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

Optimal. Leaf size=497 \[ \frac{\sqrt{a+b x} (c+d x)^{3/2} \left (378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4+28 a b^3 c^3 d+3 b^4 c^4\right )}{192 b^6 d^2 (b c-a d)}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-2 b d x \left (1287 a^2 d^2-902 a b c d+15 b^2 c^2\right )-2343 a^2 b c d^2+3003 a^3 d^3+125 a b^2 c^2 d+15 b^3 c^3\right )}{240 b^5 d^2 (b c-a d)}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4+28 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^7 d^2}+\frac{(b c-a d) \left (378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4+28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{15/2} d^{5/2}}-\frac{2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}+\frac{x^2 \sqrt{a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{15 b^3 (b c-a d)}-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}} \]

[Out]

((3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d
*x])/(128*b^7*d^2) + ((3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*Sqr
t[a + b*x]*(c + d*x)^(3/2))/(192*b^6*d^2*(b*c - a*d)) - (2*x^4*(c + d*x)^(5/2))/(3*b*(a + b*x)^(3/2)) - (2*(8*
b*c - 13*a*d)*x^3*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)*Sqrt[a + b*x]) + ((93*b*c - 143*a*d)*x^2*Sqrt[a + b*x]*(
c + d*x)^(5/2))/(15*b^3*(b*c - a*d)) - (Sqrt[a + b*x]*(c + d*x)^(5/2)*(15*b^3*c^3 + 125*a*b^2*c^2*d - 2343*a^2
*b*c*d^2 + 3003*a^3*d^3 - 2*b*d*(15*b^2*c^2 - 902*a*b*c*d + 1287*a^2*d^2)*x))/(240*b^5*d^2*(b*c - a*d)) + ((b*
c - a*d)*(3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*ArcTanh[(Sqrt[d]
*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(15/2)*d^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.564319, antiderivative size = 497, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {97, 150, 153, 147, 50, 63, 217, 206} \[ \frac{\sqrt{a+b x} (c+d x)^{3/2} \left (378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4+28 a b^3 c^3 d+3 b^4 c^4\right )}{192 b^6 d^2 (b c-a d)}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-2 b d x \left (1287 a^2 d^2-902 a b c d+15 b^2 c^2\right )-2343 a^2 b c d^2+3003 a^3 d^3+125 a b^2 c^2 d+15 b^3 c^3\right )}{240 b^5 d^2 (b c-a d)}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4+28 a b^3 c^3 d+3 b^4 c^4\right )}{128 b^7 d^2}+\frac{(b c-a d) \left (378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4+28 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{15/2} d^{5/2}}-\frac{2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}+\frac{x^2 \sqrt{a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{15 b^3 (b c-a d)}-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d
*x])/(128*b^7*d^2) + ((3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*Sqr
t[a + b*x]*(c + d*x)^(3/2))/(192*b^6*d^2*(b*c - a*d)) - (2*x^4*(c + d*x)^(5/2))/(3*b*(a + b*x)^(3/2)) - (2*(8*
b*c - 13*a*d)*x^3*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)*Sqrt[a + b*x]) + ((93*b*c - 143*a*d)*x^2*Sqrt[a + b*x]*(
c + d*x)^(5/2))/(15*b^3*(b*c - a*d)) - (Sqrt[a + b*x]*(c + d*x)^(5/2)*(15*b^3*c^3 + 125*a*b^2*c^2*d - 2343*a^2
*b*c*d^2 + 3003*a^3*d^3 - 2*b*d*(15*b^2*c^2 - 902*a*b*c*d + 1287*a^2*d^2)*x))/(240*b^5*d^2*(b*c - a*d)) + ((b*
c - a*d)*(3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*ArcTanh[(Sqrt[d]
*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(128*b^(15/2)*d^(5/2))

