\(\int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx\) [777]

Optimal result

Integrand size = 22, antiderivative size = 167 \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \left (c \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right )+d (b c-3 a d) (b c-a d) x\right )}{b^2 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {3 (b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 167, 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, 148, 65, 223, 212} \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} \left (c \left (3 a^2 d^2-2 a b c d+3 b^2 c^2\right )+d x (b c-3 a d) (b c-a d)\right )}{b^2 d^2 \sqrt {c+d x} (b c-a d)^2}-\frac {3 (a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac {2 a x^2}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)} \]

[In]

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

[Out]

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

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 148

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

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 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \int \frac {x \left (2 a c+\frac {1}{2} (-b c+3 a d) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{b (b c-a d)} \\ & = \frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \left (c \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right )+d (b c-3 a d) (b c-a d) x\right )}{b^2 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (b c+a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^2 d^2} \\ & = \frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \left (c \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right )+d (b c-3 a d) (b c-a d) x\right )}{b^2 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (b c+a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^3 d^2} \\ & = \frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \left (c \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right )+d (b c-3 a d) (b c-a d) x\right )}{b^2 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {(3 (b c+a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^3 d^2} \\ & = \frac {2 a x^2}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \left (c \left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right )+d (b c-3 a d) (b c-a d) x\right )}{b^2 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {3 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.98 \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {3 a^3 d^2 (c+d x)+b^3 c^2 x (3 c+d x)+a b^2 c \left (3 c^2-c d x-2 d^2 x^2\right )+a^2 b d \left (-2 c^2-c d x+d^2 x^2\right )}{b^2 d^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}-\frac {3 (b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{5/2} d^{5/2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(905\) vs. \(2(147)=294\).

Time = 2.23 (sec) , antiderivative size = 906, normalized size of antiderivative = 5.43

method	result	size
default	\(-\frac {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} b \,d^{4} 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^{2} b^{2} c \,d^{3} 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 \,b^{3} 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 ) b^{4} c^{3} d \,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^{4} d^{4} x -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^{2} b^{2} c^{2} d^{2} 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^{4} c^{4} x -2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x^{2}+4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x^{2}-2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{3} c^{2} d \,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^{4} 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^{3} b \,c^{2} d^{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^{2} b^{2} 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^{3} c^{4}-6 a^{3} d^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2} x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d x -6 b^{3} c^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-6 a^{3} c \,d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b \,c^{2} d -6 a \,b^{2} c^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \left (a d -b c \right )^{2} \sqrt {b d}\, \sqrt {b x +a}\, \sqrt {d x +c}\, b^{2} d^{2}}\)	\(906\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (147) = 294\).

Time = 0.32 (sec) , antiderivative size = 910, normalized size of antiderivative = 5.45 \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\left [\frac {3 \, {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3} + {\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} 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 (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + {\left (3 \, b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + 3 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (a b^{5} c^{3} d^{3} - 2 \, a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + {\left (b^{6} c^{2} d^{4} - 2 \, a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{2} + {\left (b^{6} c^{3} d^{3} - a b^{5} c^{2} d^{4} - a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x\right )}}, \frac {3 \, {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3} + {\left (b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} 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 (3 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + {\left (3 \, b^{4} c^{3} d - a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} + 3 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b^{5} c^{3} d^{3} - 2 \, a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + {\left (b^{6} c^{2} d^{4} - 2 \, a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{2} + {\left (b^{6} c^{3} d^{3} - a b^{5} c^{2} d^{4} - a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (147) = 294\).

Time = 0.39 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.04 \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {4 \, a^{3} d}{{\left (\sqrt {b d} b c {\left | b \right |} - \sqrt {b d} a d {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} + \frac {\sqrt {b x + a} {\left (\frac {{\left (b^{6} c^{2} d^{2} - 2 \, a b^{5} c d^{3} + a^{2} b^{4} d^{4}\right )} {\left (b x + a\right )}}{b^{7} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{6} c d^{4} {\left | b \right |} + a^{2} b^{5} d^{5} {\left | b \right |}} + \frac {3 \, b^{7} c^{3} d - 3 \, a b^{6} c^{2} d^{2} + 3 \, a^{2} b^{5} c d^{3} - a^{3} b^{4} d^{4}}{b^{7} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{6} c d^{4} {\left | b \right |} + a^{2} b^{5} d^{5} {\left | b \right |}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {3 \, {\left (b c + a d\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{2 \, \sqrt {b d} b d^{2} {\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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