\(\int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx\) [2213]

Optimal result

Integrand size = 24, antiderivative size = 250 \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\frac {(b d-a e)^2 (5 b B d-8 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^2 e^3}-\frac {(b d-a e) (5 b B d-8 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b^2 e^2}-\frac {(5 b B d-8 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {(b d-a e)^3 (5 b B d-8 A b e+3 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{5/2} e^{7/2}} \]

[Out]

1/4*B*(b*x+a)^(5/2)*(e*x+d)^(3/2)/b/e-1/64*(-a*e+b*d)^3*(-8*A*b*e+3*B*a*e+5*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/
2)/b^(1/2)/(e*x+d)^(1/2))/b^(5/2)/e^(7/2)-1/96*(-a*e+b*d)*(-8*A*b*e+3*B*a*e+5*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/
2)/b^2/e^2-1/24*(-8*A*b*e+3*B*a*e+5*B*b*d)*(b*x+a)^(5/2)*(e*x+d)^(1/2)/b^2/e+1/64*(-a*e+b*d)^2*(-8*A*b*e+3*B*a
*e+5*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^2/e^3

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {81, 52, 65, 223, 212} \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=-\frac {(b d-a e)^3 (3 a B e-8 A b e+5 b B d) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{5/2} e^{7/2}}+\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (3 a B e-8 A b e+5 b B d)}{64 b^2 e^3}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e) (3 a B e-8 A b e+5 b B d)}{96 b^2 e^2}-\frac {(a+b x)^{5/2} \sqrt {d+e x} (3 a B e-8 A b e+5 b B d)}{24 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e} \]

[In]

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

[Out]

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

Rule 52

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 65

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 81

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 212

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

Rule 223

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]

Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}+\frac {\left (4 A b e-B \left (\frac {5 b d}{2}+\frac {3 a e}{2}\right )\right ) \int (a+b x)^{3/2} \sqrt {d+e x} \, dx}{4 b e} \\ & = -\frac {(5 b B d-8 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {((b d-a e) (5 b B d-8 A b e+3 a B e)) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{48 b^2 e} \\ & = -\frac {(b d-a e) (5 b B d-8 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b^2 e^2}-\frac {(5 b B d-8 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}+\frac {\left ((b d-a e)^2 (5 b B d-8 A b e+3 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{64 b^2 e^2} \\ & = \frac {(b d-a e)^2 (5 b B d-8 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^2 e^3}-\frac {(b d-a e) (5 b B d-8 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b^2 e^2}-\frac {(5 b B d-8 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {\left ((b d-a e)^3 (5 b B d-8 A b e+3 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{128 b^2 e^3} \\ & = \frac {(b d-a e)^2 (5 b B d-8 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^2 e^3}-\frac {(b d-a e) (5 b B d-8 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b^2 e^2}-\frac {(5 b B d-8 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {\left ((b d-a e)^3 (5 b B d-8 A b e+3 a B e)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^3 e^3} \\ & = \frac {(b d-a e)^2 (5 b B d-8 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^2 e^3}-\frac {(b d-a e) (5 b B d-8 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b^2 e^2}-\frac {(5 b B d-8 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {\left ((b d-a e)^3 (5 b B d-8 A b e+3 a B e)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{64 b^3 e^3} \\ & = \frac {(b d-a e)^2 (5 b B d-8 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^2 e^3}-\frac {(b d-a e) (5 b B d-8 A b e+3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{96 b^2 e^2}-\frac {(5 b B d-8 A b e+3 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{24 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{3/2}}{4 b e}-\frac {(b d-a e)^3 (5 b B d-8 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{5/2} e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.93 \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (-9 a^3 B e^3+3 a^2 b e^2 (3 B d+8 A e+2 B e x)+a b^2 e \left (16 A e (4 d+7 e x)+B \left (-31 d^2+20 d e x+72 e^2 x^2\right )\right )+b^3 \left (8 A e \left (-3 d^2+2 d e x+8 e^2 x^2\right )+B \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b^2 e^3}+\frac {(b d-a e)^3 (-5 b B d+8 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{64 b^{5/2} e^{7/2}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-9*a^3*B*e^3 + 3*a^2*b*e^2*(3*B*d + 8*A*e + 2*B*e*x) + a*b^2*e*(16*A*e*(4*d + 7*
e*x) + B*(-31*d^2 + 20*d*e*x + 72*e^2*x^2)) + b^3*(8*A*e*(-3*d^2 + 2*d*e*x + 8*e^2*x^2) + B*(15*d^3 - 10*d^2*e
*x + 8*d*e^2*x^2 + 48*e^3*x^3))))/(192*b^2*e^3) + ((b*d - a*e)^3*(-5*b*B*d + 8*A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[
b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(64*b^(5/2)*e^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs. \(2(212)=424\).