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 147

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx &=-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}+\frac{2 \int \frac{x^3 (c+d x)^{3/2} \left (4 c+\frac{13 d x}{2}\right )}{(a+b x)^{3/2}} \, dx}{3 b}\\ &=-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (8 b c-13 a d) x^3 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{4 \int \frac{x^2 (c+d x)^{3/2} \left (\frac{3}{2} c (8 b c-13 a d)+\frac{1}{4} d (93 b c-143 a d) x\right )}{\sqrt{a+b x}} \, dx}{3 b^2 (b c-a d)}\\ &=-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (8 b c-13 a d) x^3 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{(93 b c-143 a d) x^2 \sqrt{a+b x} (c+d x)^{5/2}}{15 b^3 (b c-a d)}+\frac{4 \int \frac{x (c+d x)^{3/2} \left (-\frac{1}{2} a c d (93 b c-143 a d)+\frac{1}{8} d \left (15 b^2 c^2-902 a b c d+1287 a^2 d^2\right ) x\right )}{\sqrt{a+b x}} \, dx}{15 b^3 d (b c-a d)}\\ &=-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (8 b c-13 a d) x^3 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{(93 b c-143 a d) x^2 \sqrt{a+b x} (c+d x)^{5/2}}{15 b^3 (b c-a d)}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (15 b^3 c^3+125 a b^2 c^2 d-2343 a^2 b c d^2+3003 a^3 d^3-2 b d \left (15 b^2 c^2-902 a b c d+1287 a^2 d^2\right ) x\right )}{240 b^5 d^2 (b c-a d)}+\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \int \frac{(c+d x)^{3/2}}{\sqrt{a+b x}} \, dx}{96 b^5 d^2 (b c-a d)}\\ &=\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^6 d^2 (b c-a d)}-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (8 b c-13 a d) x^3 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{(93 b c-143 a d) x^2 \sqrt{a+b x} (c+d x)^{5/2}}{15 b^3 (b c-a d)}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (15 b^3 c^3+125 a b^2 c^2 d-2343 a^2 b c d^2+3003 a^3 d^3-2 b d \left (15 b^2 c^2-902 a b c d+1287 a^2 d^2\right ) x\right )}{240 b^5 d^2 (b c-a d)}+\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx}{128 b^6 d^2}\\ &=\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^7 d^2}+\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^6 d^2 (b c-a d)}-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (8 b c-13 a d) x^3 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{(93 b c-143 a d) x^2 \sqrt{a+b x} (c+d x)^{5/2}}{15 b^3 (b c-a d)}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (15 b^3 c^3+125 a b^2 c^2 d-2343 a^2 b c d^2+3003 a^3 d^3-2 b d \left (15 b^2 c^2-902 a b c d+1287 a^2 d^2\right ) x\right )}{240 b^5 d^2 (b c-a d)}+\frac{\left ((b c-a d) \left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 b^7 d^2}\\ &=\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^7 d^2}+\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^6 d^2 (b c-a d)}-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (8 b c-13 a d) x^3 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{(93 b c-143 a d) x^2 \sqrt{a+b x} (c+d x)^{5/2}}{15 b^3 (b c-a d)}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (15 