\(\int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx\) [759]

Optimal result

Integrand size = 22, antiderivative size = 271 \[ \int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {3 \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {9 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 b^2 c^2+14 a b c d-105 a^2 d^2-4 b d (b c-21 a d) x\right )}{32 b^4 d^2}+\frac {3 (b c-a d) \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{11/2} d^{5/2}} \]

[Out]

3/64*(-a*d+b*c)*(-105*a^3*d^3+35*a^2*b*c*d^2+5*a*b^2*c^2*d+b^3*c^3)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x
+c)^(1/2))/b^(11/2)/d^(5/2)-2*x^3*(d*x+c)^(3/2)/b/(b*x+a)^(1/2)+9/4*x^2*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b^2-1/32*(
d*x+c)^(3/2)*(3*b^2*c^2+14*a*b*c*d-105*a^2*d^2-4*b*d*(-21*a*d+b*c)*x)*(b*x+a)^(1/2)/b^4/d^2+3/64*(-105*a^3*d^3
+35*a^2*b*c*d^2+5*a*b^2*c^2*d+b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^5/d^2

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {99, 158, 152, 52, 65, 223, 212} \[ \int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-105 a^2 d^2-4 b d x (b c-21 a d)+14 a b c d+3 b^2 c^2\right )}{32 b^4 d^2}+\frac {3 (b c-a d) \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{11/2} d^{5/2}}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )}{64 b^5 d^2}+\frac {9 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b^2}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}} \]

[In]

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

[Out]

(3*(b^3*c^3 + 5*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 105*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^5*d^2) - (2*x^3
*(c + d*x)^(3/2))/(b*Sqrt[a + b*x]) + (9*x^2*Sqrt[a + b*x]*(c + d*x)^(3/2))/(4*b^2) - (Sqrt[a + b*x]*(c + d*x)
^(3/2)*(3*b^2*c^2 + 14*a*b*c*d - 105*a^2*d^2 - 4*b*d*(b*c - 21*a*d)*x))/(32*b^4*d^2) + (3*(b*c - a*d)*(b^3*c^3
+ 5*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 105*a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64
*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 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 158

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

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 {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {2 \int \frac {x^2 \sqrt {c+d x} \left (3 c+\frac {9 d x}{2}\right )}{\sqrt {a+b x}} \, dx}{b} \\ & = -\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {9 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b^2}+\frac {\int \frac {x \sqrt {c+d x} \left (-9 a c d+\frac {3}{4} d (b c-21 a d) x\right )}{\sqrt {a+b x}} \, dx}{2 b^2 d} \\ & = -\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {9 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 b^2 c^2+14 a b c d-105 a^2 d^2-4 b d (b c-21 a d) x\right )}{32 b^4 d^2}+\frac {\left (3 \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{64 b^4 d^2} \\ & = \frac {3 \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {9 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 b^2 c^2+14 a b c d-105 a^2 d^2-4 b d (b c-21 a d) x\right )}{32 b^4 d^2}+\frac {\left (3 (b c-a d) \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^5 d^2} \\ & = \frac {3 \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {9 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 b^2 c^2+14 a b c d-105 a^2 d^2-4 b d (b c-21 a d) x\right )}{32 b^4 d^2}+\frac {\left (3 (b c-a d) \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\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 )}{64 b^6 d^2} \\ & = \frac {3 \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {9 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 b^2 c^2+14 a b c d-105 a^2 d^2-4 b d (b c-21 a d) x\right )}{32 b^4 d^2}+\frac {\left (3 (b c-a d) \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\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 )}{64 b^6 d^2} \\ & = \frac {3 \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}+\frac {9 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b^2}-\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (3 b^2 c^2+14 a b c d-105 a^2 d^2-4 b d (b c-21 a d) x\right )}{32 b^4 d^2}+\frac {3 (b c-a d) \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{11/2} d^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.48 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (-\frac {\sqrt {d} \left (315 a^4 d^3+105 a^3 b d^2 (-3 c+d x)+a^2 b^2 d \left (13 c^2-119 c d x-42 d^2 x^2\right )-b^4 x \left (-3 c^3+2 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )+a b^3 \left (3 c^3+11 c^2 d x+44 c d^2 x^2+24 d^3 x^3\right )\right )}{\sqrt {a+b x}}+\frac {3 \sqrt {b c-a d} \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{64 b^5 d^{5/2}} \]

[In]

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

[Out]

(Sqrt[c + d*x]*(-((Sqrt[d]*(315*a^4*d^3 + 105*a^3*b*d^2*(-3*c + d*x) + a^2*b^2*d*(13*c^2 - 119*c*d*x - 42*d^2*
x^2) - b^4*x*(-3*c^3 + 2*c^2*d*x + 24*c*d^2*x^2 + 16*d^3*x^3) + a*b^3*(3*c^3 + 11*c^2*d*x + 44*c*d^2*x^2 + 24*
d^3*x^3)))/Sqrt[a + b*x]) + (3*Sqrt[b*c - a*d]*(b^3*c^3 + 5*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 105*a^3*d^3)*ArcSin
h[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(64*b^5*d^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(960\) vs. \(2(235)=470\).