Time = 1.09 (sec) , antiderivative size = 968, normalized size of antiderivative = 3.87

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {e x +d}\, \left (-40 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} d \,e^{2} x -9 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{4} e^{4}+15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{4}+48 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{2} e -48 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b \,e^{3}-36 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{3} e +18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d^{2} e^{2}-72 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d \,e^{3}+72 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{2} e^{2}+12 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b d \,e^{3}-18 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b d \,e^{2}+62 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} d^{2} e +18 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{3} e^{3}-30 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{3}+24 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b \,e^{4}-24 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{3} e -224 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a \,b^{2} e^{3} x -32 A \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d \,e^{2} x -12 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a^{2} b \,e^{3} x +20 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b^{3} d^{2} e x -144 B a \,b^{2} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-16 B \,b^{3} d \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-128 A a \,b^{2} d \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-96 B \,b^{3} e^{3} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-128 A \,b^{3} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{384 b^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, e^{3} \sqrt {b e}}\)	\(968\)

[In]

int((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/384*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-40*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^2*d*e^2*x-9*B*ln(1/2*(2*b*e*
x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*e^4+15*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^4*d^4+48*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b^3*d^2*e-48*A*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b*e^3-36*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(
b*e)^(1/2))*a*b^3*d^3*e+18*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b
^2*d^2*e^2-72*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^2*d*e^3+72*A
*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^3*d^2*e^2+12*B*ln(1/2*(2*b*e*
x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b*d*e^3-18*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2)*a^2*b*d*e^2+62*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^2*d^2*e+18*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)
*a^3*e^3-30*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b^3*d^3+24*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b*e^4-24*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*
e)^(1/2))*b^4*d^3*e-224*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^2*e^3*x-32*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2)*b^3*d*e^2*x-12*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b*e^3*x+20*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)
*b^3*d^2*e*x-144*B*a*b^2*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-16*B*b^3*d*e^2*x^2*((b*x+a)*(e*x+d))^(1/2
)*(b*e)^(1/2)-128*A*a*b^2*d*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-96*B*b^3*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(
b*e)^(1/2)-128*A*b^3*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/b^2/((b*x+a)*(e*x+d))^(1/2)/e^3/(b*e)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 766, normalized size of antiderivative = 3.06 \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\left [\frac {3 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (3 \, B a b^{3} + 2 \, A b^{4}\right )} d^{3} e + 6 \, {\left (B a^{2} b^{2} + 4 \, A a b^{3}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} b - 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (48 \, B b^{4} e^{4} x^{3} + 15 \, B b^{4} d^{3} e - {\left (31 \, B a b^{3} + 24 \, A b^{4}\right )} d^{2} e^{2} + {\left (9 \, B a^{2} b^{2} + 64 \, A a b^{3}\right )} d e^{3} - 3 \, {\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (B b^{4} d e^{3} + {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{4} d^{2} e^{2} - 2 \, {\left (5 \, B a b^{3} + 4 \, A b^{4}\right )} d e^{3} - {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{768 \, b^{3} e^{4}}, \frac {3 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (3 \, B a b^{3} + 2 \, A b^{4}\right )} d^{3} e + 6 \, {\left (B a^{2} b^{2} + 4 \, A a b^{3}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} b - 6 \, A a^{2} b^{2}\right )} d e^{3} - {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, B b^{4} e^{4} x^{3} + 15 \, B b^{4} d^{3} e - {\left (31 \, B a b^{3} + 24 \, A b^{4}\right )} d^{2} e^{2} + {\left (9 \, B a^{2} b^{2} + 64 \, A a b^{3}\right )} d e^{3} - 3 \, {\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} e^{4} + 8 \, {\left (B b^{4} d e^{3} + {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} e^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{4} d^{2} e^{2} - 2 \, {\left (5 \, B a b^{3} + 4 \, A b^{4}\right )} d e^{3} - {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} e^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{384 \, b^{3} e^{4}}\right ] \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[1/768*(3*(5*B*b^4*d^4 - 4*(3*B*a*b^3 + 2*A*b^4)*d^3*e + 6*(B*a^2*b^2 + 4*A*a*b^3)*d^2*e^2 + 4*(B*a^3*b - 6*A*
a^2*b^2)*d*e^3 - (3*B*a^4 - 8*A*a^3*b)*e^4)*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*(48*B*b^4*e^4*x^3 + 1
5*B*b^4*d^3*e - (31*B*a*b^3 + 24*A*b^4)*d^2*e^2 + (9*B*a^2*b^2 + 64*A*a*b^3)*d*e^3 - 3*(3*B*a^3*b - 8*A*a^2*b^
2)*e^4 + 8*(B*b^4*d*e^3 + (9*B*a*b^3 + 8*A*b^4)*e^4)*x^2 - 2*(5*B*b^4*d^2*e^2 - 2*(5*B*a*b^3 + 4*A*b^4)*d*e^3
- (3*B*a^2*b^2 + 56*A*a*b^3)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^4), 1/384*(3*(5*B*b^4*d^4 - 4*(3*B*a*
b^3 + 2*A*b^4)*d^3*e + 6*(B*a^2*b^2 + 4*A*a*b^3)*d^2*e^2 + 4*(B*a^3*b - 6*A*a^2*b^2)*d*e^3 - (3*B*a^4 - 8*A*a^
3*b)*e^4)*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*(48*B*b^4*e^4*x^3 + 15*B*b^4*d^3*e - (31*B*a*b^3 + 24*A*b^4)*d^2*e^2 + (9*
B*a^2*b^2 + 64*A*a*b^3)*d*e^3 - 3*(3*B*a^3*b - 8*A*a^2*b^2)*e^4 + 8*(B*b^4*d*e^3 + (9*B*a*b^3 + 8*A*b^4)*e^4)*
x^2 - 2*(5*B*b^4*d^2*e^2 - 2*(5*B*a*b^3 + 4*A*b^4)*d*e^3 - (3*B*a^2*b^2 + 56*A*a*b^3)*e^4)*x)*sqrt(b*x + a)*sq
rt(e*x + d))/(b^3*e^4)]

