\(\int \frac {x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) [806]

Optimal result

Integrand size = 22, antiderivative size = 241 \[ \int \frac {x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 a x^3}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (3 b c-a d) x^2}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \sqrt {a+b x} \left (c (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right )+2 d \left (2 b^3 c^3-a b^2 c^2 d-12 a^2 b c d^2+3 a^3 d^3\right ) x\right )}{3 b^2 d^2 (b c-a d)^4 (c+d x)^{3/2}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{5/2}} \]

[Out]

2/3*a*x^3/b/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)+2*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*(-a*d+3*b*c)*x^2/b^2/(-a*d+b*c)^2/(d*x+c)^(3/2)/(b*x+a)^(1/2)-2/3*c*(c*(a*d+b*c)*(3*a^2*d^2-14
*a*b*c*d+3*b^2*c^2)+2*d*(3*a^3*d^3-12*a^2*b*c*d^2-a*b^2*c^2*d+2*b^3*c^3)*x)*(b*x+a)^(1/2)/b^2/d^2/(-a*d+b*c)^4
/(d*x+c)^(3/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 241, 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, 150, 65, 223, 212} \[ \int \frac {x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 c \sqrt {a+b x} \left (c (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )+2 d x \left (3 a^3 d^3-12 a^2 b c d^2-a b^2 c^2 d+2 b^3 c^3\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} (b c-a d)^4}+\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac {2 a x^2 (3 b c-a d)}{b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac {2 a x^3}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[In]

Int[x^4/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(2*a*x^3)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*(3*b*c - a*d)*x^2)/(b^2*(b*c - a*d)^2*Sqrt[
a + b*x]*(c + d*x)^(3/2)) - (2*c*Sqrt[a + b*x]*(c*(b*c + a*d)*(3*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2) + 2*d*(2*b^
3*c^3 - a*b^2*c^2*d - 12*a^2*b*c*d^2 + 3*a^3*d^3)*x))/(3*b^2*d^2*(b*c - a*d)^4*(c + d*x)^(3/2)) + (2*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 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(
n + 1), x] + Dist[f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

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}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {2 \int \frac {x^2 \left (3 a c-\frac {3}{2} (b c-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^3}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (3 b c-a d) x^2}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {4 \int \frac {x \left (3 a c (3 b c-a d)-\frac {3}{4} (b c-a d)^2 x\right )}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{3 b^2 (b c-a d)^2} \\ & = \frac {2 a x^3}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (3 b c-a d) x^2}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \sqrt {a+b x} \left (c (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right )+2 d \left (2 b^3 c^3-a b^2 c^2 d-12 a^2 b c d^2+3 a^3 d^3\right ) x\right )}{3 b^2 d^2 (b c-a d)^4 (c+d x)^{3/2}}+\frac {\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b^2 d^2} \\ & = \frac {2 a x^3}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (3 b c-a d) x^2}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \sqrt {a+b x} \left (c (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right )+2 d \left (2 b^3 c^3-a b^2 c^2 d-12 a^2 b c d^2+3 a^3 d^3\right ) x\right )}{3 b^2 d^2 (b c-a d)^4 (c+d x)^{3/2}}+\frac {2 \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^3}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (3 b c-a d) x^2}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \sqrt {a+b x} \left (c (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right )+2 d \left (2 b^3 c^3-a b^2 c^2 d-12 a^2 b c d^2+3 a^3 d^3\right ) x\right )}{3 b^2 d^2 (b c-a d)^4 (c+d x)^{3/2}}+\frac {2 \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^3}{3 b (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 a (3 b c-a d) x^2}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 c \sqrt {a+b x} \left (c (b c+a d) \left (3 b^2 c^2-14 a b c d+3 a^2 d^2\right )+2 d \left (2 b^3 c^3-a b^2 c^2 d-12 a^2 b c d^2+3 a^3 d^3\right ) x\right )}{3 b^2 d^2 (b c-a d)^4 (c+d x)^{3/2}}+\frac {2 \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 = 0.33 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.83 \[ \int \frac {x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 (a+b x)^{3/2} \left (b^2 c^4 d+\frac {3 b^3 c^4 (c+d x)}{a+b x}-\frac {12 a b^2 c^3 d (c+d x)}{a+b x}-\frac {12 a^3 b c d^2 (c+d x)^2}{(a+b x)^2}+\frac {3 a^4 d^3 (c+d x)^2}{(a+b x)^2}+\frac {a^4 b d^2 (c+d x)^3}{(a+b x)^3}\right )}{3 b^2 d^2 (b c-a d)^4 (c+d x)^{3/2}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{5/2} d^{5/2}} \]

