\(\int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) [805]

Optimal result

Integrand size = 22, antiderivative size = 341 \[ \int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 a x^4}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (11 b c-5 a d) x^3}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+12 a b c d-5 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right )+d (b c-a d) \left (5 b^3 c^3-9 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{3 b^3 d^3 (b c-a d)^4 \sqrt {c+d x}}-\frac {5 (b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2} d^{7/2}} \]

[Out]

2/3*a*x^4/b/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)-5*(a*d+b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(
1/2))/b^(7/2)/d^(7/2)+2/3*a*(-5*a*d+11*b*c)*x^3/b^2/(-a*d+b*c)^2/(d*x+c)^(3/2)/(b*x+a)^(1/2)-2/3*c*(-5*a^2*d^2
+12*a*b*c*d+b^2*c^2)*x^2*(b*x+a)^(1/2)/b^2/d/(-a*d+b*c)^3/(d*x+c)^(3/2)+1/3*(c*(15*a^4*d^4-40*a^3*b*c*d^3+18*a
^2*b^2*c^2*d^2-40*a*b^3*c^3*d+15*b^4*c^4)+d*(-a*d+b*c)*(-15*a^3*d^3+35*a^2*b*c*d^2-9*a*b^2*c^2*d+5*b^3*c^3)*x)
*(b*x+a)^(1/2)/b^3/d^3/(-a*d+b*c)^4/(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {100, 155, 148, 65, 223, 212} \[ \int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 c x^2 \sqrt {a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}+\frac {\sqrt {a+b x} \left (d x (b c-a d) \left (-15 a^3 d^3+35 a^2 b c d^2-9 a b^2 c^2 d+5 b^3 c^3\right )+c \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )\right )}{3 b^3 d^3 \sqrt {c+d x} (b c-a d)^4}-\frac {5 (a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2} d^{7/2}}+\frac {2 a x^3 (11 b c-5 a d)}{3 b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac {2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[In]

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

[Out]

(2*a*x^4)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*(11*b*c - 5*a*d)*x^3)/(3*b^2*(b*c - a*d)^2*
Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b^2*c^2 + 12*a*b*c*d - 5*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*b^2*d*(b*c - a*
d)^3*(c + d*x)^(3/2)) + (Sqrt[a + b*x]*(c*(15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 +
15*a^4*d^4) + d*(b*c - a*d)*(5*b^3*c^3 - 9*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*x))/(3*b^3*d^3*(b*c - a
*d)^4*Sqrt[c + d*x]) - (5*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(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 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 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^4}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {2 \int \frac {x^3 \left (4 a c+\frac {1}{2} (-3 b c+5 a d) x\right )}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{3 b (b c-a d)} \\ & = \frac {2 a x^4}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (11 b c-5 a d) x^3}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {4 \int \frac {x^2 \left (\frac {3}{2} a c (11 b c-5 a d)-\frac {3}{4} \left (b^2 c^2-10 a b c d+5 a^2 d^2\right ) x\right )}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{3 b^2 (b c-a d)^2} \\ & = \frac {2 a x^4}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (11 b c-5 a d) x^3}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+12 a b c d-5 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {8 \int \frac {x \left (\frac {3}{2} a c \left (b^2 c^2+12 a b c d-5 a^2 d^2\right )+\frac {3}{8} \left (5 b^3 c^3-9 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{9 b^2 d (b c-a d)^3} \\ & = \frac {2 a x^4}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (11 b c-5 a d) x^3}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+12 a b c d-5 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right )+d (b c-a d) \left (5 b^3 c^3-9 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{3 b^3 d^3 (b c-a d)^4 \sqrt {c+d x}}-\frac {(5 (b c+a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b^3 d^3} \\ & = \frac {2 a x^4}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (11 b c-5 a d) x^3}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+12 a b c d-5 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right )+d (b c-a d) \left (5 b^3 c^3-9 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{3 b^3 d^3 (b c-a d)^4 \sqrt {c+d x}}-\frac {(5 (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^4 d^3} \\ & = \frac {2 a x^4}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (11 b c-5 a d) x^3}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+12 a b c d-5 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right )+d (b c-a d) \left (5 b^3 c^3-9 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{3 b^3 d^3 (b c-a d)^4 \sqrt {c+d x}}-\frac {(5 (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^4 d^3} \\ & = \frac {2 a x^4}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (11 b c-5 a d) x^3}{3 b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \left (b^2 c^2+12 a b c d-5 a^2 d^2\right ) x^2 \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right )+d (b c-a d) \left (5 b^3 c^3-9 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x\right )}{3 b^3 d^3 (b c-a d)^4 \sqrt {c+d x}}-\frac {5 (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2} d^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.96 \[ \int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {15 a^6 d^4 (c+d x)^2+20 a^5 b d^3 (-2 c+d x) (c+d x)^2+3 a^4 b^2 d^2 (c+d x)^2 \left (6 c^2-18 c d x+d^2 x^2\right )+b^6 c^4 x^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )+6 a b^5 c^3 x \left (5 c^3-8 c d^2 x^2-2 d^3 x^3\right )-2 a^3 b^3 c d \left (20 c^4+9 c^3 d x-24 c^2 d^2 x^2-6 c d^3 x^3+6 d^4 x^4\right )+3 a^2 b^4 c^2 \left (5 c^4-20 c^3 d x-29 c^2 d^2 x^2+4 c d^3 x^3+6 d^4 x^4\right )}{3 b^3 d^3 (b c-a d)^4 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {5 (b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{7/2} d^{7/2}} \]

