3.794 \(\int \frac{x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=377 \[ -\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right )}{96 b^5 d (b c-a d)}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (231 a^2 d^2+2 b d x (59 b c-99 a d)-156 a b c d+5 b^2 c^2\right )}{24 b^4 d (b c-a d)}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right )}{64 b^6 d}-\frac{5 (b c-a d) \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{13/2} d^{3/2}}-\frac{2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}} \]

[Out]

(-5*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^6*d) - (5*(b
^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*b^5*d*(b*c - a*d))
- (2*x^3*(c + d*x)^(5/2))/(3*b*(a + b*x)^(3/2)) - (2*(6*b*c - 11*a*d)*x^2*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)
*Sqrt[a + b*x]) + (Sqrt[a + b*x]*(c + d*x)^(5/2)*(5*b^2*c^2 - 156*a*b*c*d + 231*a^2*d^2 + 2*b*d*(59*b*c - 99*a
*d)*x))/(24*b^4*d*(b*c - a*d)) - (5*(b*c - a*d)*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*Arc
Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(13/2)*d^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.361397, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {97, 150, 147, 50, 63, 217, 206} \[ -\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right )}{96 b^5 d (b c-a d)}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (231 a^2 d^2+2 b d x (59 b c-99 a d)-156 a b c d+5 b^2 c^2\right )}{24 b^4 d (b c-a d)}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right )}{64 b^6 d}-\frac{5 (b c-a d) \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{13/2} d^{3/2}}-\frac{2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(c + d*x)^(5/2))/(a + b*x)^(5/2),x]

[Out]

(-5*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*b^6*d) - (5*(b
^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*b^5*d*(b*c - a*d))
- (2*x^3*(c + d*x)^(5/2))/(3*b*(a + b*x)^(3/2)) - (2*(6*b*c - 11*a*d)*x^2*(c + d*x)^(5/2))/(3*b^2*(b*c - a*d)
*Sqrt[a + b*x]) + (Sqrt[a + b*x]*(c + d*x)^(5/2)*(5*b^2*c^2 - 156*a*b*c*d + 231*a^2*d^2 + 2*b*d*(59*b*c - 99*a
*d)*x))/(24*b^4*d*(b*c - a*d)) - (5*(b*c - a*d)*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*Arc
Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(13/2)*d^(3/2))

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 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^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx &=-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}+\frac{2 \int \frac{x^2 (c+d x)^{3/2} \left (3 c+\frac{11 d x}{2}\right )}{(a+b x)^{3/2}} \, dx}{3 b}\\ &=-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (6 b c-11 a d) x^2 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{4 \int \frac{x (c+d x)^{3/2} \left (c (6 b c-11 a d)+\frac{1}{4} d (59 b c-99 a d) x\right )}{\sqrt{a+b x}} \, dx}{3 b^2 (b c-a d)}\\ &=-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (6 b c-11 a d) x^2 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (5 b^2 c^2-156 a b c d+231 a^2 d^2+2 b d (59 b c-99 a d) x\right )}{24 b^4 d (b c-a d)}-\frac{\left (5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right )\right ) \int \frac{(c+d x)^{3/2}}{\sqrt{a+b x}} \, dx}{48 b^4 d (b c-a d)}\\ &=-\frac{5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^5 d (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (6 b c-11 a d) x^2 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (5 b^2 c^2-156 a b c d+231 a^2 d^2+2 b d (59 b c-99 a d) x\right )}{24 b^4 d (b c-a d)}-\frac{\left (5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right )\right ) \int \frac{\sqrt{c+d x}}{\sqrt{a+b x}} \, dx}{64 b^5 d}\\ &=-\frac{5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^6 d}-\frac{5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^5 d (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (6 b c-11 a d) x^2 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (5 b^2 c^2-156 a b c d+231 a^2 d^2+2 b d (59 b c-99 a d) x\right )}{24 b^4 d (b c-a d)}-\frac{\left (5 (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b^6 d}\\ &=-\frac{5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^6 d}-\frac{5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^5 d (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (6 b c-11 a d) x^2 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (5 b^2 c^2-156 a b c d+231 a^2 d^2+2 b d (59 b c-99 a d) x\right )}{24 b^4 d (b c-a d)}-\frac{\left (5 (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\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 )}{64 b^7 d}\\ &=-\frac{5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^6 d}-\frac{5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^5 d (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (6 b c-11 a d) x^2 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (5 b^2 c^2-156 a b c d+231 a^2 d^2+2 b d (59 b c-99 a d) x\right )}{24 b^4 d (b c-a d)}-\frac{\left (5 (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\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 )}{64 b^7 d}\\ &=-\frac{5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{64 b^6 d}-\frac{5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt{a+b x} (c+d x)^{3/2}}{96 b^5 d (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac{2 (6 b c-11 a d) x^2 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt{a+b x}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (5 b^2 c^2-156 a b c d+231 a^2 d^2+2 b d (59 b c-99 a d) x\right )}{24 b^4 d (b c-a d)}-\frac{5 (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{13/2} d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.14846, size = 302, normalized size = 0.8 \[ \frac{\sqrt{c+d x} \left (\frac{\sqrt{d} \left (-21 a^3 b^2 d \left (83 c^2-334 c d x+33 d^2 x^2\right )+3 a^2 b^3 \left (-824 c^2 d x+5 c^3+387 c d^2 x^2+66 d^3 x^3\right )+105 a^4 b d^2 (49 c-44 d x)-3465 a^5 d^3-a b^4 x \left (483 c^2 d x-30 c^3+316 c d^2 x^2+88 d^3 x^3\right )+b^5 x^2 \left (118 c^2 d x+15 c^3+136 c d^2 x^2+48 d^3 x^3\right )\right )}{(a+b x)^{3/2}}-\frac{15 \sqrt{b c-a d} \left (-189 a^2 b c d^2+231 a^3 d^3+21 a b^2 c^2 d+b^3 c^3\right ) \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{\sqrt{\frac{b (c+d x)}{b c-a d}}}\right )}{192 b^6 d^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(c + d*x)^(5/2))/(a + b*x)^(5/2),x]