Time = 1.67 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.55

method	result	size
default	\(\frac {\sqrt {d x +c}\, \left (32 b^{4} d^{3} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-48 a \,b^{3} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+48 b^{4} c \,d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+315 \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 \,d^{4} x -420 \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 \,d^{3} x +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^{2} b^{3} c^{2} d^{2} x +12 \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^{3} d x +3 \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^{4} x +84 a^{2} b^{2} d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-88 a \,b^{3} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+4 b^{4} c^{2} d \,x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+315 \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^{4}-420 \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^{3}+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^{2}+12 \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 +3 \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}-210 a^{3} b \,d^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+238 a^{2} b^{2} c \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-22 a \,b^{3} c^{2} d x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-6 b^{4} c^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-630 a^{4} d^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+630 a^{3} b c \,d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-26 a^{2} b^{2} c^{2} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-6 a \,b^{3} c^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{128 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {b x +a}\, b^{5} d^{2}}\)	\(961\)

[In]

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

[Out]

1/128*(d*x+c)^(1/2)*(32*b^4*d^3*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-48*a*b^3*d^3*x^3*((b*x+a)*(d*x+c))^(1/
2)*(b*d)^(1/2)+48*b^4*c*d^2*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+315*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*d^4*x-420*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*d^3*x+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^2*b^3*c^2*d^2*x+12*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^
3*d*x+3*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^4*x+84*a^2*b^2*d^3*x
^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-88*a*b^3*c*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+4*b^4*c^2*d*x^2*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+315*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^4-420*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^3+90*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^2+12*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+3*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^4-210*a^3*b*d^3*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+238*a^
2*b^2*c*d^2*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-22*a*b^3*c^2*d*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*b^4*c
^3*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-630*a^4*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+630*a^3*b*c*d^2*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-26*a^2*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*a*b^3*c^3*((b*x+a)*(d*x+
c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(b*x+a)^(1/2)/b^5/d^2

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.91 \[ \int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\left [\frac {3 \, {\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d + 30 \, a^{3} b^{2} c^{2} d^{2} - 140 \, a^{4} b c d^{3} + 105 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d + 30 \, a^{2} b^{3} c^{2} d^{2} - 140 \, a^{3} b^{2} c d^{3} + 105 \, a^{4} b d^{4}\right )} x\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 (16 \, b^{5} d^{4} x^{4} - 3 \, a b^{4} c^{3} d - 13 \, a^{2} b^{3} c^{2} d^{2} + 315 \, a^{3} b^{2} c d^{3} - 315 \, a^{4} b d^{4} + 24 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} + 2 \, {\left (b^{5} c^{2} d^{2} - 22 \, a b^{4} c d^{3} + 21 \, a^{2} b^{3} d^{4}\right )} x^{2} - {\left (3 \, b^{5} c^{3} d + 11 \, a b^{4} c^{2} d^{2} - 119 \, a^{2} b^{3} c d^{3} + 105 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{256 \, {\left (b^{7} d^{3} x + a b^{6} d^{3}\right )}}, -\frac {3 \, {\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d + 30 \, a^{3} b^{2} c^{2} d^{2} - 140 \, a^{4} b c d^{3} + 105 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d + 30 \, a^{2} b^{3} c^{2} d^{2} - 140 \, a^{3} b^{2} c d^{3} + 105 \, a^{4} b d^{4}\right )} x\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 (16 \, b^{5} d^{4} x^{4} - 3 \, a b^{4} c^{3} d - 13 \, a^{2} b^{3} c^{2} d^{2} + 315 \, a^{3} b^{2} c d^{3} - 315 \, a^{4} b d^{4} + 24 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} + 2 \, {\left (b^{5} c^{2} d^{2} - 22 \, a b^{4} c d^{3} + 21 \, a^{2} b^{3} d^{4}\right )} x^{2} - {\left (3 \, b^{5} c^{3} d + 11 \, a b^{4} c^{2} d^{2} - 119 \, a^{2} b^{3} c d^{3} + 105 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{128 \, {\left (b^{7} d^{3} x + a b^{6} d^{3}\right )}}\right ] \]

[In]

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

[Out]