[In]

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

[Out]

(15*a^6*d^4*(c + d*x)^2 + 20*a^5*b*d^3*(-2*c + d*x)*(c + d*x)^2 + 3*a^4*b^2*d^2*(c + d*x)^2*(6*c^2 - 18*c*d*x
+ d^2*x^2) + b^6*c^4*x^2*(15*c^2 + 20*c*d*x + 3*d^2*x^2) + 6*a*b^5*c^3*x*(5*c^3 - 8*c*d^2*x^2 - 2*d^3*x^3) - 2
*a^3*b^3*c*d*(20*c^4 + 9*c^3*d*x - 24*c^2*d^2*x^2 - 6*c*d^3*x^3 + 6*d^4*x^4) + 3*a^2*b^4*c^2*(5*c^4 - 20*c^3*d
*x - 29*c^2*d^2*x^2 + 4*c*d^3*x^3 + 6*d^4*x^4))/(3*b^3*d^3*(b*c - a*d)^4*(a + b*x)^(3/2)*(c + d*x)^(3/2)) - (5
*(b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(b^(7/2)*d^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2747\) vs. \(2(305)=610\).

Time = 0.60 (sec) , antiderivative size = 2748, normalized size of antiderivative = 8.06

method	result	size
default	\(\text {Expression too large to display}\)	\(2748\)

[In]

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

[Out]

-1/6*(-60*a^6*c*d^5*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-60*a*b^5*c^6*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-4
5*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^3*c*d^6*x^4+30*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^4*c^2*d^5*x^4+30*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^5*c^3*d^4*x^4-45*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^6*c^4*d^3*x^4-60*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*b^2*c*d^6*x^3-30*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^3*c^2*d^5*x^3+120*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^4*c^3*d^4*x^3-30*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^5*c^4*d^3*x^3-60*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^6*c^5*d^2*x
^3+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^6*b*c*d^6*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^5*b^2*c^2*d^5*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))*a^4*b^3*c^3*d^4*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))*a^3*b^4*c^4*d^3*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^2*b^5*c^5*d^2*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))*a*b^6*c^6*d*x^2-60*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^6*b*c^2*d^5*x-30*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
*b^2*c^3*d^4*x+120*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^3*c^4*d^3
*x-30*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^4*c^5*d^2*x-60*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^5*c^6*d*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^2*b^5*c^7+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^7*d^7*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^7*c^7*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))*a^7
*c^2*d^5-30*a^6*d^6*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*b^6*c^6*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)
-30*a^6*c^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*a^2*b^4*c^6*((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^5*b^2*d^7*x^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))*b^7*c^5*d^2*x^4+30*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^6*b*d^7*x^3+30*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^7*c^6*d*x^3+30*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^7*c*d^6*x+30*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^6*c^7*x
-45*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^6*b*c^3*d^4+24*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c*d^5*x^4-36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^4*c^2*d^4*x^4-6*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*d^6*x^4-6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^6*c^4*d^2*x^4-40*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*b*d^6*x^3-40*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^6*c^5*d*x^3+80*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)*a^5*b*c^3*d^3-36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*c^4*d^2+80*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^5*d+30*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*b^2*c^4*d^3+30*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^3*c^5
*d^2-45*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^4*c^6*d-24*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^4*c^3*d^3*x^3+174*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^4*c^4*d^2*x^2-96*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^3*d^3*x^2+24*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^5*c^3*d^3*x
^4+96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*c*d^5*x^3-24*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^2
*d^4*x^3+96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^5*c^4*d^2*x^3+120*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*
b*c^2*d^4*x+174*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*c^2*d^4*x^2+36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)
*a^4*b^2*c^3*d^3*x+36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^4*d^2*x+120*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)*a^2*b^4*c^5*d*x)/((b*x+a)*(d*x+c))^(1/2)/(a*d-b*c)^4/(b*d)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(3/2)/b^3/d^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1230 vs. \(2 (305) = 610\).