[In]

Integrate[x^4/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(-2*(a + b*x)^(3/2)*(b^2*c^4*d + (3*b^3*c^4*(c + d*x))/(a + b*x) - (12*a*b^2*c^3*d*(c + d*x))/(a + b*x) - (12*
a^3*b*c*d^2*(c + d*x)^2)/(a + b*x)^2 + (3*a^4*d^3*(c + d*x)^2)/(a + b*x)^2 + (a^4*b*d^2*(c + d*x)^3)/(a + b*x)
^3))/(3*b^2*d^2*(b*c - a*d)^4*(c + d*x)^(3/2)) + (2*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. \(2088\) vs. \(2(213)=426\).

Time = 1.72 (sec) , antiderivative size = 2089, normalized size of antiderivative = 8.67

method	result	size
default	\(\text {Expression too large to display}\)	\(2089\)

[In]

int(x^4/(b*x+a)^(5/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(-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^5*b*c^2*d^4*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^4*c^2*d^4*x^4-18*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^4*b^2*c*d^5*x^3-18*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*b^5*c^4*d^2*x^3+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^4*b^2*c^4*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^3*b^3*c^2*d^4*x^3+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^4*c^3*d^3*x^3-8*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b*d^5*x^3-8*(b*d)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)*b^5*c^4*d*x^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^4*b^2*c^
2*d^4*x^2+48*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^3*c^3*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^6*c^2*d^4+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^4*c^6+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^6*d^6*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^6*c^6*x^2+48*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^2*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^4*b^2*d^6*x^4+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^6*c^4*d^2*x^4+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^5*b*d^6*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))*b^6*c^5*d*x^3+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^2*c^3*d
^3*x+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^3*c^4*d^2*x-12*(b*d)
^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^5*c*d^4*x-12*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^4*c^5*x-6*(b*d)^(1/2)*((
b*x+a)*(d*x+c))^(1/2)*a^5*d^5*x^2-6*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^5*c^5*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^6*c*d^5*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*b^5*c^6*x-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^5*b*c^3*d^3-12*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^3*c^5*d-6*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^5*c^2*d^3-6*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^3*c^5-
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^4*c^5*d*x+22*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)*a^4*b*c^3*d^2+22*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^2*c^4*d-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^4*c^4*d^2*x^2+36*(b*d)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)*a^4*b*c^2*d^3*x+24*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^2*c*d^4*x^3+24*(b*d)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)*a*b^4*c^3*d^2*x^3-12*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^3*c*d^5*x^4-12*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^5*c^3*d^3*x
^4+6*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b*c*d^4*x^2+48*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^2*c^2*d^
3*x^2+48*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^3*c^3*d^2*x^2+6*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^4*c
^4*d*x^2+36*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^3*c^4*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^2/d^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1026 vs. \(2 (213) = 426\).

