\(\int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx\) [591]

Optimal result

Integrand size = 22, antiderivative size = 218 \[ \int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^2 c^2-100 a b c d+3 a^2 d^2-2 b d (35 b c-31 a d) x\right )}{12 b d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{9/2}} \]

[Out]

1/4*(-a^2*d^2-10*a*b*c*d+35*b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(3/2)/d^(9/2)-2/3*
x^3*(b*x+a)^(1/2)/d/(d*x+c)^(3/2)-2/3*(-6*a*d+7*b*c)*x^2*(b*x+a)^(1/2)/d^2/(-a*d+b*c)/(d*x+c)^(1/2)-1/12*(105*
b^2*c^2-100*a*b*c*d+3*a^2*d^2-2*b*d*(-31*a*d+35*b*c)*x)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b/d^4/(-a*d+b*c)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {99, 155, 152, 65, 223, 212} \[ \int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {\left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2-2 b d x (35 b c-31 a d)-100 a b c d+105 b^2 c^2\right )}{12 b d^4 (b c-a d)}-\frac {2 x^2 \sqrt {a+b x} (7 b c-6 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)}-\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}} \]

[In]

Int[(x^3*Sqrt[a + b*x])/(c + d*x)^(5/2),x]

[Out]

(-2*x^3*Sqrt[a + b*x])/(3*d*(c + d*x)^(3/2)) - (2*(7*b*c - 6*a*d)*x^2*Sqrt[a + b*x])/(3*d^2*(b*c - a*d)*Sqrt[c
+ d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*(105*b^2*c^2 - 100*a*b*c*d + 3*a^2*d^2 - 2*b*d*(35*b*c - 31*a*d)*x))/(
12*b*d^4*(b*c - a*d)) + ((35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(4*b^(3/2)*d^(9/2))

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 155

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

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 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}+\frac {2 \int \frac {x^2 \left (3 a+\frac {7 b x}{2}\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 d} \\ & = -\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {4 \int \frac {x \left (-a (7 b c-6 a d)-\frac {1}{4} b (35 b c-31 a d) x\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 d^2 (b c-a d)} \\ & = -\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^2 c^2-100 a b c d+3 a^2 d^2-2 b d (35 b c-31 a d) x\right )}{12 b d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b d^4} \\ & = -\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^2 c^2-100 a b c d+3 a^2 d^2-2 b d (35 b c-31 a d) x\right )}{12 b d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\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 )}{4 b^2 d^4} \\ & = -\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^2 c^2-100 a b c d+3 a^2 d^2-2 b d (35 b c-31 a d) x\right )}{12 b d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\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 )}{4 b^2 d^4} \\ & = -\frac {2 x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2}}-\frac {2 (7 b c-6 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^2 c^2-100 a b c d+3 a^2 d^2-2 b d (35 b c-31 a d) x\right )}{12 b d^4 (b c-a d)}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (3 a^2 d^2 (c+d x)^2+b^2 c \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )-2 a b d \left (50 c^3+69 c^2 d x+12 c d^2 x^2-3 d^3 x^3\right )\right )}{12 b d^4 (-b c+a d) (c+d x)^{3/2}}+\frac {\left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{9/2}} \]

[In]

Integrate[(x^3*Sqrt[a + b*x])/(c + d*x)^(5/2),x]

[Out]

(Sqrt[a + b*x]*(3*a^2*d^2*(c + d*x)^2 + b^2*c*(105*c^3 + 140*c^2*d*x + 21*c*d^2*x^2 - 6*d^3*x^3) - 2*a*b*d*(50
*c^3 + 69*c^2*d*x + 12*c*d^2*x^2 - 3*d^3*x^3)))/(12*b*d^4*(-(b*c) + a*d)*(c + d*x)^(3/2)) + ((35*b^2*c^2 - 10*
a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(3/2)*d^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(985\) vs. \(2(186)=372\).

Time = 1.52 (sec) , antiderivative size = 986, normalized size of antiderivative = 4.52

method	result	size
default	\(-\frac {\left (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^{3} d^{5} x^{2}+27 \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 c \,d^{4} x^{2}-135 \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^{2} c^{2} d^{3} x^{2}+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 ) b^{3} c^{3} d^{2} x^{2}-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a b \,d^{4} x^{3}+12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} c \,d^{3} x^{3}+6 \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} c \,d^{4} x +54 \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 \,c^{2} d^{3} x -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 \,b^{2} c^{3} d^{2} x +210 \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^{3} c^{4} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d^{4} x^{2}+48 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{3} x^{2}-42 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d^{2} x^{2}+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^{3} c^{2} d^{3}+27 \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 \,c^{3} d^{2}-135 \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^{2} c^{4} 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 ) b^{3} c^{5}-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c \,d^{3} x +276 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d^{2} x -280 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c^{2} d^{2}+200 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{3} d -210 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{4}\right ) \sqrt {b x +a}}{24 \left (a d -b c \right ) \sqrt {b d}\, b \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{4} \left (d x +c \right )^{\frac {3}{2}}}\)	\(986\)

