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

Optimal. Leaf size=241 \[ -\frac{2 c \sqrt{a+b x} \left (2 d x \left (-12 a^2 b c d^2+3 a^3 d^3-a b^2 c^2 d+2 b^3 c^3\right )+c (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} (b c-a d)^4}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac{2 a x^2 (3 b c-a d)}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^3}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.199557, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {98, 150, 145, 63, 217, 206} \[ -\frac{2 c \sqrt{a+b x} \left (2 d x \left (-12 a^2 b c d^2+3 a^3 d^3-a b^2 c^2 d+2 b^3 c^3\right )+c (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} (b c-a d)^4}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac{2 a x^2 (3 b c-a d)}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^3}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[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 98

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_))^(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 145

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)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), 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 63

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 217

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]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \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 \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 \operatorname{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 \operatorname{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]  time = 5.45093, size = 194, normalized size = 0.8 \[ \frac{2}{3} \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{a^4}{b^2 (a+b x)^2 (b c-a d)^3}-\frac{4 a^3 (a d-3 b c)}{b^2 (a+b x) (b c-a d)^4}-\frac{4 c^3 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^4}-\frac{c^4}{d^2 (c+d x)^2 (a d-b c)^3}\right )+\frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.031, size = 2089, normalized size = 8.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 12.9227, size = 4101, normalized size = 17.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 3.9871, size = 1235, normalized size = 5.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*sqrt(b*d)*a^3*b^5*c^3 - 14*sqrt(b*d)*a^4*b^4*c^2*d + 10*sqrt(b*d
)*a^5*b^3*c*d^2 - 2*sqrt(b*d)*a^6*b^2*d^3 - 12*sqrt(b*d)*(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 + 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*
b^2*c*d - 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b*d^2 + 6*sqrt(b*d
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b*c - 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x
+ a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*d)/((b^4*c^3*abs(b) - 3*a*b^3*c^2*d*abs(b) + 3*a^2*b^2*c*d^2
*abs(b) - 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) - 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))