\(\int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx\) [715]

Optimal result

Integrand size = 20, antiderivative size = 217 \[ \int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=-\frac {5 (b c-a d)^2 (b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d}-\frac {5 (b c-a d) (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac {(b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b d}-\frac {5 (b c-a d)^3 (b c+7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{3/2}} \]

[Out]

-5/64*(-a*d+b*c)^3*(7*a*d+b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(9/2)/d^(3/2)-5/96*(-a*d
+b*c)*(7*a*d+b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b^3/d-1/24*(7*a*d+b*c)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/b^2/d+1/4*(d*
x+c)^(7/2)*(b*x+a)^(1/2)/b/d-5/64*(-a*d+b*c)^2*(7*a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223, 212} \[ \int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=-\frac {5 (b c-a d)^3 (7 a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{3/2}}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2 (7 a d+b c)}{64 b^4 d}-\frac {5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d) (7 a d+b c)}{96 b^3 d}-\frac {\sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{24 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b d} \]

[In]

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

[Out]

(-5*(b*c - a*d)^2*(b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^4*d) - (5*(b*c - a*d)*(b*c + 7*a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))/(96*b^3*d) - ((b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*b^2*d) + (Sqrt[a + b*x]
*(c + d*x)^(7/2))/(4*b*d) - (5*(b*c - a*d)^3*(b*c + 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d
*x])])/(64*b^(9/2)*d^(3/2))

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 {\sqrt {a+b x} (c+d x)^{7/2}}{4 b d}-\frac {(b c+7 a d) \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}} \, dx}{8 b d} \\ & = -\frac {(b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b d}-\frac {(5 (b c-a d) (b c+7 a d)) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{48 b^2 d} \\ & = -\frac {5 (b c-a d) (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac {(b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b d}-\frac {\left (5 (b c-a d)^2 (b c+7 a d)\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{64 b^3 d} \\ & = -\frac {5 (b c-a d)^2 (b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d}-\frac {5 (b c-a d) (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac {(b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b d}-\frac {\left (5 (b c-a d)^3 (b c+7 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^4 d} \\ & = -\frac {5 (b c-a d)^2 (b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d}-\frac {5 (b c-a d) (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac {(b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b d}-\frac {\left (5 (b c-a d)^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 )}{64 b^5 d} \\ & = -\frac {5 (b c-a d)^2 (b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d}-\frac {5 (b c-a d) (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac {(b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b d}-\frac {\left (5 (b c-a d)^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 )}{64 b^5 d} \\ & = -\frac {5 (b c-a d)^2 (b c+7 a d) \sqrt {a+b x} \sqrt {c+d x}}{64 b^4 d}-\frac {5 (b c-a d) (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^3 d}-\frac {(b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b d}-\frac {5 (b c-a d)^3 (b c+7 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82 \[ \int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^3 d^3+5 a^2 b d^2 (53 c+14 d x)-a b^2 d \left (191 c^2+172 c d x+56 d^2 x^2\right )+b^3 \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^4 d}-\frac {5 (b c-a d)^3 (b c+7 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{9/2} d^{3/2}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^3*d^3 + 5*a^2*b*d^2*(53*c + 14*d*x) - a*b^2*d*(191*c^2 + 172*c*d*x + 56*d
^2*x^2) + b^3*(15*c^3 + 118*c^2*d*x + 136*c*d^2*x^2 + 48*d^3*x^3)))/(192*b^4*d) - (5*(b*c - a*d)^3*(b*c + 7*a*
d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(9/2)*d^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(573\) vs. \(2(179)=358\).

Time = 0.57 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.65

method	result	size
default	\(\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (96 b^{3} d^{3} x^{3} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-112 a \,b^{2} d^{3} x^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+272 b^{3} c \,d^{2} x^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+105 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{4}-300 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c \,d^{3}+270 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2} d^{2}-60 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{3} d -15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{4}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x -344 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x +236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{2} d x -210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} d^{3}+530 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2}-382 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d +30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{3}\right )}{384 b^{4} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}\)	\(574\)

[In]

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

[Out]

1/384*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(96*b^3*d^3*x^3*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-112*a*b^2*d^3*x^2*(b*d)^
(1/2)*((b*x+a)*(d*x+c))^(1/2)+272*b^3*c*d^2*x^2*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(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^4*d^4-300*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*c*d^3+270*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^2*c^2*d^2-60*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^3*c^3*d-15*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^4*c^4+140*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b*d^3*x-344*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^2*c*d^2*x+236*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)*b^3*c^2*d*x-210*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*d^3+530*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)*a^2*b*c*d^2-382*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^2*c^2*d+30*((b*x+a)*(d*x+c))^(1/2)*(b*d)
^(1/2)*b^3*c^3)/b^4/d/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.51 \[ \int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\left [-\frac {15 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 191 \, a b^{3} c^{2} d^{2} + 265 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{5} d^{2}}, \frac {15 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d - 191 \, a b^{3} c^{2} d^{2} + 265 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{5} d^{2}}\right ] \]

[In]

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

[Out]

[-1/768*(15*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*a^4*d^4)*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*(48*b^4*d^4*x^3 + 15*b^4*c^3*d - 191*a*b^3*c^2*d^2 + 265*a^2*b^2*c*d^3 - 105*a^3*b*d^4
+ 8*(17*b^4*c*d^3 - 7*a*b^3*d^4)*x^2 + 2*(59*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x)*sqrt(b*x + a)*s
qrt(d*x + c))/(b^5*d^2), 1/384*(15*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*a^4*d^4)
*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*(48*b^4*d^4*x^3 + 15*b^4*c^3*d - 191*a*b^3*c^2*d^2 + 265*a^2*b^2*c*d^3 - 105*a^3*b*
d^4 + 8*(17*b^4*c*d^3 - 7*a*b^3*d^4)*x^2 + 2*(59*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x)*sqrt(b*x +
a)*sqrt(d*x + c))/(b^5*d^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[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. 639 vs. \(2 (179) = 358\).

Time = 0.36 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.94 \[ \int \frac {x (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {\frac {16 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} c d {\left | b \right |}}{b^{2}} + \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {b^{12} c d^{5} - 25 \, a b^{11} d^{6}}{b^{14} d^{6}}\right )} - \frac {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}}\right )} + \frac {3 \, {\left (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}\right )}}{b^{14} d^{6}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (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}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} d^{2} {\left | b \right |}}{b^{2}} + \frac {48 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} c^{2} {\left | b \right |}}{b^{3}}}{192 \, b} \]

[In]

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

[Out]

1/192*(16*(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*d*abs(b)/b^2 + (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))*d^2*abs(b)/b^2 + 48*(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(-sqr
t(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d))*c^2*abs(b)/b^3)/b

Mupad [F(-1)]

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

[In]

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

[Out]

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