[In]

int(x^3*(b*x+a)^(1/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*(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^3*d^5*x^2+27*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*c*d^4*x^2-135*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^2*c^2*d^3*x^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))*b^3*c^3*d^2*x^2-12*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b*d^4*x^3+12*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^2*c*d^3*x^3+6*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*c*d^4*x+54*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*c^2*d^3*x-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*b^2*c^3*d^2*x
+210*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^3*c^4*d*x-6*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)*a^2*d^4*x^2+48*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*d^3*x^2-42*(b*d)^(1/2)*((b*x+a)*
(d*x+c))^(1/2)*b^2*c^2*d^2*x^2+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
^3*c^2*d^3+27*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*c^3*d^2-135*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^2*c^4*d+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))*b^3*c^5-12*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*c*d
^3*x+276*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^2*d^2*x-280*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^3*d*x
-6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*c^2*d^2+200*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^3*d-210*(b*d)
^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^4)*(b*x+a)^(1/2)/(a*d-b*c)/(b*d)^(1/2)/b/((b*x+a)*(d*x+c))^(1/2)/d^4/(d*x
+c)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (186) = 372\).

Time = 0.45 (sec) , antiderivative size = 866, normalized size of antiderivative = 3.97 \[ \int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (35 \, b^{3} c^{5} - 45 \, a b^{2} c^{4} d + 9 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (35 \, b^{3} c^{3} d^{2} - 45 \, a b^{2} c^{2} d^{3} + 9 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{4} d - 45 \, a b^{2} c^{3} d^{2} + 9 \, a^{2} b c^{2} d^{3} + a^{3} c 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 (105 \, b^{3} c^{4} d - 100 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - 6 \, {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x^{3} + 3 \, {\left (7 \, b^{3} c^{2} d^{3} - 8 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (70 \, b^{3} c^{3} d^{2} - 69 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{3} c^{3} d^{5} - a b^{2} c^{2} d^{6} + {\left (b^{3} c d^{7} - a b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{3} c^{2} d^{6} - a b^{2} c d^{7}\right )} x\right )}}, -\frac {3 \, {\left (35 \, b^{3} c^{5} - 45 \, a b^{2} c^{4} d + 9 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (35 \, b^{3} c^{3} d^{2} - 45 \, a b^{2} c^{2} d^{3} + 9 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{4} d - 45 \, a b^{2} c^{3} d^{2} + 9 \, a^{2} b c^{2} d^{3} + a^{3} c 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 (105 \, b^{3} c^{4} d - 100 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - 6 \, {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x^{3} + 3 \, {\left (7 \, b^{3} c^{2} d^{3} - 8 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (70 \, b^{3} c^{3} d^{2} - 69 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (b^{3} c^{3} d^{5} - a b^{2} c^{2} d^{6} + {\left (b^{3} c d^{7} - a b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{3} c^{2} d^{6} - a b^{2} c d^{7}\right )} x\right )}}\right ] \]

[In]

integrate(x^3*(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[-1/48*(3*(35*b^3*c^5 - 45*a*b^2*c^4*d + 9*a^2*b*c^3*d^2 + a^3*c^2*d^3 + (35*b^3*c^3*d^2 - 45*a*b^2*c^2*d^3 +
9*a^2*b*c*d^4 + a^3*d^5)*x^2 + 2*(35*b^3*c^4*d - 45*a*b^2*c^3*d^2 + 9*a^2*b*c^2*d^3 + a^3*c*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*(105*b^3*c^4*d - 100*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - 6*(b^3*c*d^4 - a*b^2
*d^5)*x^3 + 3*(7*b^3*c^2*d^3 - 8*a*b^2*c*d^4 + a^2*b*d^5)*x^2 + 2*(70*b^3*c^3*d^2 - 69*a*b^2*c^2*d^3 + 3*a^2*b
*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*c^3*d^5 - a*b^2*c^2*d^6 + (b^3*c*d^7 - a*b^2*d^8)*x^2 + 2*(b^3*c^
2*d^6 - a*b^2*c*d^7)*x), -1/24*(3*(35*b^3*c^5 - 45*a*b^2*c^4*d + 9*a^2*b*c^3*d^2 + a^3*c^2*d^3 + (35*b^3*c^3*d
^2 - 45*a*b^2*c^2*d^3 + 9*a^2*b*c*d^4 + a^3*d^5)*x^2 + 2*(35*b^3*c^4*d - 45*a*b^2*c^3*d^2 + 9*a^2*b*c^2*d^3 +
a^3*c*d^4)*x)*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*(105*b^3*c^4*d - 100*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - 6*(b^3*c*d^4 -
a*b^2*d^5)*x^3 + 3*(7*b^3*c^2*d^3 - 8*a*b^2*c*d^4 + a^2*b*d^5)*x^2 + 2*(70*b^3*c^3*d^2 - 69*a*b^2*c^2*d^3 + 3*
a^2*b*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*c^3*d^5 - a*b^2*c^2*d^6 + (b^3*c*d^7 - a*b^2*d^8)*x^2 + 2*(b
^3*c^2*d^6 - a*b^2*c*d^7)*x)]