Sympy [F]

\[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\int \left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}} \sqrt {d + e x}\, dx \]

[In]

integrate((b*x+a)**(3/2)*(B*x+A)*(e*x+d)**(1/2),x)

[Out]

Integral((A + B*x)*(a + b*x)**(3/2)*sqrt(d + e*x), x)

Maxima [F(-2)]

Exception generated. \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (212) = 424\).

Time = 0.44 (sec) , antiderivative size = 1071, normalized size of antiderivative = 4.28 \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

1/192*(8*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*
b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) - 3*(b^3*d^3 + a*b^2*d^2*e +
3*a^2*b*d*e^2 - 5*a^3*e^3)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e
)*b*e^2))*A*abs(b) + (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^3 + (b^12*d
*e^5 - 25*a*b^11*e^6)/(b^14*e^6)) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)/(b^14*e^6)) + 3*(5*b
^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)/(b^14*e^6))*sqrt(b*x + a) + 3*(5*b^4*d^4
+ 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^
2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b^2*e^3))*B*abs(b) - 192*((b^2*d - a*b*e)*log(abs(-sqrt(b*e)*sqrt(b*
x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b*e) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a))*
A*a^2*abs(b)/b^2 + 16*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*
d*e^3 - 13*a*b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) - 3*(b^3*d^3 +
a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e
)))/(sqrt(b*e)*b*e^2))*B*a*abs(b)/b + 48*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + 2*a + (b*d*e - 5*a*e^2)
/e^2)*sqrt(b*x + a) + (b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b
*x + a)*b*e - a*b*e)))/(sqrt(b*e)*e))*B*a^2*abs(b)/b^3 + 96*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + 2*a
+ (b*d*e - 5*a*e^2)/e^2)*sqrt(b*x + a) + (b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*log(abs(-sqrt(b*e)*sqrt(b*x + a
) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*e))*A*a*abs(b)/b^2)/b

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{3/2} (A+B x) \sqrt {d+e x} \, dx=\int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {d+e\,x} \,d x \]

[In]

int((A + B*x)*(a + b*x)^(3/2)*(d + e*x)^(1/2),x)

[Out]

int((A + B*x)*(a + b*x)^(3/2)*(d + e*x)^(1/2), x)