3.2225 \(\int (a+b x)^{5/2} (A+B x) \sqrt{d+e x} \, dx\)

Optimal. Leaf size=304 \[ -\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (3 a B e-10 A b e+7 b B d)}{128 b^2 e^4}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (3 a B e-10 A b e+7 b B d)}{192 b^2 e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e) (3 a B e-10 A b e+7 b B d)}{240 b^2 e^2}+\frac{(b d-a e)^4 (3 a B e-10 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{5/2} e^{9/2}}-\frac{(a+b x)^{7/2} \sqrt{d+e x} (3 a B e-10 A b e+7 b B d)}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e} \]

[Out]

-((b*d - a*e)^3*(7*b*B*d - 10*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^2*e^4) + ((b*d - a*e)^2*(7*
b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(192*b^2*e^3) - ((b*d - a*e)*(7*b*B*d - 10*A*b*e +
3*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(240*b^2*e^2) - ((7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(7/2)*Sqrt[d
+ e*x])/(40*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(5*b*e) + ((b*d - a*e)^4*(7*b*B*d - 10*A*b*e + 3*a*B
*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(5/2)*e^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.240317, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {80, 50, 63, 217, 206} \[ -\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3 (3 a B e-10 A b e+7 b B d)}{128 b^2 e^4}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2 (3 a B e-10 A b e+7 b B d)}{192 b^2 e^3}-\frac{(a+b x)^{5/2} \sqrt{d+e x} (b d-a e) (3 a B e-10 A b e+7 b B d)}{240 b^2 e^2}+\frac{(b d-a e)^4 (3 a B e-10 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{5/2} e^{9/2}}-\frac{(a+b x)^{7/2} \sqrt{d+e x} (3 a B e-10 A b e+7 b B d)}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d - a*e)^3*(7*b*B*d - 10*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^2*e^4) + ((b*d - a*e)^2*(7*
b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(192*b^2*e^3) - ((b*d - a*e)*(7*b*B*d - 10*A*b*e +
3*a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(240*b^2*e^2) - ((7*b*B*d - 10*A*b*e + 3*a*B*e)*(a + b*x)^(7/2)*Sqrt[d
+ e*x])/(40*b^2*e) + (B*(a + b*x)^(7/2)*(d + e*x)^(3/2))/(5*b*e) + ((b*d - a*e)^4*(7*b*B*d - 10*A*b*e + 3*a*B
*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(5/2)*e^(9/2))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int (a+b x)^{5/2} (A+B x) \sqrt{d+e x} \, dx &=\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac{\left (5 A b e-B \left (\frac{7 b d}{2}+\frac{3 a e}{2}\right )\right ) \int (a+b x)^{5/2} \sqrt{d+e x} \, dx}{5 b e}\\ &=-\frac{(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}-\frac{((b d-a e) (7 b B d-10 A b e+3 a B e)) \int \frac{(a+b x)^{5/2}}{\sqrt{d+e x}} \, dx}{80 b^2 e}\\ &=-\frac{(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{240 b^2 e^2}-\frac{(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac{\left ((b d-a e)^2 (7 b B d-10 A b e+3 a B e)\right ) \int \frac{(a+b x)^{3/2}}{\sqrt{d+e x}} \, dx}{96 b^2 e^2}\\ &=\frac{(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{192 b^2 e^3}-\frac{(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{240 b^2 e^2}-\frac{(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}-\frac{\left ((b d-a e)^3 (7 b B d-10 A b e+3 a B e)\right ) \int \frac{\sqrt{a+b x}}{\sqrt{d+e x}} \, dx}{128 b^2 