[Out]

(Sqrt[c + d*x]*((Sqrt[d]*(-3465*a^5*d^3 + 105*a^4*b*d^2*(49*c - 44*d*x) - 21*a^3*b^2*d*(83*c^2 - 334*c*d*x + 3
3*d^2*x^2) + b^5*x^2*(15*c^3 + 118*c^2*d*x + 136*c*d^2*x^2 + 48*d^3*x^3) + 3*a^2*b^3*(5*c^3 - 824*c^2*d*x + 38
7*c*d^2*x^2 + 66*d^3*x^3) - a*b^4*x*(-30*c^3 + 483*c^2*d*x + 316*c*d^2*x^2 + 88*d^3*x^3)))/(a + b*x)^(3/2) - (
15*Sqrt[b*c - a*d]*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/
Sqrt[b*c - a*d]])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(192*b^6*d^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.029, size = 1366, 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^3*(d*x+c)^(5/2)/(b*x+a)^(5/2),x)

[Out]

1/384*(d*x+c)^(1/2)*(272*x^4*b^5*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+60*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)*x*a*b^4*c^3-3486*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^2*c^2*d-12600*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*d^3+6300*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^2*d^2-9240*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^4*b*d^3+10290*(b*
d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b*c*d^2-6300*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*d^3+3150*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^2*d^2-1386*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^3*b^2*d^3-300*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^5*c^3*d-600*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^3*d+396*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^2*b^3*d^3+
236*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*b^5*c^2*d+2322*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a^2*b^3*c*d
^2+14028*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^3*b^2*c*d^2-632*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a*b^4
*c*d^2-966*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a*b^4*c^2*d-4944*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*
b^3*c^2*d+6930*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*d^4-176*x^4
*a*b^4*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6300*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*d^3+3150*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^2+96*x^5*b^5*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-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*a*b^5*c^4-300*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+30*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*b^5*c^3+30*(b*d)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)*a^2*b^3*c^3+3465*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*d^4+3465*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^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*b^6*c^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))*a^2*b^4*c^4-6930*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^5*d^
3)/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(b*x+a)^(3/2)/b^6/d

