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

Optimal. Leaf size=341 \[ -\frac{2 c x^2 \sqrt{a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}+\frac{\sqrt{a+b x} \left (d x (b c-a d) \left (35 a^2 b c d^2-15 a^3 d^3-9 a b^2 c^2 d+5 b^3 c^3\right )+c \left (18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4-40 a b^3 c^3 d+15 b^4 c^4\right )\right )}{3 b^3 d^3 \sqrt{c+d x} (b c-a d)^4}-\frac{5 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{7/2} d^{7/2}}+\frac{2 a x^3 (11 b c-5 a d)}{3 b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.34881, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {98, 150, 143, 63, 217, 206} \[ -\frac{2 c x^2 \sqrt{a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}+\frac{\sqrt{a+b x} \left (d x (b c-a d) \left (35 a^2 b c d^2-15 a^3 d^3-9 a b^2 c^2 d+5 b^3 c^3\right )+c \left (18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4-40 a b^3 c^3 d+15 b^4 c^4\right )\right )}{3 b^3 d^3 \sqrt{c+d x} (b c-a d)^4}-\frac{5 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{7/2} d^{7/2}}+\frac{2 a x^3 (11 b c-5 a d)}{3 b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[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 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 143

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[n, 1] &&  !SumSimplerQ[m, 1])

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^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 \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)) \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^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)) \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^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 [C]  time = 8.29523, size = 1498, normalized size = 4.39 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

x^4/(b*d*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (((-4*a*b*c + (5*a*(b*c + a*d))/2)*((-2*x^3)/(3*(b*c - a*d)*(a + b
*x)^(3/2)*(c + d*x)^(3/2)) + (2*c*((-2*a^2)/(b^2*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (2*((2*((b*c*(b*
c - a*d))/2 + (a*d*(b*c + 3*a*d))/2)*Sqrt[a + b*x])/(3*d*(-(b*c) + a*d)*(c + d*x)^(3/2)) + (4*(-(b*((b*c)/2 -
(3*a*d)/2)*(b*c - a*d))/2 + (a*b*d*(b*c + 3*a*d))/2)*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(-(b*c) + a*d)*Sqrt[c + d
*x])))/(b^2*(b*c - a*d))))/(b*c - a*d)))/b - (5*a^3*(b*c + a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))
)^(5/2)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*((2520*b*(c + d*x))/(a*d*Sqrt[(b*(c + d*x))/(b*c - a*d)]) - (1330*b*(c
+ d*x))/(d*(a + b*x)*Sqrt[(b*(c + d*x))/(b*c - a*d)]) - (1050*b*(a + b*x)*(c + d*x))/(a^2*d*Sqrt[(b*(c + d*x)
)/(b*c - a*d)]) + (196*b*(a + b*x)^2*(c + d*x))/(a^3*d*Sqrt[(b*(c + d*x))/(b*c - a*d)]) + 1568*Sqrt[(b*(c + d*
x))/(b*c - a*d)] + (1575*(b*c - a*d)^2*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(a^2*d^2) + (1995*(b*c - a*d)^2*Sqrt[(
b*(c + d*x))/(b*c - a*d)])/(d^2*(a + b*x)^2) - (3780*(b*c - a*d)^2*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(a*d^2*(a
+ b*x)) - (294*(b*c - a*d)^2*(a + b*x)*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(a^3*d^2) - 504*(1 + (b*x)/a)*Sqrt[(b*
(c + d*x))/(b*c - a*d)] + 336*(1 + (b*x)/a)^2*Sqrt[(b*(c + d*x))/(b*c - a*d)] - 56*(1 + (b*x)/a)^3*Sqrt[(b*(c
+ d*x))/(b*c - a*d)] - (1995*ArcSin[Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]])/((d*(a + b*x))/(-(b*c) + a*d))^(5/2)
+ (3780*(a + b*x)*ArcSin[Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]])/(a*((d*(a + b*x))/(-(b*c) + a*d))^(5/2)) - (1575
*(a + b*x)^2*ArcSin[Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]])/(a^2*((d*(a + b*x))/(-(b*c) + a*d))^(5/2)) + (294*(a
+ b*x)^3*ArcSin[Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]])/(a^3*((d*(a + b*x))/(-(b*c) + a*d))^(5/2)) - (168*d^2*(a
+ b*x)^2*Hypergeometric2F1[3/2, 9/2, 11/2, (d*(a + b*x))/(-(b*c) + a*d)])/(b*c - a*d)^2 + (392*d^2*(a + b*x)^3
*Hypergeometric2F1[3/2, 9/2, 11/2, (d*(a + b*x))/(-(b*c) + a*d)])/(a*(b*c - a*d)^2) - (280*d^2*(a + b*x)^4*Hyp
ergeometric2F1[3/2, 9/2, 11/2, (d*(a + b*x))/(-(b*c) + a*d)])/(a^2*(b*c - a*d)^2) + (56*d^2*(a + b*x)^5*Hyperg
eometric2F1[3/2, 9/2, 11/2, (d*(a + b*x))/(-(b*c) + a*d)])/(a^3*(b*c - a*d)^2) - (96*d*(a + b*x)*Hypergeometri
cPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (d*(a + b*x))/(-(b*c) + a*d)])/(-(b*c) + a*d) + (288*d*(a + b*x)^2*Hyperge
ometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (d*(a + b*x))/(-(b*c) + a*d)])/(a*(-(b*c) + a*d)) - (288*d*(a + b*x)
^3*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (d*(a + b*x))/(-(b*c) + a*d)])/(a^2*(-(b*c) + a*d)) + (96*
d*(a + b*x)^4*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (d*(a + b*x))/(-(b*c) + a*d)])/(a^3*(-(b*c) + a
*d))))/(504*b^3*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x]))/(b*d)