e^3}\\ &=-\frac{(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{128 b^2 e^4}+\frac{(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{192 b^2 e^3}-\frac{(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{240 b^2 e^2}-\frac{(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac{\left ((b d-a e)^4 (7 b B d-10 A b e+3 a B e)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{256 b^2 e^4}\\ &=-\frac{(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{128 b^2 e^4}+\frac{(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{192 b^2 e^3}-\frac{(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{240 b^2 e^2}-\frac{(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac{\left ((b d-a e)^4 (7 b B d-10 A b e+3 a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{128 b^3 e^4}\\ &=-\frac{(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{128 b^2 e^4}+\frac{(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{192 b^2 e^3}-\frac{(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{240 b^2 e^2}-\frac{(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac{\left ((b d-a e)^4 (7 b B d-10 A b e+3 a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{128 b^3 e^4}\\ &=-\frac{(b d-a e)^3 (7 b B d-10 A b e+3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{128 b^2 e^4}+\frac{(b d-a e)^2 (7 b B d-10 A b e+3 a B e) (a+b x)^{3/2} \sqrt{d+e x}}{192 b^2 e^3}-\frac{(b d-a e) (7 b B d-10 A b e+3 a B e) (a+b x)^{5/2} \sqrt{d+e x}}{240 b^2 e^2}-\frac{(7 b B d-10 A b e+3 a B e) (a+b x)^{7/2} \sqrt{d+e x}}{40 b^2 e}+\frac{B (a+b x)^{7/2} (d+e x)^{3/2}}{5 b e}+\frac{(b d-a e)^4 (7 b B d-10 A b e+3 a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{128 b^{5/2} e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 2.86998, size = 359, normalized size = 1.18 \[ \frac{(a+b x)^{7/2} (d+e x)^{3/2} \left (\frac{7 \left (-\frac{3 a B e}{2}+5 A b e-\frac{7}{2} b B d\right ) \left (48 b^4 e^4 (a+b x)^4 \sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}}+b (b d-a e) \left (8 b^3 e^3 (a+b x)^3 \sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}}-10 b^3 e^2 (a+b x)^2 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}}+15 b^3 e (a+b x) (b d-a e)^{5/2} \sqrt{\frac{b (d+e x)}{b d-a e}}-15 b^3 \sqrt{e} \sqrt{a+b x} (b d-a e)^3 \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )\right )}{192 b^4 e^4 (a+b x)^4 (b d-a e)^{3/2} \left (\frac{b (d+e x)}{b d-a e}\right )^{3/2}}+7 B\right )}{35 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(7/2)*(d + e*x)^(3/2)*(7*B + (7*((-7*b*B*d)/2 + 5*A*b*e - (3*a*B*e)/2)*(48*b^4*e^4*Sqrt[b*d - a*e]*
(a + b*x)^4*Sqrt[(b*(d + e*x))/(b*d - a*e)] + b*(b*d - a*e)*(15*b^3*e*(b*d - a*e)^(5/2)*(a + b*x)*Sqrt[(b*(d +
e*x))/(b*d - a*e)] - 10*b^3*e^2*(b*d - a*e)^(3/2)*(a + b*x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*b^3*e^3*Sqr
t[b*d - a*e]*(a + b*x)^3*Sqrt[(b*(d + e*x))/(b*d - a*e)] - 15*b^3*Sqrt[e]*(b*d - a*e)^3*Sqrt[a + b*x]*ArcSinh[
(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])))/(192*b^4*e^4*(b*d - a*e)^(3/2)*(a + b*x)^4*((b*(d + e*x))/(b*d - a
*e))^(3/2))))/(35*b*e)

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Maple [B]  time = 0.017, size = 1631, normalized size = 5.4 \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)^(5/2)*(B*x+A)*(e*x+d)^(1/2),x)

[Out]

