\(\int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx\) [776]

Optimal result

Integrand size = 22, antiderivative size = 250 \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 c (b c+a d) x^2 \sqrt {a+b x}}{b d (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right )-2 b d \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) x\right )}{4 b^3 d^3 (b c-a d)^2}+\frac {3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{7/2}} \]

[Out]

3/4*(5*a^2*d^2+6*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^(7/2)/d^(7/2)+2*a*x
^3/b/(-a*d+b*c)/(b*x+a)^(1/2)/(d*x+c)^(1/2)-2*c*(a*d+b*c)*x^2*(b*x+a)^(1/2)/b/d/(-a*d+b*c)^2/(d*x+c)^(1/2)-1/4
*((a*d+b*c)*(15*a^2*d^2-22*a*b*c*d+15*b^2*c^2)-2*b*d*(5*a^2*d^2-2*a*b*c*d+5*b^2*c^2)*x)*(b*x+a)^(1/2)*(d*x+c)^
(1/2)/b^3/d^3/(-a*d+b*c)^2

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 250, 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 = {100, 155, 152, 65, 223, 212} \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )-2 b d x \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )\right )}{4 b^3 d^3 (b c-a d)^2}+\frac {2 a x^3}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (a d+b c)}{b d \sqrt {c+d x} (b c-a d)^2} \]

[In]

Int[x^4/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

(2*a*x^3)/(b*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) - (2*c*(b*c + a*d)*x^2*Sqrt[a + b*x])/(b*d*(b*c - a*d)^2
*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*((b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d + 15*a^2*d^2) - 2*b*d*(5*
b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x))/(4*b^3*d^3*(b*c - a*d)^2) + (3*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTan
h[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(7/2)*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 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 a x^3}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 \int \frac {x^2 \left (3 a c+\frac {1}{2} (-b c+5 a d) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{b (b c-a d)} \\ & = \frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 c (b c+a d) x^2 \sqrt {a+b x}}{b d (b c-a d)^2 \sqrt {c+d x}}+\frac {4 \int \frac {x \left (a c (b c+a d)+\frac {1}{4} \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b d (b c-a d)^2} \\ & = \frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 c (b c+a d) x^2 \sqrt {a+b x}}{b d (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right )-2 b d \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) x\right )}{4 b^3 d^3 (b c-a d)^2}+\frac {\left (3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^3 d^3} \\ & = \frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 c (b c+a d) x^2 \sqrt {a+b x}}{b d (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right )-2 b d \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) x\right )}{4 b^3 d^3 (b c-a d)^2}+\frac {\left (3 \left (5 b^2 c^2+6 a b c d+5 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^4 d^3} \\ & = \frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 c (b c+a d) x^2 \sqrt {a+b x}}{b d (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right )-2 b d \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) x\right )}{4 b^3 d^3 (b c-a d)^2}+\frac {\left (3 \left (5 b^2 c^2+6 a b c d+5 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^4 d^3} \\ & = \frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 c (b c+a d) x^2 \sqrt {a+b x}}{b d (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right )-2 b d \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) x\right )}{4 b^3 d^3 (b c-a d)^2}+\frac {3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.99 \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {-15 a^4 d^3 (c+d x)+a^3 b d^2 \left (7 c^2+2 c d x-5 d^2 x^2\right )+b^4 c^2 x \left (-15 c^2-5 c d x+2 d^2 x^2\right )+a b^3 c \left (-15 c^3+2 c^2 d x+5 c d^2 x^2-4 d^3 x^3\right )+a^2 b^2 d \left (7 c^3+10 c^2 d x+5 c d^2 x^2+2 d^3 x^3\right )}{4 b^3 d^3 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{7/2} d^{7/2}} \]

[In]

Integrate[x^4/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

(-15*a^4*d^3*(c + d*x) + a^3*b*d^2*(7*c^2 + 2*c*d*x - 5*d^2*x^2) + b^4*c^2*x*(-15*c^2 - 5*c*d*x + 2*d^2*x^2) +
a*b^3*c*(-15*c^3 + 2*c^2*d*x + 5*c*d^2*x^2 - 4*d^3*x^3) + a^2*b^2*d*(7*c^3 + 10*c^2*d*x + 5*c*d^2*x^2 + 2*d^3
*x^3))/(4*b^3*d^3*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x]) + (3*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[
(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(4*b^(7/2)*d^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1368\) vs. \(2(222)=444\).