Time = 0.81 (sec) , antiderivative size = 2066, normalized size of antiderivative = 8.57 \[ \int \frac {x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(x^4/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*
a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^
2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b
^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2
*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*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*(3*a^2*b^4*c^5*d - 11*
a^3*b^3*c^4*d^2 - 11*a^4*b^2*c^3*d^3 + 3*a^5*b*c^2*d^4 + 4*(b^6*c^4*d^2 - 3*a*b^5*c^3*d^3 - 3*a^3*b^3*c*d^5 +
a^4*b^2*d^6)*x^3 + 3*(b^6*c^5*d - a*b^5*c^4*d^2 - 8*a^2*b^4*c^3*d^3 - 8*a^3*b^3*c^2*d^4 - a^4*b^2*c*d^5 + a^5*
b*d^6)*x^2 + 6*(a*b^5*c^5*d - 3*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 - 3*a^4*b^2*c^2*d^4 + a^5*b*c*d^5)*x)*sqrt
(b*x + a)*sqrt(d*x + c))/(a^2*b^7*c^6*d^3 - 4*a^3*b^6*c^5*d^4 + 6*a^4*b^5*c^4*d^5 - 4*a^5*b^4*c^3*d^6 + a^6*b^
3*c^2*d^7 + (b^9*c^4*d^5 - 4*a*b^8*c^3*d^6 + 6*a^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^4 + 2*(b^9*c
^5*d^4 - 3*a*b^8*c^4*d^5 + 2*a^2*b^7*c^3*d^6 + 2*a^3*b^6*c^2*d^7 - 3*a^4*b^5*c*d^8 + a^5*b^4*d^9)*x^3 + (b^9*c
^6*d^3 - 9*a^2*b^7*c^4*d^5 + 16*a^3*b^6*c^3*d^6 - 9*a^4*b^5*c^2*d^7 + a^6*b^3*d^9)*x^2 + 2*(a*b^8*c^6*d^3 - 3*
a^2*b^7*c^5*d^4 + 2*a^3*b^6*c^4*d^5 + 2*a^4*b^5*c^3*d^6 - 3*a^5*b^4*c^2*d^7 + a^6*b^3*c*d^8)*x), -1/3*(3*(a^2*
b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d^3
+ 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3
+ 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 -
9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*
a^5*b*c^2*d^4 + a^6*c*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*(3*a^2*b^4*c^5*d - 11*a^3*b^3*c^4*d^2 - 11*a^4*b^2*c^3
*d^3 + 3*a^5*b*c^2*d^4 + 4*(b^6*c^4*d^2 - 3*a*b^5*c^3*d^3 - 3*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^3 + 3*(b^6*c^5*d
- a*b^5*c^4*d^2 - 8*a^2*b^4*c^3*d^3 - 8*a^3*b^3*c^2*d^4 - a^4*b^2*c*d^5 + a^5*b*d^6)*x^2 + 6*(a*b^5*c^5*d - 3*
a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 - 3*a^4*b^2*c^2*d^4 + a^5*b*c*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b^
7*c^6*d^3 - 4*a^3*b^6*c^5*d^4 + 6*a^4*b^5*c^4*d^5 - 4*a^5*b^4*c^3*d^6 + a^6*b^3*c^2*d^7 + (b^9*c^4*d^5 - 4*a*b
^8*c^3*d^6 + 6*a^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^4 + 2*(b^9*c^5*d^4 - 3*a*b^8*c^4*d^5 + 2*a^2
*b^7*c^3*d^6 + 2*a^3*b^6*c^2*d^7 - 3*a^4*b^5*c*d^8 + a^5*b^4*d^9)*x^3 + (b^9*c^6*d^3 - 9*a^2*b^7*c^4*d^5 + 16*
a^3*b^6*c^3*d^6 - 9*a^4*b^5*c^2*d^7 + a^6*b^3*d^9)*x^2 + 2*(a*b^8*c^6*d^3 - 3*a^2*b^7*c^5*d^4 + 2*a^3*b^6*c^4*
d^5 + 2*a^4*b^5*c^3*d^6 - 3*a^5*b^4*c^2*d^7 + a^6*b^3*c*d^8)*x)]

Sympy [F]

\[ \int \frac {x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x^{4}}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**4/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Integral(x**4/((a + b*x)**(5/2)*(c + d*x)**(5/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^4/(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. 896 vs. \(2 (213) = 426\).