b^3 c^3+125 a b^2 c^2 d-2343 a^2 b c d^2+3003 a^3 d^3-2 b d \left (15 b^2 c^2-902 a b c d+1287 a^2 d^2\right ) x\right )}{240 b^5 d^2 (b c-a d)}+\frac{\left ((b c-a d) \left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{128 b^8 d^2}\\ &=\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^7 d^2}+\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^6 d^2 (b c-a d)}-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (8 b c-13 a d) x^3 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{(93 b c-143 a d) x^2 \sqrt{a+b x} (c+d x)^{5/2}}{15 b^3 (b c-a d)}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (15 b^3 c^3+125 a b^2 c^2 d-2343 a^2 b c d^2+3003 a^3 d^3-2 b d \left (15 b^2 c^2-902 a b c d+1287 a^2 d^2\right ) x\right )}{240 b^5 d^2 (b c-a d)}+\frac{\left ((b c-a d) \left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{128 b^8 d^2}\\ &=\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{128 b^7 d^2}+\frac{\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt{a+b x} (c+d x)^{3/2}}{192 b^6 d^2 (b c-a d)}-\frac{2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (8 b c-13 a d) x^3 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{(93 b c-143 a d) x^2 \sqrt{a+b x} (c+d x)^{5/2}}{15 b^3 (b c-a d)}-\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (15 b^3 c^3+125 a b^2 c^2 d-2343 a^2 b c d^2+3003 a^3 d^3-2 b d \left (15 b^2 c^2-902 a b c d+1287 a^2 d^2\right ) x\right )}{240 b^5 d^2 (b c-a d)}+\frac{(b c-a d) \left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{15/2} d^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.85165, size = 391, normalized size = 0.79 \[ \frac{\sqrt{c+d x} \left (\frac{\sqrt{d} \left (21 a^4 b^2 d^2 \left (1304 c^2-4642 c d x+429 d^2 x^2\right )-6 a^3 b^3 d \left (-6441 c^2 d x+65 c^3+2673 c d^2 x^2+429 d^3 x^3\right )+a^2 b^4 \left (7404 c^2 d^2 x^2-750 c^3 d x-45 c^4+4378 c d^3 x^3+1144 d^4 x^4\right )+2310 a^5 b d^3 (26 d x-31 c)+45045 a^6 d^4-2 a b^5 x \left (917 c^2 d^2 x^2+165 c^3 d x+45 c^4+944 c d^3 x^3+312 d^4 x^4\right )+3 b^6 x^2 \left (248 c^2 d^2 x^2+10 c^3 d x-15 c^4+336 c d^3 x^3+128 d^4 x^4\right )\right )}{(a+b x)^{3/2}}+\frac{15 \sqrt{b c-a d} \left (378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4+28 a b^3 c^3 d+3 b^4 c^4\right ) \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{1920 b^7 d^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x]*((Sqrt[d]*(45045*a^6*d^4 + 2310*a^5*b*d^3*(-31*c + 26*d*x) + 21*a^4*b^2*d^2*(1304*c^2 - 4642*c*
d*x + 429*d^2*x^2) - 6*a^3*b^3*d*(65*c^3 - 6441*c^2*d*x + 2673*c*d^2*x^2 + 429*d^3*x^3) + 3*b^6*x^2*(-15*c^4 +
10*c^3*d*x + 248*c^2*d^2*x^2 + 336*c*d^3*x^3 + 128*d^4*x^4) - 2*a*b^5*x*(45*c^4 + 165*c^3*d*x + 917*c^2*d^2*x
^2 + 944*c*d^3*x^3 + 312*d^4*x^4) + a^2*b^4*(-45*c^4 - 750*c^3*d*x + 7404*c^2*d^2*x^2 + 4378*c*d^3*x^3 + 1144*
d^4*x^4)))/(a + b*x)^(3/2) + (15*Sqrt[b*c - a*d]*(3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*
b*c*d^3 + 3003*a^4*d^4)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(1
920*b^7*d^(5/2))