Time = 0.59 (sec) , antiderivative size = 1369, normalized size of antiderivative = 5.48

method	result	size
default	\(\text {Expression too large to display}\)	\(1369\)

[In]

int(x^4/(b*x+a)^(3/2)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/8*(-30*a^4*d^4*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*b^4*c^4*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*a^4
*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*a*b^3*c^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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))*a^4*b*d^5*x^2+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^5*c^4*d*x^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^4*b*c^2*d^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*b^2*c^3*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^4*d-10*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b*d^4*x^2-10*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^4*c^3*d*
x^2+14*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b*c^2*d^2+4*a^2*b^2*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)
+4*b^4*c^2*d^2*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-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^3*b^2*c*d^4*x^2-6*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^3*c^2*d^3*x^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
*b^4*c^3*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^4*b*c*d^4*x-1
8*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*x-18*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*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))*a*b^4*c^4*d*x+14*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^
2*c^3*d+10*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^3*c^2*d^2*x^2+4*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b*c
*d^3*x+4*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^3*c^3*d*x+10*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^2*c*d^
3*x^2-8*a*b^3*c*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+20*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*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))*a^5*d^5*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^5*c^5*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))*a^5*c*d^4+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))*a*b^4*c^5)/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(a*d-b*c)^2/(b*x+a)^(1/2)/(d*x+c)^(1/2)/
b^3/d^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (223) = 446\).

Time = 0.53 (sec) , antiderivative size = 1200, normalized size of antiderivative = 4.80 \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\left [\frac {3 \, {\left (5 \, a b^{4} c^{5} - 4 \, a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} - 4 \, a^{4} b c^{2} d^{3} + 5 \, a^{5} c d^{4} + {\left (5 \, b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} - 2 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x^{2} + {\left (5 \, b^{5} c^{5} + a b^{4} c^{4} d - 6 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} + 5 \, a^{5} d^{5}\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 \, a b^{4} c^{4} d - 7 \, a^{2} b^{3} c^{3} d^{2} - 7 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 2 \, {\left (b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{3} + 5 \, {\left (b^{5} c^{3} d^{2} - a b^{4} c^{2} d^{3} - a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + {\left (15 \, b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} - 10 \, a^{2} b^{3} c^{2} d^{3} - 2 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a b^{6} c^{3} d^{4} - 2 \, a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{2} + {\left (b^{7} c^{3} d^{4} - a b^{6} c^{2} d^{5} - a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x\right )}}, -\frac {3 \, {\left (5 \, a b^{4} c^{5} - 4 \, a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} - 4 \, a^{4} b c^{2} d^{3} + 5 \, a^{5} c d^{4} + {\left (5 \, b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} - 2 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x^{2} + {\left (5 \, b^{5} c^{5} + a b^{4} c^{4} d - 6 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} + 5 \, a^{5} d^{5}\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 \, a b^{4} c^{4} d - 7 \, a^{2} b^{3} c^{3} d^{2} - 7 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 2 \, {\left (b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{3} + 5 \, {\left (b^{5} c^{3} d^{2} - a b^{4} c^{2} d^{3} - a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + {\left (15 \, b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} - 10 \, a^{2} b^{3} c^{2} d^{3} - 2 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a b^{6} c^{3} d^{4} - 2 \, a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{2} + {\left (b^{7} c^{3} d^{4} - a b^{6} c^{2} d^{5} - a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x\right )}}\right ] \]

[In]

