\(\int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx\) [714]

Optimal result

Integrand size = 22, antiderivative size = 314 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^5 d^2}+\frac {(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac {3 (b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{7/2}}{5 b d}+\frac {(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{11/2} d^{5/2}} \]

[Out]

1/128*(-a*d+b*c)^3*(63*a^2*d^2+14*a*b*c*d+3*b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1
1/2)/d^(5/2)+1/192*(-a*d+b*c)*(63*a^2*d^2+14*a*b*c*d+3*b^2*c^2)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b^4/d^2+1/240*(63*
a^2*d^2+14*a*b*c*d+3*b^2*c^2)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/b^3/d^2-3/40*(3*a*d+b*c)*(d*x+c)^(7/2)*(b*x+a)^(1/2)
/b^2/d^2+1/5*x*(d*x+c)^(7/2)*(b*x+a)^(1/2)/b/d+1/128*(-a*d+b*c)^2*(63*a^2*d^2+14*a*b*c*d+3*b^2*c^2)*(b*x+a)^(1
/2)*(d*x+c)^(1/2)/b^5/d^2

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 314, 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} \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {(b c-a d)^3 \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{11/2} d^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2 \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{128 b^5 d^2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (b c-a d) \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{192 b^4 d^2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right )}{240 b^3 d^2}-\frac {3 \sqrt {a+b x} (c+d x)^{7/2} (3 a d+b c)}{40 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{7/2}}{5 b d} \]

[In]

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

[Out]

((b*c - a*d)^2*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*b^5*d^2) + ((b*c - a*d)
*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(192*b^4*d^2) + ((3*b^2*c^2 + 14*a*b*c*d
+ 63*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(240*b^3*d^2) - (3*(b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/
(40*b^2*d^2) + (x*Sqrt[a + b*x]*(c + d*x)^(7/2))/(5*b*d) + ((b*c - a*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2
)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(11/2)*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 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 \sqrt {a+b x} (c+d x)^{7/2}}{5 b d}+\frac {\int \frac {(c+d x)^{5/2} \left (-a c-\frac {3}{2} (b c+3 a d) x\right )}{\sqrt {a+b x}} \, dx}{5 b d} \\ & = -\frac {3 (b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{7/2}}{5 b d}+\frac {\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}} \, dx}{80 b^2 d^2} \\ & = \frac {\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac {3 (b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right )\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{96 b^3 d^2} \\ & = \frac {(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac {3 (b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{128 b^4 d^2} \\ & = \frac {(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^5 d^2}+\frac {(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac {3 (b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^5 d^2} \\ & = \frac {(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^5 d^2}+\frac {(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac {3 (b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 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^6 d^2} \\ & = \frac {(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^5 d^2}+\frac {(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac {3 (b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{7/2}}{5 b d}+\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 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^6 d^2} \\ & = \frac {(b c-a d)^2 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^5 d^2}+\frac {(b c-a d) \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{240 b^3 d^2}-\frac {3 (b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 b^2 d^2}+\frac {x \sqrt {a+b x} (c+d x)^{7/2}}{5 b d}+\frac {(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{11/2} d^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (945 a^4 d^4-210 a^3 b d^3 (11 c+3 d x)+2 a^2 b^2 d^2 \left (782 c^2+749 c d x+252 d^2 x^2\right )-2 a b^3 d \left (45 c^3+481 c^2 d x+592 c d^2 x^2+216 d^3 x^3\right )+b^4 \left (-45 c^4+30 c^3 d x+744 c^2 d^2 x^2+1008 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^5 d^2}+\frac {(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{11/2} d^{5/2}} \]

[In]

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(945*a^4*d^4 - 210*a^3*b*d^3*(11*c + 3*d*x) + 2*a^2*b^2*d^2*(782*c^2 + 749*c*d*x
+ 252*d^2*x^2) - 2*a*b^3*d*(45*c^3 + 481*c^2*d*x + 592*c*d^2*x^2 + 216*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^5*d^2) + ((b*c - a*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63
*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(11/2)*d^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(787\) vs. \(2(270)=540\).