-1/3840*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(375*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b
*d)/(b*e)^(1/2))*a*b^4*d^4*e-2016*B*x^3*a*b^3*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+1100*A*(b*e)^(1/
2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*b^3*d^2*e^2-768*B*x^4*b^4*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)
-960*A*x^3*b^4*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-300*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/
2)*a^3*b*e^4-300*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*b^4*d^3*e+150*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*
x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b*e^5-450*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*
d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*d^3*e^2+75*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1
/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b*d*e^4+150*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e
)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2*d^2*e^3-96*B*x^3*b^4*d*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-1
40*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*b^4*d^3*e-120*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)
*a^3*b*d*e^3+112*B*x^2*b^4*d^2*e^2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-2720*A*x^2*a*b^3*e^4*(b*e*x^2+a
*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-160*A*x^2*b^4*d*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-1488*B*x^2*a
^2*b^2*e^4*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-600*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)
*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2*d*e^4+900*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)
^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*d^2*e^3-2360*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*b^2*e^4+
200*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*b^4*d^2*e^2-60*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/
2)*x*a^3*b*e^4+444*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a*b^3*d^2*e^2-720*A*(b*e)^(1/2)*(b*e*x^2+a*
e*x+b*d*x+a*d)^(1/2)*x*a*b^3*d*e^3-436*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*a^2*b^2*d*e^3-352*B*x^2
*a*b^3*d*e^3*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)-45*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2
)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^5*e^5-105*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2
)+a*e+b*d)/(b*e)^(1/2))*b^5*d^5-680*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*b^3*d^3*e+692*B*(b*e)^(1/2
)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*b^2*d^2*e^2-1460*A*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*b^2*d
*e^3+90*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^4*e^4+210*B*(b*e)^(1/2)*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2
)*b^4*d^4+150*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^5*d^4*e-
600*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^4*d^3*e^2)/b^2/(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)/e^4/(b*e)^(1/2)

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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)^(5/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78804, size = 2329, normalized size = 7.66 \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)^(5/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[-1/7680*(15*(7*B*b^5*d^5 - 5*(5*B*a*b^4 + 2*A*b^5)*d^4*e + 10*(3*B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 - 10*(B*a^3*b
^2 + 6*A*a^2*b^3)*d^2*e^3 - 5*(B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 10*A*a^4*b)*e^5)*sqrt(b*e)*log(8*b^2*
e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b
^2*d*e + a*b*e^2)*x) - 4*(384*B*b^5*e^5*x^4 - 105*B*b^5*d^4*e + 10*(34*B*a*b^4 + 15*A*b^5)*d^3*e^2 - 2*(173*B*
a^2*b^3 + 275*A*a*b^4)*d^2*e^3 + 10*(6*B*a^3*b^2 + 73*A*a^2*b^3)*d*e^4 - 15*(3*B*a^4*b - 10*A*a^3*b^2)*e^5 + 4
8*(B*b^5*d*e^4 + (21*B*a*b^4 + 10*A*b^5)*e^5)*x^3 - 8*(7*B*b^5*d^2*e^3 - 2*(11*B*a*b^4 + 5*A*b^5)*d*e^4 - (93*
B*a^2*b^3 + 170*A*a*b^4)*e^5)*x^2 + 2*(35*B*b^5*d^3*e^2 - (111*B*a*b^4 + 50*A*b^5)*d^2*e^3 + (109*B*a^2*b^3 +
180*A*a*b^4)*d*e^4 + 5*(3*B*a^3*b^2 + 118*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^5), -1/3840*(
15*(7*B*b^5*d^5 - 5*(5*B*a*b^4 + 2*A*b^5)*d^4*e + 10*(3*B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 - 10*(B*a^3*b^2 + 6*A*a
^2*b^3)*d^2*e^3 - 5*(B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 10*A*a^4*b)*e^5)*sqrt(-b*e)*arctan(1/2*(2*b*e*x
+ b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) - 2*(384
*B*b^5*e^5*x^4 - 105*B*b^5*d^4*e + 10*(34*B*a*b^4 + 15*A*b^5)*d^3*e^2 - 2*(173*B*a^2*b^3 + 275*A*a*b^4)*d^2*e^
3 + 10*(6*B*a^3*b^2 + 73*A*a^2*b^3)*d*e^4 - 15*(3*B*a^4*b - 10*A*a^3*b^2)*e^5 + 48*(B*b^5*d*e^4 + (21*B*a*b^4
+ 10*A*b^5)*e^5)*x^3 - 8*(7*B*b^5*d^2*e^3 - 2*(11*B*a*b^4 + 5*A*b^5)*d*e^4 - (93*B*a^2*b^3 + 170*A*a*b^4)*e^5)
*x^2 + 2*(35*B*b^5*d^3*e^2 - (111*B*a*b^4 + 50*A*b^5)*d^2*e^3 + (109*B*a^2*b^3 + 180*A*a*b^4)*d*e^4 + 5*(3*B*a
^3*b^2 + 118*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^5)]