Time = 1.15 (sec) , antiderivative size = 2474, normalized size of antiderivative = 7.26 \[ \int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/12*(15*(a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c^4*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*d
^5 + (b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x
^4 + 2*(b^7*c^6*d - 2*a*b^6*c^5*d^2 - a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 - a^4*b^3*c^2*d^5 - 2*a^5*b^2*c*d^6
+ a^6*b*d^7)*x^3 + (b^7*c^7 + a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 7*a^3*b^4*c^4*d^3 + 7*a^4*b^3*c^3*d^4 - 9*a^5*
b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)*x^2 + 2*(a*b^6*c^7 - 2*a^2*b^5*c^6*d - a^3*b^4*c^5*d^2 + 4*a^4*b^3*c^4*d^
3 - a^5*b^2*c^3*d^4 - 2*a^6*b*c^2*d^5 + a^7*c*d^6)*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^2*b^5
*c^6*d - 40*a^3*b^4*c^5*d^2 + 18*a^4*b^3*c^4*d^3 - 40*a^5*b^2*c^3*d^4 + 15*a^6*b*c^2*d^5 + 3*(b^7*c^4*d^3 - 4*
a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^4 + 4*(5*b^7*c^5*d^2 - 12*a*b^6*c^4*d^3 +
3*a^2*b^5*c^3*d^4 + 3*a^3*b^4*c^2*d^5 - 12*a^4*b^3*c*d^6 + 5*a^5*b^2*d^7)*x^3 + 3*(5*b^7*c^6*d - 29*a^2*b^5*c
^4*d^3 + 16*a^3*b^4*c^3*d^4 - 29*a^4*b^3*c^2*d^5 + 5*a^6*b*d^7)*x^2 + 6*(5*a*b^6*c^6*d - 10*a^2*b^5*c^5*d^2 -
3*a^3*b^4*c^4*d^3 - 3*a^4*b^3*c^3*d^4 - 10*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a
^2*b^8*c^6*d^4 - 4*a^3*b^7*c^5*d^5 + 6*a^4*b^6*c^4*d^6 - 4*a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + (b^10*c^4*d^6 -
4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^4 + 2*(b^10*c^5*d^5 - 3*a*b^9*c^4*d^6
+ 2*a^2*b^8*c^3*d^7 + 2*a^3*b^7*c^2*d^8 - 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^3 + (b^10*c^6*d^4 - 9*a^2*b^8*c^4
*d^6 + 16*a^3*b^7*c^3*d^7 - 9*a^4*b^6*c^2*d^8 + a^6*b^4*d^10)*x^2 + 2*(a*b^9*c^6*d^4 - 3*a^2*b^8*c^5*d^5 + 2*a
^3*b^7*c^4*d^6 + 2*a^4*b^6*c^3*d^7 - 3*a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9)*x), 1/6*(15*(a^2*b^5*c^7 - 3*a^3*b^4*c
^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c^4*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*d^5 + (b^7*c^5*d^2 - 3*a*b^6*c^4*d^3
+ 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^4 + 2*(b^7*c^6*d - 2*a*b^6*c^5*d^2
- a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 - a^4*b^3*c^2*d^5 - 2*a^5*b^2*c*d^6 + a^6*b*d^7)*x^3 + (b^7*c^7 + a*b^6*
c^6*d - 9*a^2*b^5*c^5*d^2 + 7*a^3*b^4*c^4*d^3 + 7*a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)
*x^2 + 2*(a*b^6*c^7 - 2*a^2*b^5*c^6*d - a^3*b^4*c^5*d^2 + 4*a^4*b^3*c^4*d^3 - a^5*b^2*c^3*d^4 - 2*a^6*b*c^2*d^
5 + a^7*c*d^6)*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^2*b^5*c^6*d - 40*a^3*b^4*c^5*d^2 + 18*a^4*b^3*c^4*d^3 - 40*a
^5*b^2*c^3*d^4 + 15*a^6*b*c^2*d^5 + 3*(b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 + a
^4*b^3*d^7)*x^4 + 4*(5*b^7*c^5*d^2 - 12*a*b^6*c^4*d^3 + 3*a^2*b^5*c^3*d^4 + 3*a^3*b^4*c^2*d^5 - 12*a^4*b^3*c*d
^6 + 5*a^5*b^2*d^7)*x^3 + 3*(5*b^7*c^6*d - 29*a^2*b^5*c^4*d^3 + 16*a^3*b^4*c^3*d^4 - 29*a^4*b^3*c^2*d^5 + 5*a^
6*b*d^7)*x^2 + 6*(5*a*b^6*c^6*d - 10*a^2*b^5*c^5*d^2 - 3*a^3*b^4*c^4*d^3 - 3*a^4*b^3*c^3*d^4 - 10*a^5*b^2*c^2*
d^5 + 5*a^6*b*c*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^8*c^6*d^4 - 4*a^3*b^7*c^5*d^5 + 6*a^4*b^6*c^4*d^6
- 4*a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + (b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9
+ a^4*b^6*d^10)*x^4 + 2*(b^10*c^5*d^5 - 3*a*b^9*c^4*d^6 + 2*a^2*b^8*c^3*d^7 + 2*a^3*b^7*c^2*d^8 - 3*a^4*b^6*c*
d^9 + a^5*b^5*d^10)*x^3 + (b^10*c^6*d^4 - 9*a^2*b^8*c^4*d^6 + 16*a^3*b^7*c^3*d^7 - 9*a^4*b^6*c^2*d^8 + a^6*b^4
*d^10)*x^2 + 2*(a*b^9*c^6*d^4 - 3*a^2*b^8*c^5*d^5 + 2*a^3*b^7*c^4*d^6 + 2*a^4*b^6*c^3*d^7 - 3*a^5*b^5*c^2*d^8
+ a^6*b^4*c*d^9)*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^5/(b*x+a)^(5/2)/(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. 1249 vs. \(2 (305) = 610\).

