\(\int x^2 \sqrt {a+b x} (c+d x)^{3/2} \, dx\) [557]

Optimal result

Integrand size = 22, antiderivative size = 315 \[ \int x^2 \sqrt {a+b x} (c+d x)^{3/2} \, dx=\frac {(b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^3}+\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}-\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{7/2}} \]

[Out]

1/48*(7*a^2*d^2+6*a*b*c*d+3*b^2*c^2)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/b^3/d^2-1/40*(7*a*d+5*b*c)*(b*x+a)^(3/2)*(d*x
+c)^(5/2)/b^2/d^2+1/5*x*(b*x+a)^(3/2)*(d*x+c)^(5/2)/b/d-1/128*(-a*d+b*c)^3*(7*a^2*d^2+6*a*b*c*d+3*b^2*c^2)*arc
tanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(9/2)/d^(7/2)+1/64*(-a*d+b*c)*(7*a^2*d^2+6*a*b*c*d+3*b^2*c
^2)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^4/d^2+1/128*(-a*d+b*c)^2*(7*a^2*d^2+6*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1/2)*(d*x+
c)^(1/2)/b^4/d^3

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65, 223, 212} \[ \int x^2 \sqrt {a+b x} (c+d x)^{3/2} \, dx=-\frac {\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{7/2}}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)}{64 b^4 d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^2}{128 b^4 d^3}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right )}{48 b^3 d^2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+5 b c)}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d} \]

[In]

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

[Out]

((b*c - a*d)^2*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^4*d^3) + ((b*c - a*d)*(
3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(64*b^4*d^2) + ((3*b^2*c^2 + 6*a*b*c*d + 7*a
^2*d^2)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(48*b^3*d^2) - ((5*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(40*
b^2*d^2) + (x*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(5*b*d) - ((b*c - a*d)^3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*Ar
cTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(9/2)*d^(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 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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 {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}+\frac {\int \sqrt {a+b x} (c+d x)^{3/2} \left (-a c-\frac {1}{2} (5 b c+7 a d) x\right ) \, dx}{5 b d} \\ & = -\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \int \sqrt {a+b x} (c+d x)^{3/2} \, dx}{16 b^2 d^2} \\ & = \frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}+\frac {\left ((b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{32 b^3 d^2} \\ & = \frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^4 d^2} \\ & = \frac {(b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^3}+\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}-\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^4 d^3} \\ & = \frac {(b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^3}+\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}-\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{128 b^5 d^3} \\ & = \frac {(b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^3}+\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}-\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 b^5 d^3} \\ & = \frac {(b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^3}+\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}-\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.78 \[ \int x^2 \sqrt {a+b x} (c+d x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+10 a^3 b d^3 (19 c+7 d x)-2 a^2 b^2 d^2 \left (18 c^2+61 c d x+28 d^2 x^2\right )+6 a b^3 d \left (-5 c^3+3 c^2 d x+16 c d^2 x^2+8 d^3 x^3\right )+3 b^4 \left (15 c^4-10 c^3 d x+8 c^2 d^2 x^2+176 c d^3 x^3+128 d^4 x^4\right )\right )}{1920 b^4 d^3}-\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{7/2}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^4*d^4 + 10*a^3*b*d^3*(19*c + 7*d*x) - 2*a^2*b^2*d^2*(18*c^2 + 61*c*d*x +
28*d^2*x^2) + 6*a*b^3*d*(-5*c^3 + 3*c^2*d*x + 16*c*d^2*x^2 + 8*d^3*x^3) + 3*b^4*(15*c^4 - 10*c^3*d*x + 8*c^2*d
^2*x^2 + 176*c*d^3*x^3 + 128*d^4*x^4)))/(1920*b^4*d^3) - ((b*c - a*d)^3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*Ar
cTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(9/2)*d^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(271)=542\).

Time = 0.53 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.50

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (768 b^{4} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+96 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+1056 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-112 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+192 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+48 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{5} d^{5}-225 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} b c \,d^{4}+90 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b^{2} c^{2} d^{3}+30 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{3} c^{3} d^{2}+45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{4} c^{4} d -45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{5} c^{5}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b \,d^{4} x -244 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x +36 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x -60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{3} d x -210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{4} d^{4}+380 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b c \,d^{3}-72 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}-60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{3} d +90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{4}\right )}{3840 d^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{4} \sqrt {b d}}\)	\(788\)

[In]

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

[Out]

