\(\int \frac {x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx\) [741]

Optimal result

Integrand size = 22, antiderivative size = 175 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=-\frac {2 c x^2 \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} ((5 b c-3 a d) (3 b c+a d)-2 b d (5 b c-a d) x)}{4 b^2 d^3 (b c-a d)}+\frac {3 \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {100, 152, 65, 223, 212} \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {3 \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} ((5 b c-3 a d) (a d+3 b c)-2 b d x (5 b c-a d))}{4 b^2 d^3 (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x}}{d \sqrt {c+d x} (b c-a d)} \]

[In]

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

[Out]

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

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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 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 c x^2 \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}+\frac {2 \int \frac {x \left (2 a c+\frac {1}{2} (5 b c-a d) x\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{d (b c-a d)} \\ & = -\frac {2 c x^2 \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} ((5 b c-3 a d) (3 b c+a d)-2 b d (5 b c-a d) x)}{4 b^2 d^3 (b c-a d)}+\frac {\left (3 \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^2 d^3} \\ & = -\frac {2 c x^2 \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} ((5 b c-3 a d) (3 b c+a d)-2 b d (5 b c-a d) x)}{4 b^2 d^3 (b c-a d)}+\frac {\left (3 \left (5 b^2 c^2+2 a b c d+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 )}{4 b^3 d^3} \\ & = -\frac {2 c x^2 \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} ((5 b c-3 a d) (3 b c+a d)-2 b d (5 b c-a d) x)}{4 b^2 d^3 (b c-a d)}+\frac {\left (3 \left (5 b^2 c^2+2 a b c d+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 )}{4 b^3 d^3} \\ & = -\frac {2 c x^2 \sqrt {a+b x}}{d (b c-a d) \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} ((5 b c-3 a d) (3 b c+a d)-2 b d (5 b c-a d) x)}{4 b^2 d^3 (b c-a d)}+\frac {3 \left (5 b^2 c^2+2 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^{5/2} d^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.11 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {-\sqrt {b} \sqrt {d} \sqrt {a+b x} \left (3 a^2 d^2 (c+d x)+2 a b d \left (2 c^2+c d x-d^2 x^2\right )+b^2 c \left (-15 c^2-5 c d x+2 d^2 x^2\right )\right )-3 \left (5 b^3 c^3-3 a b^2 c^2 d-a^2 b c d^2-a^3 d^3\right ) \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{7/2} (-b c+a d) \sqrt {c+d x}} \]

[In]

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

[Out]

(-(Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*(3*a^2*d^2*(c + d*x) + 2*a*b*d*(2*c^2 + c*d*x - d^2*x^2) + b^2*c*(-15*c^2 - 5
*c*d*x + 2*d^2*x^2))) - 3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - a^3*d^3)*Sqrt[c + d*x]*ArcTanh[(Sqrt[d]*S
qrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(5/2)*d^(7/2)*(-(b*c) + a*d)*Sqrt[c + d*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(672\) vs. \(2(151)=302\).

Time = 1.67 (sec) , antiderivative size = 673, normalized size of antiderivative = 3.85

method	result	size
default	\(\frac {\sqrt {b x +a}\, \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^{4} 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 ) a^{2} b c \,d^{3} x +9 \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^{2} x -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^{3} c^{3} d x +4 a b \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-4 b^{2} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b 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^{3} c \,d^{3}+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^{2} b \,c^{2} d^{2}+9 \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 -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^{3} c^{4}-6 a^{2} d^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-4 a b c \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+10 b^{2} c^{2} d x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-6 a^{2} c \,d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-8 a b \,c^{2} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+30 b^{2} c^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{8 \left (a d -b c \right ) \sqrt {b d}\, b^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{3} \sqrt {d x +c}}\)	\(673\)

[In]

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

[Out]

1/8*(b*x+a)^(1/2)*(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*d^4*x+3*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^2*b*c*d^3*x+9*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^2*x-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^3*c^3*d*x+4*a*b*d^3*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-4*b^2*
c*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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*d^3+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^2*b*c^
2*d^2+9*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-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^3*c^4-6*a^2*d^3*x*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)-4*a*b*c*d^2*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+10*b^2*c^2*d*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)-6*a^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-8*a*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*b^2*c^3*(
(b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(a*d-b*c)/(b*d)^(1/2)/b^2/((b*x+a)*(d*x+c))^(1/2)/d^3/(d*x+c)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (152) = 304\).

