\(\int x \sqrt {a+b x} (c+d x)^{5/2} \, dx\) [567]

Optimal result

Integrand size = 20, antiderivative size = 268 \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=-\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {(b c-a d)^4 (3 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\frac {(7 a d+3 b c) (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (7 a d+3 b c) (b c-a d)^3}{128 b^4 d^2}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (7 a d+3 b c) (b c-a d)^2}{64 b^4 d}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 b^3 d}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+3 b c)}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d} \]

[In]

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

[Out]

-1/128*((b*c - a*d)^3*(3*b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(b^4*d^2) - ((b*c - a*d)^2*(3*b*c + 7*a*d)*
(a + b*x)^(3/2)*Sqrt[c + d*x])/(64*b^4*d) - ((b*c - a*d)*(3*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(48*
b^3*d) - ((3*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(5/2))/(40*b^2*d) + ((a + b*x)^(3/2)*(c + d*x)^(7/2))/(5*b
*d) + ((b*c - a*d)^4*(3*b*c + 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(9/2)*d^
(5/2))

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 {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {(3 b c+7 a d) \int \sqrt {a+b x} (c+d x)^{5/2} \, dx}{10 b d} \\ & = -\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {((b c-a d) (3 b c+7 a d)) \int \sqrt {a+b x} (c+d x)^{3/2} \, dx}{16 b^2 d} \\ & = -\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {\left ((b c-a d)^2 (3 b c+7 a d)\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{32 b^3 d} \\ & = -\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}-\frac {\left ((b c-a d)^3 (3 b c+7 a d)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^4 d} \\ & = -\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^4 (3 b c+7 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^4 d^2} \\ & = -\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^4 (3 b c+7 a d)\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^2} \\ & = -\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^4 (3 b c+7 a d)\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^2} \\ & = -\frac {(b c-a d)^3 (3 b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^2}-\frac {(b c-a d)^2 (3 b c+7 a d) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d}-\frac {(3 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d}+\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 b d}+\frac {(b c-a d)^4 (3 b c+7 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.86 \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+10 a^3 b d^3 (34 c+7 d x)-2 a^2 b^2 d^2 \left (173 c^2+111 c d x+28 d^2 x^2\right )+2 a b^3 d \left (30 c^3+109 c^2 d x+88 c d^2 x^2+24 d^3 x^3\right )+b^4 \left (-45 c^4+30 c^3 d x+744 c^2 d^2 x^2+1008 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^4 d^2}+\frac {(b c-a d)^4 (3 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{5/2}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^4*d^4 + 10*a^3*b*d^3*(34*c + 7*d*x) - 2*a^2*b^2*d^2*(173*c^2 + 111*c*d*x
+ 28*d^2*x^2) + 2*a*b^3*d*(30*c^3 + 109*c^2*d*x + 88*c*d^2*x^2 + 24*d^3*x^3) + b^4*(-45*c^4 + 30*c^3*d*x + 744
*c^2*d^2*x^2 + 1008*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^4*d^2) + ((b*c - a*d)^4*(3*b*c + 7*a*d)*ArcTanh[(Sqrt[d
]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(9/2)*d^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(224)=448\).

Time = 1.51 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.94

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}+2016 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}+352 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+1488 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}-375 \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}+450 \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}-150 \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}-75 \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 -444 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x +436 \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}+680 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b c \,d^{3}-692 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}+120 \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^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{4} \sqrt {b d}}\)	\(788\)

[In]

int(x*(d*x+c)^(5/2)*(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)+2016*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)+352*a*b^3*c*d^3*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1488*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^5-375*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+450*l
n(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^3-150*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^2-75*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-444*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)*a^2*b^2*c*d^3*x+436*((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+680*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)*a^3*b*c*d^3-692*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^2*c^2*d^2+120*((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^2/((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 = 2.63 \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\left [\frac {15 \, {\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d + 60 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 340 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (21 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} + 22 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} + 109 \, a b^{4} c^{2} d^{3} - 111 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{5} d^{3}}, -\frac {15 \, {\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d + 60 \, a b^{4} c^{3} d^{2} - 346 \, a^{2} b^{3} c^{2} d^{3} + 340 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (21 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} + 22 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} + 109 \, a b^{4} c^{2} d^{3} - 111 \, a^{2} b^{3} c d^{4} + 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{5} d^{3}}\right ] \]

[In]

integrate(x*(d*x+c)^(5/2)*(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/7680*(15*(3*b^5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b^2*c^2*d^3 - 25*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 + 60*a*b^4*c^3*d^2 - 346*a^2*b^3
*c^2*d^3 + 340*a^3*b^2*c*d^4 - 105*a^4*b*d^5 + 48*(21*b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(93*b^5*c^2*d^3 + 22*a*b^
4*c*d^4 - 7*a^2*b^3*d^5)*x^2 + 2*(15*b^5*c^3*d^2 + 109*a*b^4*c^2*d^3 - 111*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^3), -1/3840*(15*(3*b^5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b
^2*c^2*d^3 - 25*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 + 60*a*b^4*
c^3*d^2 - 346*a^2*b^3*c^2*d^3 + 340*a^3*b^2*c*d^4 - 105*a^4*b*d^5 + 48*(21*b^5*c*d^4 + a*b^4*d^5)*x^3 + 8*(93*
b^5*c^2*d^3 + 22*a*b^4*c*d^4 - 7*a^2*b^3*d^5)*x^2 + 2*(15*b^5*c^3*d^2 + 109*a*b^4*c^2*d^3 - 111*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^3)]

Sympy [F]

\[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\int x \sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}\, dx \]

[In]

integrate(x*(d*x+c)**(5/2)*(b*x+a)**(1/2),x)

[Out]

Integral(x*sqrt(a + b*x)*(c + d*x)**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x*(d*x+c)^(5/2)*(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. 1522 vs. \(2 (224) = 448\).

Time = 0.49 (sec) , antiderivative size = 1522, normalized size of antiderivative = 5.68 \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate(x*(d*x+c)^(5/2)*(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))*c^2*abs(b)/b + 160*(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*d*abs(b)/b^2 + 20*(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*d*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^2*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^2*abs(b)/b + 480*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x +
a) + (b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*
b*d)))/(sqrt(b*d)*d))*a*c^2*abs(b)/b^3)/b

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {a+b x} (c+d x)^{5/2} \, dx=\int x\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2} \,d x \]

[In]

int(x*(a + b*x)^(1/2)*(c + d*x)^(5/2),x)

[Out]

int(x*(a + b*x)^(1/2)*(c + d*x)^(5/2), x)