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Maple [B]  time = 0.038, size = 1762, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840*(d*x+c)^(1/2)*(45045*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*d^5+90090*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*d^5-86625
*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^4+47250*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^3-5250*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^2+90*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*b^6
*c^4+90*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^4*c^4-768*x^6*b^6*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-90
*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^5-375*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+5148*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)*x^3*a^3*b^3*d^4+45045*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^5-
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^2*b^7*c^5-45*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^2*b^5*c^5-90090*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)*a^6*d^4-60*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*b^6*c^3*d+180*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a*b
^5*c^4+780*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^3*c^3*d-375*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^4*d-750*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^4*d-86625*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*d^4+47250*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^2*d^3-5250*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^
3*d^2-18018*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^4*b^2*d^4-173250*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*d^4+94500*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^2*d^3-10500*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^3*d^2-120120*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^5*b*d^4+143220*(b*d)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)*a^5*b*c*d^3-54768*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^2*c^2*d^2+1248*x^5*a*b^5*d^4*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2016*x^5*b^6*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2288*x^4*a^2*b^4*d^
4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1488*x^4*b^6*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1500*(b*d)^(1/2
)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*b^4*c^3*d+32076*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^3*b^3*c*d^3-14808*(b
*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^2*b^4*c^2*d^2+194964*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^4*b^2*c*d
^3-77292*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^3*b^3*c^2*d^2-8756*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^
2*b^4*c*d^3+3668*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a*b^5*c^2*d^2+660*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)
*x^2*a*b^5*c^3*d+3776*x^4*a*b^5*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)
/(b*x+a)^(3/2)/b^7/d^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*(d*x+c)^(5/2)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 22.7656, size = 3009, normalized size = 6.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/7680*(15*(3*a^2*b^5*c^5 + 25*a^3*b^4*c^4*d + 350*a^4*b^3*c^3*d^2 - 3150*a^5*b^2*c^2*d^3 + 5775*a^6*b*c*d^4
- 3003*a^7*d^5 + (3*b^7*c^5 + 25*a*b^6*c^4*d + 350*a^2*b^5*c^3*d^2 - 3150*a^3*b^4*c^2*d^3 + 5775*a^4*b^3*c*d^
4 - 3003*a^5*b^2*d^5)*x^2 + 2*(3*a*b^6*c^5 + 25*a^2*b^5*c^4*d + 350*a^3*b^4*c^3*d^2 - 3150*a^4*b^3*c^2*d^3 + 5
775*a^5*b^2*c*d^4 - 3003*a^6*b*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*(384*b^7*d^5*x^6 - 45*a^2*
b^5*c^4*d - 390*a^3*b^4*c^3*d^2 + 27384*a^4*b^3*c^2*d^3 - 71610*a^5*b^2*c*d^4 + 45045*a^6*b*d^5 + 48*(21*b^7*c
*d^4 - 13*a*b^6*d^5)*x^5 + 8*(93*b^7*c^2*d^3 - 236*a*b^6*c*d^4 + 143*a^2*b^5*d^5)*x^4 + 2*(15*b^7*c^3*d^2 - 91
7*a*b^6*c^2*d^3 + 2189*a^2*b^5*c*d^4 - 1287*a^3*b^4*d^5)*x^3 - 3*(15*b^7*c^4*d + 110*a*b^6*c^3*d^2 - 2468*a^2*
b^5*c^2*d^3 + 5346*a^3*b^4*c*d^4 - 3003*a^4*b^3*d^5)*x^2 - 6*(15*a*b^6*c^4*d + 125*a^2*b^5*c^3*d^2 - 6441*a^3*
b^4*c^2*d^3 + 16247*a^4*b^3*c*d^4 - 10010*a^5*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^10*d^3*x^2 + 2*a*b^9
*d^3*x + a^2*b^8*d^3), -1/3840*(15*(3*a^2*b^5*c^5 + 25*a^3*b^4*c^4*d + 350*a^4*b^3*c^3*d^2 - 3150*a^5*b^2*c^2*
d^3 + 5775*a^6*b*c*d^4 - 3003*a^7*d^5 + (3*b^7*c^5 + 25*a*b^6*c^4*d + 350*a^2*b^5*c^3*d^2 - 3150*a^3*b^4*c^2*d
^3 + 5775*a^4*b^3*c*d^4 - 3003*a^5*b^2*d^5)*x^2 + 2*(3*a*b^6*c^5 + 25*a^2*b^5*c^4*d + 350*a^3*b^4*c^3*d^2 - 31
50*a^4*b^3*c^2*d^3 + 5775*a^5*b^2*c*d^4 - 3003*a^6*b*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*(384*b^7*d^5*x^6 - 45*a
^2*b^5*c^4*d - 390*a^3*b^4*c^3*d^2 + 27384*a^4*b^3*c^2*d^3 - 71610*a^5*b^2*c*d^4 + 45045*a^6*b*d^5 + 48*(21*b^
7*c*d^4 - 13*a*b^6*d^5)*x^5 + 8*(93*b^7*c^2*d^3 - 236*a*b^6*c*d^4 + 143*a^2*b^5*d^5)*x^4 + 2*(15*b^7*c^3*d^2 -
917*a*b^6*c^2*d^3 + 2189*a^2*b^5*c*d^4 - 1287*a^3*b^4*d^5)*x^3 - 3*(15*b^7*c^4*d + 110*a*b^6*c^3*d^2 - 2468*a
^2*b^5*c^2*d^3 + 5346*a^3*b^4*c*d^4 - 3003*a^4*b^3*d^5)*x^2 - 6*(15*a*b^6*c^4*d + 125*a^2*b^5*c^3*d^2 - 6441*a
^3*b^4*c^2*d^3 + 16247*a^4*b^3*c*d^4 - 10010*a^5*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^10*d^3*x^2 + 2*a*
b^9*d^3*x + a^2*b^8*d^3)]