Time = 0.87 (sec) , antiderivative size = 1249, normalized size of antiderivative = 3.66 \[ \int \frac {x^5}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*((b*x + a)*(3*(b^13*c^7*d^4*abs(b) - 7*a*b^12*c^6*d^5*abs(b) + 21*a^2*b^11*c^5*d^6*abs(b) - 35*a^3*b^10*c^
4*d^7*abs(b) + 35*a^4*b^9*c^3*d^8*abs(b) - 21*a^5*b^8*c^2*d^9*abs(b) + 7*a^6*b^7*c*d^10*abs(b) - a^7*b^6*d^11*
abs(b))*(b*x + a)/(b^16*c^7*d^5 - 7*a*b^15*c^6*d^6 + 21*a^2*b^14*c^5*d^7 - 35*a^3*b^13*c^4*d^8 + 35*a^4*b^12*c
^3*d^9 - 21*a^5*b^11*c^2*d^10 + 7*a^6*b^10*c*d^11 - a^7*b^9*d^12) + 2*(10*b^14*c^8*d^3*abs(b) - 60*a*b^13*c^7*
d^4*abs(b) + 150*a^2*b^12*c^6*d^5*abs(b) - 220*a^3*b^11*c^5*d^6*abs(b) + 225*a^4*b^10*c^4*d^7*abs(b) - 168*a^5
*b^9*c^3*d^8*abs(b) + 84*a^6*b^8*c^2*d^9*abs(b) - 24*a^7*b^7*c*d^10*abs(b) + 3*a^8*b^6*d^11*abs(b))/(b^16*c^7*
d^5 - 7*a*b^15*c^6*d^6 + 21*a^2*b^14*c^5*d^7 - 35*a^3*b^13*c^4*d^8 + 35*a^4*b^12*c^3*d^9 - 21*a^5*b^11*c^2*d^1
0 + 7*a^6*b^10*c*d^11 - a^7*b^9*d^12)) + 3*(5*b^15*c^9*d^2*abs(b) - 35*a*b^14*c^8*d^3*abs(b) + 100*a^2*b^13*c^
7*d^4*abs(b) - 160*a^3*b^12*c^6*d^5*abs(b) + 170*a^4*b^11*c^5*d^6*abs(b) - 136*a^5*b^10*c^4*d^7*abs(b) + 84*a^
6*b^9*c^3*d^8*abs(b) - 36*a^7*b^8*c^2*d^9*abs(b) + 9*a^8*b^7*c*d^10*abs(b) - a^9*b^6*d^11*abs(b))/(b^16*c^7*d^
5 - 7*a*b^15*c^6*d^6 + 21*a^2*b^14*c^5*d^7 - 35*a^3*b^13*c^4*d^8 + 35*a^4*b^12*c^3*d^9 - 21*a^5*b^11*c^2*d^10
+ 7*a^6*b^10*c*d^11 - a^7*b^9*d^12))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 4/3*(15*a^4*b^5*c^3
*d - 37*a^5*b^4*c^2*d^2 + 29*a^6*b^3*c*d^3 - 7*a^7*b^2*d^4 - 30*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +
a)*b*d - a*b*d))^2*a^4*b^3*c^2*d + 42*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b
^2*c*d^2 - 12*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b*d^3 + 15*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b*c*d - 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^4*a^5*d^2)/((sqrt(b*d)*b^4*c^3*abs(b) - 3*sqrt(b*d)*a*b^3*c^2*d*abs(b) + 3*sqrt(b*d)*a^2*b^
2*c*d^2*abs(b) - sqrt(b*d)*a^3*b*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)^3) + 5/2*(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^2*d^3*abs(b))

Mupad [F(-1)]

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

[In]

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

[Out]

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