Time = 0.33 (sec) , antiderivative size = 630, normalized size of antiderivative = 3.60 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\left [\frac {3 \, {\left (5 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - a^{2} b c^{2} d^{2} - a^{3} c d^{3} + {\left (5 \, b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (15 \, b^{3} c^{3} d - 4 \, a b^{2} c^{2} d^{2} - 3 \, a^{2} b c d^{3} - 2 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{2} + {\left (5 \, b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} - 3 \, a^{2} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (b^{4} c^{2} d^{4} - a b^{3} c d^{5} + {\left (b^{4} c d^{5} - a b^{3} d^{6}\right )} x\right )}}, -\frac {3 \, {\left (5 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d - a^{2} b c^{2} d^{2} - a^{3} c d^{3} + {\left (5 \, b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (15 \, b^{3} c^{3} d - 4 \, a b^{2} c^{2} d^{2} - 3 \, a^{2} b c d^{3} - 2 \, {\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{2} + {\left (5 \, b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} - 3 \, a^{2} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (b^{4} c^{2} d^{4} - a b^{3} c d^{5} + {\left (b^{4} c d^{5} - a b^{3} d^{6}\right )} x\right )}}\right ] \]

[In]

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

[Out]

[1/16*(3*(5*b^3*c^4 - 3*a*b^2*c^3*d - a^2*b*c^2*d^2 - a^3*c*d^3 + (5*b^3*c^3*d - 3*a*b^2*c^2*d^2 - a^2*b*c*d^3
- a^3*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*(15*b^3*c^3*d - 4*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 -
2*(b^3*c*d^3 - a*b^2*d^4)*x^2 + (5*b^3*c^2*d^2 - 2*a*b^2*c*d^3 - 3*a^2*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/
(b^4*c^2*d^4 - a*b^3*c*d^5 + (b^4*c*d^5 - a*b^3*d^6)*x), -1/8*(3*(5*b^3*c^4 - 3*a*b^2*c^3*d - a^2*b*c^2*d^2 -
a^3*c*d^3 + (5*b^3*c^3*d - 3*a*b^2*c^2*d^2 - a^2*b*c*d^3 - a^3*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*(15*b^3*c^3*d
- 4*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 - 2*(b^3*c*d^3 - a*b^2*d^4)*x^2 + (5*b^3*c^2*d^2 - 2*a*b^2*c*d^3 - 3*a^2*b*
d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*c^2*d^4 - a*b^3*c*d^5 + (b^4*c*d^5 - a*b^3*d^6)*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^3/(d*x+c)^(3/2)/(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. 305 vs. \(2 (152) = 304\).

Time = 0.34 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.74 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{6} c d^{4} {\left | b \right |} - a b^{5} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c d^{5} - a b^{8} d^{6}} - \frac {5 \, b^{7} c^{2} d^{3} {\left | b \right |} + 2 \, a b^{6} c d^{4} {\left | b \right |} - 7 \, a^{2} b^{5} d^{5} {\left | b \right |}}{b^{9} c d^{5} - a b^{8} d^{6}}\right )} - \frac {15 \, b^{8} c^{3} d^{2} {\left | b \right |} - 9 \, a b^{7} c^{2} d^{3} {\left | b \right |} - 3 \, a^{2} b^{6} c d^{4} {\left | b \right |} + 5 \, a^{3} b^{5} d^{5} {\left | b \right |}}{b^{9} c d^{5} - a b^{8} d^{6}}\right )} \sqrt {b x + a}}{4 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {3 \, {\left (5 \, b^{2} c^{2} {\left | b \right |} + 2 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt {b d} b^{3} d^{3}} \]

[In]

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

[Out]

1/4*((b*x + a)*(2*(b^6*c*d^4*abs(b) - a*b^5*d^5*abs(b))*(b*x + a)/(b^9*c*d^5 - a*b^8*d^6) - (5*b^7*c^2*d^3*abs
(b) + 2*a*b^6*c*d^4*abs(b) - 7*a^2*b^5*d^5*abs(b))/(b^9*c*d^5 - a*b^8*d^6)) - (15*b^8*c^3*d^2*abs(b) - 9*a*b^7
*c^2*d^3*abs(b) - 3*a^2*b^6*c*d^4*abs(b) + 5*a^3*b^5*d^5*abs(b))/(b^9*c*d^5 - a*b^8*d^6))*sqrt(b*x + a)/sqrt(b
^2*c + (b*x + a)*b*d - a*b*d) - 3/4*(5*b^2*c^2*abs(b) + 2*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^3*d^3)

Mupad [F(-1)]

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

[In]

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

[Out]

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