[1/256*(3*(a*b^4*c^4 + 4*a^2*b^3*c^3*d + 30*a^3*b^2*c^2*d^2 - 140*a^4*b*c*d^3 + 105*a^5*d^4 + (b^5*c^4 + 4*a*b
^4*c^3*d + 30*a^2*b^3*c^2*d^2 - 140*a^3*b^2*c*d^3 + 105*a^4*b*d^4)*x)*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*(16*b^5*d^4*x^4 - 3*a*b^4*c^3*d - 13*a^2*b^3*c^2*d^2 + 315*a^3*b^2*c*d^3 - 315*a^4*b*d^4 + 24*(b^5*c*d^3
- a*b^4*d^4)*x^3 + 2*(b^5*c^2*d^2 - 22*a*b^4*c*d^3 + 21*a^2*b^3*d^4)*x^2 - (3*b^5*c^3*d + 11*a*b^4*c^2*d^2 - 1
19*a^2*b^3*c*d^3 + 105*a^3*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^7*d^3*x + a*b^6*d^3), -1/128*(3*(a*b^4*
c^4 + 4*a^2*b^3*c^3*d + 30*a^3*b^2*c^2*d^2 - 140*a^4*b*c*d^3 + 105*a^5*d^4 + (b^5*c^4 + 4*a*b^4*c^3*d + 30*a^2
*b^3*c^2*d^2 - 140*a^3*b^2*c*d^3 + 105*a^4*b*d^4)*x)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sq
rt(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*(16*b^5*d^4*x^4 - 3*a*b^4*c^3*d
- 13*a^2*b^3*c^2*d^2 + 315*a^3*b^2*c*d^3 - 315*a^4*b*d^4 + 24*(b^5*c*d^3 - a*b^4*d^4)*x^3 + 2*(b^5*c^2*d^2 -
22*a*b^4*c*d^3 + 21*a^2*b^3*d^4)*x^2 - (3*b^5*c^3*d + 11*a*b^4*c^2*d^2 - 119*a^2*b^3*c*d^3 + 105*a^3*b^2*d^4)*
x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^7*d^3*x + a*b^6*d^3)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^3*(d*x+c)^(3/2)/(b*x+a)^(3/2),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 [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.52 \[ \int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{64} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b x + a\right )} d {\left | b \right |}}{b^{7}} + \frac {3 \, b^{28} c d^{6} {\left | b \right |} - 11 \, a b^{27} d^{7} {\left | b \right |}}{b^{34} d^{6}}\right )} + \frac {b^{29} c^{2} d^{5} {\left | b \right |} - 58 \, a b^{28} c d^{6} {\left | b \right |} + 105 \, a^{2} b^{27} d^{7} {\left | b \right |}}{b^{34} d^{6}}\right )} - \frac {3 \, b^{30} c^{3} d^{4} {\left | b \right |} + 15 \, a b^{29} c^{2} d^{5} {\left | b \right |} - 279 \, a^{2} b^{28} c d^{6} {\left | b \right |} + 325 \, a^{3} b^{27} d^{7} {\left | b \right |}}{b^{34} d^{6}}\right )} \sqrt {b x + a} + \frac {4 \, {\left (a^{3} b^{2} c^{2} d {\left | b \right |} - 2 \, a^{4} b c d^{2} {\left | b \right |} + a^{5} d^{3} {\left | b \right |}\right )}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt {b d} b^{5}} - \frac {3 \, {\left (b^{4} c^{4} {\left | b \right |} + 4 \, a b^{3} c^{3} d {\left | b \right |} + 30 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - 140 \, a^{3} b c d^{3} {\left | b \right |} + 105 \, a^{4} d^{4} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{128 \, \sqrt {b d} b^{6} d^{2}} \]

[In]

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

[Out]

1/64*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(2*(b*x + a)*d*abs(b)/b^7 + (3*b^28*c*d^6*a
bs(b) - 11*a*b^27*d^7*abs(b))/(b^34*d^6)) + (b^29*c^2*d^5*abs(b) - 58*a*b^28*c*d^6*abs(b) + 105*a^2*b^27*d^7*a
bs(b))/(b^34*d^6)) - (3*b^30*c^3*d^4*abs(b) + 15*a*b^29*c^2*d^5*abs(b) - 279*a^2*b^28*c*d^6*abs(b) + 325*a^3*b
^27*d^7*abs(b))/(b^34*d^6))*sqrt(b*x + a) + 4*(a^3*b^2*c^2*d*abs(b) - 2*a^4*b*c*d^2*abs(b) + a^5*d^3*abs(b))/(
(b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)*sqrt(b*d)*b^5) - 3/128*(b^
4*c^4*abs(b) + 4*a*b^3*c^3*d*abs(b) + 30*a^2*b^2*c^2*d^2*abs(b) - 140*a^3*b*c*d^3*abs(b) + 105*a^4*d^4*abs(b))
*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b^6*d^2)

Mupad [F(-1)]

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

[In]

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

[Out]

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