*e^2 - a
^6*b^7*e^3)*x^2 + 5*(a^4*b^9*d^3 - 3*a^5*b^8*d^2*e + 3*a^6*b^7*d*e^2 - a^7*b^6*e^3)*x), -1/1920*(15*(10*B*a^5*
b*d*e^4 - (7*B*a^6 + 3*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (7*B*a*b^5 + 3*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 -
(7*B*a^2*b^4 + 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (7*B*a^3*b^3 + 3*A*a^2*b^4)*e^5)*x^3 + 10*(10*B
*a^3*b^3*d*e^4 - (7*B*a^4*b^2 + 3*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (7*B*a^5*b + 3*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)) + (96*(B*a*b^6 + 4*A*b^7)*d^
5 - 16*(8*B*a^2*b^5 + 57*A*a*b^6)*d^4*e - 12*(B*a^3*b^4 - 46*A*a^2*b^5)*d^3*e^2 - 6*(6*B*a^4*b^3 - A*a^3*b^4)*
d^2*e^3 + 5*(37*B*a^5*b^2 + 3*A*a^4*b^3)*d*e^4 - 15*(7*B*a^6*b + 3*A*a^5*b^2)*e^5 + 15*(10*B*b^7*d^2*e^3 - (17
*B*a*b^6 + 3*A*b^7)*d*e^4 + (7*B*a^2*b^5 + 3*A*a*b^6)*e^5)*x^4 + 10*(118*B*b^7*d^3*e^2 - 3*(99*B*a*b^6 - A*b^7
)*d^2*e^3 + 6*(43*B*a^2*b^5 - 4*A*a*b^6)*d*e^4 - (79*B*a^3*b^4 - 21*A*a^2*b^5)*e^5)*x^3 + 2*(680*B*b^7*d^4*e -
2*(661*B*a*b^6 - 186*A*b^7)*d^3*e^2 + 3*(97*B*a^2*b^5 - 357*A*a*b^6)*d^2*e^3 + (799*B*a^3*b^4 + 891*A*a^2*b^5
)*d*e^4 - 64*(7*B*a^4*b^3 + 3*A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^5 - 8*(43*B*a*b^6 - 63*A*b^7)*d^4*e + 2*(B*
a^2*b^5 - 636*A*a*b^6)*d^3*e^2 - 3*(29*B*a^3*b^4 - 279*A*a^2*b^5)*d^2*e^3 + 2*(217*B*a^4*b^3 + 18*A*a^3*b^4)*d
*e^4 - 35*(7*B*a^5*b^2 + 3*A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^3 - 3*a^6*b^7*d^2*e + 3*a^7*b^6*d*e^2
- a^8*b^5*e^3 + (b^13*d^3 - 3*a*b^12*d^2*e + 3*a^2*b^11*d*e^2 - a^3*b^10*e^3)*x^5 + 5*(a*b^12*d^3 - 3*a^2*b^11
*d^2*e + 3*a^3*b^10*d*e^2 - a^4*b^9*e^3)*x^4 + 10*(a^2*b^11*d^3 - 3*a^3*b^10*d^2*e + 3*a^4*b^9*d*e^2 - a^5*b^8
*e^3)*x^3 + 10*(a^3*b^10*d^3 - 3*a^4*b^9*d^2*e + 3*a^5*b^8*d*e^2 - a^6*b^7*e^3)*x^2 + 5*(a^4*b^9*d^3 - 3*a^5*b
^8*d^2*e + 3*a^6*b^7*d*e^2 - a^7*b^6*e^3)*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 = 1.2781, size = 1087, normalized size = 3.47 \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]
-1/128*(10*B*b*d*e^4 - 7*B*a*e^5 - 3*A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^2 - 2*a*b^5
*d*e + a^2*b^4*e^2)*sqrt(-b^2*d + a*b*e)) - 1/1920*(150*(x*e + d)^(9/2)*B*b^5*d*e^4 + 580*(x*e + d)^(7/2)*B*b^
5*d^2*e^4 - 1280*(x*e + d)^(5/2)*B*b^5*d^3*e^4 + 700*(x*e + d)^(3/2)*B*b^5*d^4*e^4 - 150*sqrt(x*e + d)*B*b^5*d
^5*e^4 - 105*(x*e + d)^(9/2)*B*a*b^4*e^5 - 45*(x*e + d)^(9/2)*A*b^5*e^5 - 1370*(x*e + d)^(7/2)*B*a*b^4*d*e^5 +
210*(x*e + d)^(7/2)*A*b^5*d*e^5 + 3456*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5 + 384*(x*e + d)^(5/2)*A*b^5*d^2*e^5 -
2590*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 - 210*(x*e + d)^(3/2)*A*b^5*d^3*e^5 + 705*sqrt(x*e + d)*B*a*b^4*d^4*e^5 +
45*sqrt(x*e + d)*A*b^5*d^4*e^5 + 790*(x*e + d)^(7/2)*B*a^2*b^3*e^6 - 210*(x*e + d)^(7/2)*A*a*b^4*e^6 - 3072*(
x*e + d)^(5/2)*B*a^2*b^3*d*e^6 - 768*(x*e + d)^(5/2)*A*a*b^4*d*e^6 + 3570*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^6 +
630*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6 - 1320*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6 - 180*sqrt(x*e + d)*A*a*b^4*d^3*e^6
+ 896*(x*e + d)^(5/2)*B*a^3*b^2*e^7 + 384*(x*e + d)^(5/2)*A*a^2*b^3*e^7 - 2170*(x*e + d)^(3/2)*B*a^3*b^2*d*e^
7 - 630*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7 + 1230*sqrt(x*e + d)*B*a^3*b^2*d^2*e^7 + 270*sqrt(x*e + d)*A*a^2*b^3*d
^2*e^7 + 490*(x*e + d)^(3/2)*B*a^4*b*e^8 + 210*(x*e + d)^(3/2)*A*a^3*b^2*e^8 - 570*sqrt(x*e + d)*B*a^4*b*d*e^8
- 180*sqrt(x*e + d)*A*a^3*b^2*d*e^8 + 105*sqrt(x*e + d)*B*a^5*e^9 + 45*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^2 -
2*a*b^5*d*e + a^2*b^4*e^2)*((x*e + d)*b - b*d + a*e)^5)