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(768*b^4*d^4*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+96*a*b^3*d^4*x^3*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1056*b^4*c*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-112*a^2*b^2*d^4*x^2*((b*
x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+192*a*b^3*c*d^3*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+48*b^4*c^2*d^2*x^2*((b
*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2
))*a^5*d^5-225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b*c*d^4+90*ln(1
/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^2*c^2*d^3+30*ln(1/2*(2*b*d*x+2*(
(b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^3*c^3*d^2+45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^4*c^4*d-45*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)
+a*d+b*c)/(b*d)^(1/2))*b^5*c^5+140*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b*d^4*x-244*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)*a^2*b^2*c*d^3*x+36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^3*c^2*d^2*x-60*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)*b^4*c^3*d*x-210*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*d^4+380*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/
2)*a^3*b*c*d^3-72*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^2*c^2*d^2-60*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a
*b^3*c^3*d+90*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^4*c^4)/d^3/((b*x+a)*(d*x+c))^(1/2)/b^4/(b*d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.23 \[ \int x^2 \sqrt {a+b x} (c+d x)^{3/2} \, dx=\left [-\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} + 45 \, b^{5} c^{4} d - 30 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} + 190 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (11 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (3 \, b^{5} c^{2} d^{3} + 12 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (15 \, b^{5} c^{3} d^{2} - 9 \, a b^{4} c^{2} d^{3} + 61 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{5} d^{4}}, \frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (384 \, b^{5} d^{5} x^{4} + 45 \, b^{5} c^{4} d - 30 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} + 190 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (11 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (3 \, b^{5} c^{2} d^{3} + 12 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (15 \, b^{5} c^{3} d^{2} - 9 \, a b^{4} c^{2} d^{3} + 61 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{5} d^{4}}\right ] \]

[In]

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

[Out]

[-1/7680*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 7*a^5*d^5)*
sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*
sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(384*b^5*d^5*x^4 + 45*b^5*c^4*d - 30*a*b^4*c^3*d^2 - 36*a^2*b^3*c
^2*d^3 + 190*a^3*b^2*c*d^4 - 105*a^4*b*d^5 + 48*(11*b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(3*b^5*c^2*d^3 + 12*a*b^4*c
*d^4 - 7*a^2*b^3*d^5)*x^2 - 2*(15*b^5*c^3*d^2 - 9*a*b^4*c^2*d^3 + 61*a^2*b^3*c*d^4 - 35*a^3*b^2*d^5)*x)*sqrt(b
*x + a)*sqrt(d*x + c))/(b^5*d^4), 1/3840*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^
3 + 15*a^4*b*c*d^4 - 7*a^5*d^5)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x
+ c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(384*b^5*d^5*x^4 + 45*b^5*c^4*d - 30*a*b^4*c^3*d^2 -
36*a^2*b^3*c^2*d^3 + 190*a^3*b^2*c*d^4 - 105*a^4*b*d^5 + 48*(11*b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(3*b^5*c^2*d^3
+ 12*a*b^4*c*d^4 - 7*a^2*b^3*d^5)*x^2 - 2*(15*b^5*c^3*d^2 - 9*a*b^4*c^2*d^3 + 61*a^2*b^3*c*d^4 - 35*a^3*b^2*d
^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^4)]

Sympy [F]

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

[In]

integrate(x**2*(d*x+c)**(3/2)*(b*x+a)**(1/2),x)

[Out]

Integral(x**2*sqrt(a + b*x)*(c + d*x)**(3/2), x)

Maxima [F(-2)]

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

[In]

integrate(x^2*(d*x+c)^(3/2)*(b*x+a)^(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(a*d-b*c>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1170 vs. \(2 (271) = 542\).

Time = 0.43 (sec) , antiderivative size = 1170, normalized size of antiderivative = 3.71 \[ \int x^2 \sqrt {a+b x} (c+d x)^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/1920*(80*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*
a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - 5*a^3*d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b
*d)*b*d^2))*a*c*abs(b)/b^2 + 10*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^
3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6
)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3
*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a
) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*c*abs(b)/b + 10*(sqrt(b^2*c + (b*x + a)*b*d - a
*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4
+ 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 -
93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^
3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*
a*d*abs(b)/b^2 + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*c*
d^7 - 41*a*b^19*d^8)/(b^23*d^8)) - (7*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^23*d^8)) + 5*(7*b^
22*c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a^2*b^20*c*d^7 - 447*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*
d^4 + 12*a*b^22*c^3*d^5 + 18*a^2*b^21*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8))*sqrt(b*x + a
) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*a^5*d^5)*log(
abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^4))*d*abs(b)/b)/b

Mupad [F(-1)]

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

[In]

int(x^2*(a + b*x)^(1/2)*(c + d*x)^(3/2),x)

[Out]

int(x^2*(a + b*x)^(1/2)*(c + d*x)^(3/2), x)