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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)**(5/2)*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 3.07847, size = 1882, normalized size = 6.19 \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)^(5/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

1/1920*(10*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^2 + (b^7*d*e^5 - 17*a
*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^8) + 3*(5*b^9*d^3*e^3 + a*b^
8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^4 - 4*a*b^3*d^3*e - 2*a^2*b^
2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a
)*b*e - a*b*e)))/b^(3/2))*A*abs(b) + (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x
+ a)/b^3 + (b^13*d*e^7 - 31*a*b^12*e^8)*e^(-8)/b^15) - (7*b^14*d^2*e^6 + 16*a*b^13*d*e^7 - 263*a^2*b^12*e^8)*
e^(-8)/b^15) + 5*(7*b^15*d^3*e^5 + 9*a*b^14*d^2*e^6 + 9*a^2*b^13*d*e^7 - 121*a^3*b^12*e^8)*e^(-8)/b^15)*(b*x +
a) - 15*(7*b^16*d^4*e^4 + 2*a*b^15*d^3*e^5 - 2*a^3*b^13*d*e^7 - 7*a^4*b^12*e^8)*e^(-8)/b^15)*sqrt(b*x + a) -
15*(7*b^5*d^5 - 5*a*b^4*d^4*e - 2*a^2*b^3*d^3*e^2 - 2*a^3*b^2*d^2*e^3 - 5*a^4*b*d*e^4 + 7*a^5*e^5)*e^(-9/2)*lo
g(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(5/2))*B*abs(b) + 20*(sqrt(b^2*
d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e^(-2)/b^4 + (b*d*e - a*e^2)*e^(-4)/b^4) + (b^2*d^2 - 2*
a*b*d*e + a^2*e^2)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(
7/2))*A*a^2*abs(b)/b^2 + 20*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^2 +
(b^7*d*e^5 - 17*a*b^6*e^6)*e^(-6)/b^8) - (5*b^8*d^2*e^4 + 6*a*b^7*d*e^5 - 59*a^2*b^6*e^6)*e^(-6)/b^8) + 3*(5*b
^9*d^3*e^3 + a*b^8*d^2*e^4 - a^2*b^7*d*e^5 - 5*a^3*b^6*e^6)*e^(-6)/b^8)*sqrt(b*x + a) + 3*(5*b^4*d^4 - 4*a*b^3
*d^3*e - 2*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + 5*a^4*e^4)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt
(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*B*a*abs(b)/b + (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(
2*(b*x + a)*(4*(b*x + a)*e^(-2)/b^6 + (b*d*e^3 - 7*a*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^2 - a^2*e^4)*e^(-6)/b^6)
- 3*(b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2
*d + (b*x + a)*b*e - a*b*e)))/b^(11/2))*B*a^2*abs(b)/b^3 + 2*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a
)*(2*(b*x + a)*(4*(b*x + a)*e^(-2)/b^6 + (b*d*e^3 - 7*a*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^2 - a^2*e^4)*e^(-6)/b^
6) - 3*(b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(
b^2*d + (b*x + a)*b*e - a*b*e)))/b^(11/2))*A*a*abs(b)/b^2)/b