Sympy [F]

\[ \int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\int \frac {x^{3} \sqrt {a + b x}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**3*(b*x+a)**(1/2)/(d*x+c)**(5/2),x)

[Out]

Integral(x**3*sqrt(a + b*x)/(c + d*x)**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^3*(b*x+a)^(1/2)/(d*x+c)^(5/2),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. 407 vs. \(2 (186) = 372\).

Time = 0.37 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.87 \[ \int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{5} c d^{6} {\left | b \right |} - a b^{4} d^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{6} c d^{7} - a b^{5} d^{8}} - \frac {7 \, b^{6} c^{2} d^{5} {\left | b \right |} - 2 \, a b^{5} c d^{6} {\left | b \right |} - 5 \, a^{2} b^{4} d^{7} {\left | b \right |}}{b^{6} c d^{7} - a b^{5} d^{8}}\right )} - \frac {4 \, {\left (35 \, b^{7} c^{3} d^{4} {\left | b \right |} - 45 \, a b^{6} c^{2} d^{5} {\left | b \right |} + 9 \, a^{2} b^{5} c d^{6} {\left | b \right |} + 3 \, a^{3} b^{4} d^{7} {\left | b \right |}\right )}}{b^{6} c d^{7} - a b^{5} d^{8}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (35 \, b^{8} c^{4} d^{3} {\left | b \right |} - 80 \, a b^{7} c^{3} d^{4} {\left | b \right |} + 54 \, a^{2} b^{6} c^{2} d^{5} {\left | b \right |} - 8 \, a^{3} b^{5} c d^{6} {\left | b \right |} - a^{4} b^{4} d^{7} {\left | b \right |}\right )}}{b^{6} c d^{7} - a b^{5} d^{8}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {{\left (35 \, b^{2} c^{2} {\left | b \right |} - 10 \, a b c d {\left | b \right |} - a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt {b d} b^{2} d^{4}} \]

[In]

integrate(x^3*(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

1/12*((3*(b*x + a)*(2*(b^5*c*d^6*abs(b) - a*b^4*d^7*abs(b))*(b*x + a)/(b^6*c*d^7 - a*b^5*d^8) - (7*b^6*c^2*d^5
*abs(b) - 2*a*b^5*c*d^6*abs(b) - 5*a^2*b^4*d^7*abs(b))/(b^6*c*d^7 - a*b^5*d^8)) - 4*(35*b^7*c^3*d^4*abs(b) - 4
5*a*b^6*c^2*d^5*abs(b) + 9*a^2*b^5*c*d^6*abs(b) + 3*a^3*b^4*d^7*abs(b))/(b^6*c*d^7 - a*b^5*d^8))*(b*x + a) - 3
*(35*b^8*c^4*d^3*abs(b) - 80*a*b^7*c^3*d^4*abs(b) + 54*a^2*b^6*c^2*d^5*abs(b) - 8*a^3*b^5*c*d^6*abs(b) - a^4*b
^4*d^7*abs(b))/(b^6*c*d^7 - a*b^5*d^8))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 1/4*(35*b^2*c^2*
abs(b) - 10*a*b*c*d*abs(b) - a^2*d^2*abs(b))*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^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sqrt {a+b x}}{(c+d x)^{5/2}} \, dx=\int \frac {x^3\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]

[In]

int((x^3*(a + b*x)^(1/2))/(c + d*x)^(5/2),x)

[Out]

int((x^3*(a + b*x)^(1/2))/(c + d*x)^(5/2), x)