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

[Out]

Timed out

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Giac [B]  time = 5.89035, size = 1530, normalized size = 3.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)*d^2*abs(b)/b^9 + (21*b^45
*c*d^9*abs(b) - 61*a*b^44*d^10*abs(b))/(b^53*d^8)) + (93*b^46*c^2*d^8*abs(b) - 866*a*b^45*c*d^9*abs(b) + 1253*
a^2*b^44*d^10*abs(b))/(b^53*d^8)) + 5*(3*b^47*c^3*d^7*abs(b) - 481*a*b^46*c^2*d^8*abs(b) + 2201*a^2*b^45*c*d^9
*abs(b) - 2107*a^3*b^44*d^10*abs(b))/(b^53*d^8))*(b*x + a) - 15*(3*b^48*c^4*d^6*abs(b) + 28*a*b^47*c^3*d^7*abs
(b) - 1158*a^2*b^46*c^2*d^8*abs(b) + 3372*a^3*b^45*c*d^9*abs(b) - 2373*a^4*b^44*d^10*abs(b))/(b^53*d^8))*sqrt(
b*x + a) + 4/3*(12*sqrt(b*d)*a^3*b^7*c^5*abs(b) - 67*sqrt(b*d)*a^4*b^6*c^4*d*abs(b) + 148*sqrt(b*d)*a^5*b^5*c^
3*d^2*abs(b) - 162*sqrt(b*d)*a^6*b^4*c^2*d^3*abs(b) + 88*sqrt(b*d)*a^7*b^3*c*d^4*abs(b) - 19*sqrt(b*d)*a^8*b^2
*d^5*abs(b) - 24*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^5*c^4*abs(b
) + 108*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^4*c^3*d*abs(b) - 180
*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^3*c^2*d^2*abs(b) + 132*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^2*c*d^3*abs(b) - 36*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b*d^4*abs(b) + 12*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^3*c^3*abs(b) - 45*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^2*c^2*d*abs(b) + 54*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*b*c*d^2*abs(b) - 21*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b
*x + a)*b*d - a*b*d))^4*a^6*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*b^8) - 1/256*(3*sqrt(b*d)*b^5*c^5*abs(b) + 25*sqrt(b*d)*a*b^4*c^4*d*abs(b) + 350*sqrt(b*d)*a^
2*b^3*c^3*d^2*abs(b) - 3150*sqrt(b*d)*a^3*b^2*c^2*d^3*abs(b) + 5775*sqrt(b*d)*a^4*b*c*d^4*abs(b) - 3003*sqrt(b
*d)*a^5*d^5*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(b^9*d^3)