integrate(x^4/(b*x+a)^(3/2)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(3*(5*a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 - 4*a^4*b*c^2*d^3 + 5*a^5*c*d^4 + (5*b^5*c^4*d - 4
*a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x^2 + (5*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3
*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 + 5*a^5*d^5)*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*a*
b^4*c^4*d - 7*a^2*b^3*c^3*d^2 - 7*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 2*(b^5*c^2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*
d^5)*x^3 + 5*(b^5*c^3*d^2 - a*b^4*c^2*d^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (15*b^5*c^4*d - 2*a*b^4*c^3*d^2
- 10*a^2*b^3*c^2*d^3 - 2*a^3*b^2*c*d^4 + 15*a^4*b*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*b^6*c^3*d^4 - 2*a^2
*b^5*c^2*d^5 + a^3*b^4*c*d^6 + (b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^2 + (b^7*c^3*d^4 - a*b^6*c^2*d^5
- a^2*b^5*c*d^6 + a^3*b^4*d^7)*x), -1/8*(3*(5*a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 - 4*a^4*b*c^2*d^
3 + 5*a^5*c*d^4 + (5*b^5*c^4*d - 4*a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x^2 + (5
*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 + 5*a^5*d^5)*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*a*b^4*c^4*d - 7*a^2*b^3*c^3*d^2 - 7*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 2*(b^5*c^2*d^3 - 2*a*b^4*c
*d^4 + a^2*b^3*d^5)*x^3 + 5*(b^5*c^3*d^2 - a*b^4*c^2*d^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (15*b^5*c^4*d -
2*a*b^4*c^3*d^2 - 10*a^2*b^3*c^2*d^3 - 2*a^3*b^2*c*d^4 + 15*a^4*b*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*b^6*
c^3*d^4 - 2*a^2*b^5*c^2*d^5 + a^3*b^4*c*d^6 + (b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^2 + (b^7*c^3*d^4 -
a*b^6*c^2*d^5 - a^2*b^5*c*d^6 + a^3*b^4*d^7)*x)]

Sympy [F]

\[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {x^{4}}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(x**4/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Integral(x**4/((a + b*x)**(3/2)*(c + d*x)**(3/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^4/(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. 478 vs. \(2 (223) = 446\).

Time = 0.44 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.91 \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {4 \, a^{4} d}{{\left (\sqrt {b d} b^{2} c {\left | b \right |} - \sqrt {b d} a b d {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} + \frac {{\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{10} c^{2} d^{4} {\left | b \right |} - 2 \, a b^{9} c d^{5} {\left | b \right |} + a^{2} b^{8} d^{6} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{14} c^{2} d^{5} - 2 \, a b^{13} c d^{6} + a^{2} b^{12} d^{7}} - \frac {5 \, b^{11} c^{3} d^{3} {\left | b \right |} + a b^{10} c^{2} d^{4} {\left | b \right |} - 17 \, a^{2} b^{9} c d^{5} {\left | b \right |} + 11 \, a^{3} b^{8} d^{6} {\left | b \right |}}{b^{14} c^{2} d^{5} - 2 \, a b^{13} c d^{6} + a^{2} b^{12} d^{7}}\right )} - \frac {15 \, b^{12} c^{4} d^{2} {\left | b \right |} - 12 \, a b^{11} c^{3} d^{3} {\left | b \right |} - 6 \, a^{2} b^{10} c^{2} d^{4} {\left | b \right |} + 20 \, a^{3} b^{9} c d^{5} {\left | b \right |} - 9 \, a^{4} b^{8} d^{6} {\left | b \right |}}{b^{14} c^{2} d^{5} - 2 \, a b^{13} c d^{6} + a^{2} b^{12} d^{7}}\right )} \sqrt {b x + a}}{4 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {3 \, {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{8 \, \sqrt {b d} b^{2} d^{3} {\left | b \right |}} \]

[In]

integrate(x^4/(b*x+a)^(3/2)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-4*a^4*d/((sqrt(b*d)*b^2*c*abs(b) - sqrt(b*d)*a*b*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)) + 1/4*((b*x + a)*(2*(b^10*c^2*d^4*abs(b) - 2*a*b^9*c*d^5*abs(b) + a^2*b^8*d
^6*abs(b))*(b*x + a)/(b^14*c^2*d^5 - 2*a*b^13*c*d^6 + a^2*b^12*d^7) - (5*b^11*c^3*d^3*abs(b) + a*b^10*c^2*d^4*
abs(b) - 17*a^2*b^9*c*d^5*abs(b) + 11*a^3*b^8*d^6*abs(b))/(b^14*c^2*d^5 - 2*a*b^13*c*d^6 + a^2*b^12*d^7)) - (1
5*b^12*c^4*d^2*abs(b) - 12*a*b^11*c^3*d^3*abs(b) - 6*a^2*b^10*c^2*d^4*abs(b) + 20*a^3*b^9*c*d^5*abs(b) - 9*a^4
*b^8*d^6*abs(b))/(b^14*c^2*d^5 - 2*a*b^13*c*d^6 + a^2*b^12*d^7))*sqrt(b*x + a)/sqrt(b^2*c + (b*x + a)*b*d - a*
b*d) - 3/8*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*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^2*d^3*abs(b))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {x^4}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]

[In]

int(x^4/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x)

[Out]

int(x^4/((a + b*x)^(3/2)*(c + d*x)^(3/2)), x)