Time = 0.58 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.51

method	result	size
default	\(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (-768 b^{4} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+864 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}-1008 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+2368 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}+945 \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}-2625 \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}+2250 \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}-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^{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}+1260 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b \,d^{4} x -2996 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x +1924 \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 -1890 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{4} d^{4}+4620 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b c \,d^{3}-3128 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}+180 \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 b^{5} d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}\)	\(788\)

[In]

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

[Out]

-1/3840*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(-768*b^4*d^4*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+864*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)-1008*a^2*b^2*d^4*x^2*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+2368*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)+945*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^5-2625*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
+2250*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^3-450*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+1260*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b*d^4*x-2996*((b*x+a)*(
d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^2*c*d^3*x+1924*((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-1890*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*d^4+4620*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)*a^3*b*c*d^3-3128*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^2*c^2*d^2+180*((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)/b^5/d^2/((b*x+a)*(d*x+c))^(1/2)
/(b*d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.25 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\left [-\frac {15 \, {\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d - 90 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 2310 \, a^{3} b^{2} c d^{4} + 945 \, a^{4} b d^{5} + 144 \, {\left (7 \, b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 63 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} - 481 \, a b^{4} c^{2} d^{3} + 749 \, a^{2} b^{3} c d^{4} - 315 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{6} d^{3}}, -\frac {15 \, {\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (384 \, b^{5} d^{5} x^{4} - 45 \, b^{5} c^{4} d - 90 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 2310 \, a^{3} b^{2} c d^{4} + 945 \, a^{4} b d^{5} + 144 \, {\left (7 \, b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (93 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 63 \, a^{2} b^{3} d^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{5} c^{3} d^{2} - 481 \, a b^{4} c^{2} d^{3} + 749 \, a^{2} b^{3} c d^{4} - 315 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{6} d^{3}}\right ] \]

[In]

integrate(x^2*(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 + 30*a^2*b^3*c^3*d^2 - 150*a^3*b^2*c^2*d^3 + 175*a^4*b*c*d^4 - 63*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 - 90*a*b^4*c^3*d^2 + 1564*a^
2*b^3*c^2*d^3 - 2310*a^3*b^2*c*d^4 + 945*a^4*b*d^5 + 144*(7*b^5*c*d^4 - 3*a*b^4*d^5)*x^3 + 8*(93*b^5*c^2*d^3 -
148*a*b^4*c*d^4 + 63*a^2*b^3*d^5)*x^2 + 2*(15*b^5*c^3*d^2 - 481*a*b^4*c^2*d^3 + 749*a^2*b^3*c*d^4 - 315*a^3*b
^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*d^3), -1/3840*(15*(3*b^5*c^5 + 5*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2
- 150*a^3*b^2*c^2*d^3 + 175*a^4*b*c*d^4 - 63*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 - 90*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2*d^3 - 2310*a^3*b^2*c*d^4 + 945*a^4*b*d^5 + 144*(7*b^5*c*d^4 - 3*a*b^4
*d^5)*x^3 + 8*(93*b^5*c^2*d^3 - 148*a*b^4*c*d^4 + 63*a^2*b^3*d^5)*x^2 + 2*(15*b^5*c^3*d^2 - 481*a*b^4*c^2*d^3
+ 749*a^2*b^3*c*d^4 - 315*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*d^3)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^2*(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. 885 vs. \(2 (270) = 540\).

Time = 0.40 (sec) , antiderivative size = 885, normalized size of antiderivative = 2.82 \[ \int \frac {x^2 (c+d x)^{5/2}}{\sqrt {a+b x}} \, dx=\frac {\frac {80 \, {\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^{2} {\left | b \right |}}{b^{2}} + \frac {20 \, {\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 )} 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 (4 \, {\left (b x + a\right )} {\left (6 \, {\left (b x + a\right )} {\left (\frac {8 \, {\left (b x + a\right )}}{b^{4}} + \frac {b^{20} c d^{7} - 41 \, a b^{19} d^{8}}{b^{23} d^{8}}\right )} - \frac {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}}\right )} + \frac {5 \, {\left (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}\right )}}{b^{23} d^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (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}\right )}}{b^{23} d^{8}}\right )} \sqrt {b x + a} - \frac {15 \, {\left (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}\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^{3} d^{4}}\right )} d^{2} {\left | b \right |}}{b^{2}}}{1920 \, b} \]

[In]

integrate(x^2*(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^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^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^2)/b

Mupad [F(-1)]

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

[In]

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

[Out]

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