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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^3*(d*x+c)^(5/2)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 16.7632, size = 2336, normalized size = 6.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d*x+c)^(5/2)/(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/768*(15*(a^2*b^4*c^4 + 20*a^3*b^3*c^3*d - 210*a^4*b^2*c^2*d^2 + 420*a^5*b*c*d^3 - 231*a^6*d^4 + (b^6*c^4 +
20*a*b^5*c^3*d - 210*a^2*b^4*c^2*d^2 + 420*a^3*b^3*c*d^3 - 231*a^4*b^2*d^4)*x^2 + 2*(a*b^5*c^4 + 20*a^2*b^4*c
^3*d - 210*a^3*b^3*c^2*d^2 + 420*a^4*b^2*c*d^3 - 231*a^5*b*d^4)*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*(48*b^6*d^4*x^5 + 15*a^2*b^4*c^3*d - 1743*a^3*b^3*c^2*d^2 + 5145*a^4*b^2*c*d^3 - 3465*a^5*b*d^4 + 8*(17*b^6*
c*d^3 - 11*a*b^5*d^4)*x^4 + 2*(59*b^6*c^2*d^2 - 158*a*b^5*c*d^3 + 99*a^2*b^4*d^4)*x^3 + 3*(5*b^6*c^3*d - 161*a
*b^5*c^2*d^2 + 387*a^2*b^4*c*d^3 - 231*a^3*b^3*d^4)*x^2 + 6*(5*a*b^5*c^3*d - 412*a^2*b^4*c^2*d^2 + 1169*a^3*b^
3*c*d^3 - 770*a^4*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^9*d^2*x^2 + 2*a*b^8*d^2*x + a^2*b^7*d^2), 1/384*
(15*(a^2*b^4*c^4 + 20*a^3*b^3*c^3*d - 210*a^4*b^2*c^2*d^2 + 420*a^5*b*c*d^3 - 231*a^6*d^4 + (b^6*c^4 + 20*a*b^
5*c^3*d - 210*a^2*b^4*c^2*d^2 + 420*a^3*b^3*c*d^3 - 231*a^4*b^2*d^4)*x^2 + 2*(a*b^5*c^4 + 20*a^2*b^4*c^3*d - 2
10*a^3*b^3*c^2*d^2 + 420*a^4*b^2*c*d^3 - 231*a^5*b*d^4)*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*(48*b^6*d^4*x^5 + 15*a^2*
b^4*c^3*d - 1743*a^3*b^3*c^2*d^2 + 5145*a^4*b^2*c*d^3 - 3465*a^5*b*d^4 + 8*(17*b^6*c*d^3 - 11*a*b^5*d^4)*x^4 +
2*(59*b^6*c^2*d^2 - 158*a*b^5*c*d^3 + 99*a^2*b^4*d^4)*x^3 + 3*(5*b^6*c^3*d - 161*a*b^5*c^2*d^2 + 387*a^2*b^4*
c*d^3 - 231*a^3*b^3*d^4)*x^2 + 6*(5*a*b^5*c^3*d - 412*a^2*b^4*c^2*d^2 + 1169*a^3*b^3*c*d^3 - 770*a^4*b^2*d^4)*
x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^9*d^2*x^2 + 2*a*b^8*d^2*x + a^2*b^7*d^2)]

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

[Out]

Timed out

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Giac [B]  time = 2.36174, size = 1382, normalized size = 3.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(d*x+c)^(5/2)/(b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)*d^2*abs(b)/b^8 + (17*b^32*c*d
^7*abs(b) - 41*a*b^31*d^8*abs(b))/(b^39*d^6)) + (59*b^33*c^2*d^6*abs(b) - 430*a*b^32*c*d^7*abs(b) + 515*a^2*b^
31*d^8*abs(b))/(b^39*d^6)) + 3*(5*b^34*c^3*d^5*abs(b) - 279*a*b^33*c^2*d^6*abs(b) + 975*a^2*b^32*c*d^7*abs(b)
- 765*a^3*b^31*d^8*abs(b))/(b^39*d^6))*sqrt(b*x + a) + 5/128*(sqrt(b*d)*b^4*c^4*abs(b) + 20*sqrt(b*d)*a*b^3*c^
3*d*abs(b) - 210*sqrt(b*d)*a^2*b^2*c^2*d^2*abs(b) + 420*sqrt(b*d)*a^3*b*c*d^3*abs(b) - 231*sqrt(b*d)*a^4*d^4*a
bs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(b^8*d^2) - 4/3*(9*sqrt(b*d)*a^2
*b^7*c^5*abs(b) - 52*sqrt(b*d)*a^3*b^6*c^4*d*abs(b) + 118*sqrt(b*d)*a^4*b^5*c^3*d^2*abs(b) - 132*sqrt(b*d)*a^5
*b^4*c^2*d^3*abs(b) + 73*sqrt(b*d)*a^6*b^3*c*d^4*abs(b) - 16*sqrt(b*d)*a^7*b^2*d^5*abs(b) - 18*sqrt(b*d)*(sqrt
(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^5*c^4*abs(b) + 84*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^4*c^3*d*abs(b) - 144*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*d^2*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^5*b^2*c*d^3*abs(b) - 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^6*b*d^4*abs(b) + 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^2*b^3*c^3*abs(b) - 36*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^2*c^2*d*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*c*d^2*abs(b) - 18*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^3*ab
s(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^7)