Time = 0.64 (sec) , antiderivative size = 896, normalized size of antiderivative = 3.72 \[ \int \frac {x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b x + a} {\left (\frac {4 \, {\left (b^{10} c^{7} d^{2} - 6 \, a b^{9} c^{6} d^{3} + 12 \, a^{2} b^{8} c^{5} d^{4} - 10 \, a^{3} b^{7} c^{4} d^{5} + 3 \, a^{4} b^{6} c^{3} d^{6}\right )} {\left (b x + a\right )}}{b^{10} c^{7} d^{3} {\left | b \right |} - 7 \, a b^{9} c^{6} d^{4} {\left | b \right |} + 21 \, a^{2} b^{8} c^{5} d^{5} {\left | b \right |} - 35 \, a^{3} b^{7} c^{4} d^{6} {\left | b \right |} + 35 \, a^{4} b^{6} c^{3} d^{7} {\left | b \right |} - 21 \, a^{5} b^{5} c^{2} d^{8} {\left | b \right |} + 7 \, a^{6} b^{4} c d^{9} {\left | b \right |} - a^{7} b^{3} d^{10} {\left | b \right |}} + \frac {3 \, {\left (b^{11} c^{8} d - 8 \, a b^{10} c^{7} d^{2} + 22 \, a^{2} b^{9} c^{6} d^{3} - 28 \, a^{3} b^{8} c^{5} d^{4} + 17 \, a^{4} b^{7} c^{4} d^{5} - 4 \, a^{5} b^{6} c^{3} d^{6}\right )}}{b^{10} c^{7} d^{3} {\left | b \right |} - 7 \, a b^{9} c^{6} d^{4} {\left | b \right |} + 21 \, a^{2} b^{8} c^{5} d^{5} {\left | b \right |} - 35 \, a^{3} b^{7} c^{4} d^{6} {\left | b \right |} + 35 \, a^{4} b^{6} c^{3} d^{7} {\left | b \right |} - 21 \, a^{5} b^{5} c^{2} d^{8} {\left | b \right |} + 7 \, a^{6} b^{4} c d^{9} {\left | b \right |} - a^{7} b^{3} d^{10} {\left | b \right |}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {8 \, {\left (6 \, a^{3} b^{5} c^{3} d - 14 \, a^{4} b^{4} c^{2} d^{2} + 10 \, a^{5} b^{3} c d^{3} - 2 \, a^{6} b^{2} d^{4} - 12 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} c^{2} d + 15 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{2} c d^{2} - 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{5} b d^{3} + 6 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b c d - 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{4} d^{2}\right )}}{3 \, {\left (\sqrt {b d} b^{3} c^{3} {\left | b \right |} - 3 \, \sqrt {b d} a b^{2} c^{2} d {\left | b \right |} + 3 \, \sqrt {b d} a^{2} b c d^{2} {\left | b \right |} - \sqrt {b d} a^{3} d^{3} {\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 )}^{3}} - \frac {\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 )}{\sqrt {b d} b d^{2} {\left | b \right |}} \]

[In]

integrate(x^4/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

-2/3*sqrt(b*x + a)*(4*(b^10*c^7*d^2 - 6*a*b^9*c^6*d^3 + 12*a^2*b^8*c^5*d^4 - 10*a^3*b^7*c^4*d^5 + 3*a^4*b^6*c^
3*d^6)*(b*x + a)/(b^10*c^7*d^3*abs(b) - 7*a*b^9*c^6*d^4*abs(b) + 21*a^2*b^8*c^5*d^5*abs(b) - 35*a^3*b^7*c^4*d^
6*abs(b) + 35*a^4*b^6*c^3*d^7*abs(b) - 21*a^5*b^5*c^2*d^8*abs(b) + 7*a^6*b^4*c*d^9*abs(b) - a^7*b^3*d^10*abs(b
)) + 3*(b^11*c^8*d - 8*a*b^10*c^7*d^2 + 22*a^2*b^9*c^6*d^3 - 28*a^3*b^8*c^5*d^4 + 17*a^4*b^7*c^4*d^5 - 4*a^5*b
^6*c^3*d^6)/(b^10*c^7*d^3*abs(b) - 7*a*b^9*c^6*d^4*abs(b) + 21*a^2*b^8*c^5*d^5*abs(b) - 35*a^3*b^7*c^4*d^6*abs
(b) + 35*a^4*b^6*c^3*d^7*abs(b) - 21*a^5*b^5*c^2*d^8*abs(b) + 7*a^6*b^4*c*d^9*abs(b) - a^7*b^3*d^10*abs(b)))/(
b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + 8/3*(6*a^3*b^5*c^3*d - 14*a^4*b^4*c^2*d^2 + 10*a^5*b^3*c*d^3 - 2*a^6*b^
2*d^4 - 12*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^3*c^2*d + 15*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^2*c*d^2 - 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2*a^5*b*d^3 + 6*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*
b*c*d - 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*d^2)/((sqrt(b*d)*b^3*c^3*abs(b
) - 3*sqrt(b*d)*a*b^2*c^2*d*abs(b) + 3*sqrt(b*d)*a^2*b*c*d^2*abs(b) - sqrt(b*d)*a^3*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) - 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. \[ \int \frac {x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x^4}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]

[In]

int(x^4/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x)

[Out]

int(x^4/((a + b*x)^(5/2)*(c + d*x)^(5/2)), x)