________________________________________________________________________________________

Maple [B]  time = 0.035, size = 2748, normalized size = 8.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(-30*x^2*b^6*c^6*((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)-3
0*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))*x^4*a^5*b^2*d^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))*x^4*b^7*c^5*d^2+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))*x^3*a^6*b*d
^7+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))*x^3*b^7*c^6*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))*x*a^7*c*d^6+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))*x*a*b^6*c^7-45*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+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-30*x^2
*a^6*d^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-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))*x^3*a*b^6*c^5*d^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))
*x^2*a^6*b*c*d^6-135*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))*x^2*a^5*b^2*c
^2*d^5+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))*x^2*a^4*b^3*c^3*d^4+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))*x^2*a^3*b^4*c^4*d^3-135*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))*x^2*a^2*b^5*c^5*d^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))*x^2*a*b^6*c^6*d-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))*x*a^6*b*c^2*d^5-30*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))*x*a^5*b^2*c^3*d^4+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))*x*a^4*b^3*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))
*x*a^3*b^4*c^5*d^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))*x*a^2*b^5*c^
6*d-40*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*b^6*c^5*d+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-40
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*a^5*b*d^6-6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^4*a^4*b^2*d^6-6*((b
*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^4*b^6*c^4*d^2-60*x*a^6*c*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-60*x*a*b^5
*c^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-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))*x^4*a^4*b^3*c*d^6+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))*x^4
*a^3*b^4*c^2*d^5+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))*x^4*a^2*b^5*c^
3*d^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))*x^4*a*b^6*c^4*d^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))*x^3*a^5*b^2*c*d^6-30*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))*x^3*a^4*b^3*c^2*d^5+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))*x^3*a^3*b^4*c^3*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))*x^2*a^7*d^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))*x^2*b^7*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*c^2*d^5+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^7-30*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))*x^3*a^2*b^5*c^4*d^3+24*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)*x^4*a^3*b^3*c*d^5-36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^4*a^2*b^4*c^2*d^4+174*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a^4*b^2*c^2*d^4-96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a^3*b^3*c^3*d^3+174
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a^2*b^4*c^4*d^2+96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*a*b^5*c^4*
d^2+120*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^2*b^4*c^5*d-24*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*a^2*b^4
*c^3*d^3+24*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^4*a*b^5*c^3*d^3+96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*a
^4*b^2*c*d^5-24*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*a^3*b^3*c^2*d^4+120*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)*x*a^5*b*c^2*d^4+36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^4*b^2*c^3*d^3+36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)*x*a^3*b^3*c^4*d^2)/((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^3/b^3

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 21.184, size = 4933, normalized size = 14.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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

[Out]

Timed out

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Giac [B]  time = 3.36162, size = 1717, normalized size = 5.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*sqrt(b*d)*a
^4*b^5*c^3 - 37*sqrt(b*d)*a^5*b^4*c^2*d + 29*sqrt(b*d)*a^6*b^3*c*d^2 - 7*sqrt(b*d)*a^7*b^2*d^3 - 30*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^3*c^2 + 42*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^2*c*d - 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^6*b*d^2 + 15*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*b*c - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^5*d)/((
b^5*c^3*abs(b) - 3*a*b^4*c^2*d*abs(b) + 3*a^2*b^3*c*d^2*abs(b) - a^3*b^2*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*(sqrt(b*d)*b*c + sqrt(b*d)*a*d)*log((sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(b^3*d^4*abs(b))