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

Optimal. Leaf size=388 \[ \frac{b \sqrt{c+d x} \left (581 a^2 b c d^2-5 a^3 d^3-1715 a b^2 c^2 d+1155 b^3 c^3\right )}{64 a^6 c \sqrt{a+b x}}+\frac{b \sqrt{c+d x} (b c-a d) \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{64 a^5 c (a+b x)^{3/2}}+\frac{\sqrt{c+d x} (b c-a d) \left (5 a^2 d^2-156 a b c d+231 b^2 c^2\right )}{64 a^4 c x (a+b x)^{3/2}}-\frac{5 (b c-a d) \left (21 a^2 b c d^2+a^3 d^3-189 a b^2 c^2 d+231 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{13/2} c^{3/2}}-\frac{\sqrt{c+d x} (99 b c-59 a d) (b c-a d)}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{11 c \sqrt{c+d x} (b c-a d)}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}} \]

[Out]

(b*(b*c - a*d)*(385*b^2*c^2 - 238*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x])/(64*a^5*c*(a + b*x)^(3/2)) + (11*c*(b*c
- a*d)*Sqrt[c + d*x])/(24*a^2*x^3*(a + b*x)^(3/2)) - ((99*b*c - 59*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(96*a^3*x^2
*(a + b*x)^(3/2)) + ((b*c - a*d)*(231*b^2*c^2 - 156*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x])/(64*a^4*c*x*(a + b*x)^
(3/2)) + (b*(1155*b^3*c^3 - 1715*a*b^2*c^2*d + 581*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt[c + d*x])/(64*a^6*c*Sqrt[a +
b*x]) - (c*(c + d*x)^(3/2))/(4*a*x^4*(a + b*x)^(3/2)) - (5*(b*c - a*d)*(231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2
*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(13/2)*c^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.525801, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {98, 149, 151, 152, 12, 93, 208} \[ \frac{b \sqrt{c+d x} \left (581 a^2 b c d^2-5 a^3 d^3-1715 a b^2 c^2 d+1155 b^3 c^3\right )}{64 a^6 c \sqrt{a+b x}}+\frac{b \sqrt{c+d x} (b c-a d) \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{64 a^5 c (a+b x)^{3/2}}+\frac{\sqrt{c+d x} (b c-a d) \left (5 a^2 d^2-156 a b c d+231 b^2 c^2\right )}{64 a^4 c x (a+b x)^{3/2}}-\frac{5 (b c-a d) \left (21 a^2 b c d^2+a^3 d^3-189 a b^2 c^2 d+231 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{13/2} c^{3/2}}-\frac{\sqrt{c+d x} (99 b c-59 a d) (b c-a d)}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{11 c \sqrt{c+d x} (b c-a d)}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(b*c - a*d)*(385*b^2*c^2 - 238*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x])/(64*a^5*c*(a + b*x)^(3/2)) + (11*c*(b*c
- a*d)*Sqrt[c + d*x])/(24*a^2*x^3*(a + b*x)^(3/2)) - ((99*b*c - 59*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(96*a^3*x^2
*(a + b*x)^(3/2)) + ((b*c - a*d)*(231*b^2*c^2 - 156*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x])/(64*a^4*c*x*(a + b*x)^
(3/2)) + (b*(1155*b^3*c^3 - 1715*a*b^2*c^2*d + 581*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt[c + d*x])/(64*a^6*c*Sqrt[a +
b*x]) - (c*(c + d*x)^(3/2))/(4*a*x^4*(a + b*x)^(3/2)) - (5*(b*c - a*d)*(231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2
*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(13/2)*c^(3/2))

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 149

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] && IntegerQ[m]

Rule 151

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx &=-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}-\frac{\int \frac{\sqrt{c+d x} \left (\frac{11}{2} c (b c-a d)+4 d (b c-a d) x\right )}{x^4 (a+b x)^{5/2}} \, dx}{4 a}\\ &=\frac{11 c (b c-a d) \sqrt{c+d x}}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}-\frac{\int \frac{-\frac{1}{4} c (99 b c-59 a d) (b c-a d)-2 d (11 b c-6 a d) (b c-a d) x}{x^3 (a+b x)^{5/2} \sqrt{c+d x}} \, dx}{12 a^2}\\ &=\frac{11 c (b c-a d) \sqrt{c+d x}}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{(99 b c-59 a d) (b c-a d) \sqrt{c+d x}}{96 a^3 x^2 (a+b x)^{3/2}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}+\frac{\int \frac{-\frac{3}{8} c (b c-a d) \left (231 b^2 c^2-156 a b c d+5 a^2 d^2\right )-\frac{3}{4} b c d (99 b c-59 a d) (b c-a d) x}{x^2 (a+b x)^{5/2} \sqrt{c+d x}} \, dx}{24 a^3 c}\\ &=\frac{11 c (b c-a d) \sqrt{c+d x}}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{(99 b c-59 a d) (b c-a d) \sqrt{c+d x}}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{(b c-a d) \left (231 b^2 c^2-156 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^4 c x (a+b x)^{3/2}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}-\frac{\int \frac{-\frac{15}{16} c (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )-\frac{3}{4} b c d (b c-a d) \left (231 b^2 c^2-156 a b c d+5 a^2 d^2\right ) x}{x (a+b x)^{5/2} \sqrt{c+d x}} \, dx}{24 a^4 c^2}\\ &=\frac{b (b c-a d) \left (385 b^2 c^2-238 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^5 c (a+b x)^{3/2}}+\frac{11 c (b c-a d) \sqrt{c+d x}}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{(99 b c-59 a d) (b c-a d) \sqrt{c+d x}}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{(b c-a d) \left (231 b^2 c^2-156 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^4 c x (a+b x)^{3/2}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}-\frac{\int \frac{-\frac{45}{32} c (b c-a d)^2 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )-\frac{9}{16} b c d (b c-a d)^2 \left (385 b^2 c^2-238 a b c d+5 a^2 d^2\right ) x}{x (a+b x)^{3/2} \sqrt{c+d x}} \, dx}{36 a^5 c^2 (b c-a d)}\\ &=\frac{b (b c-a d) \left (385 b^2 c^2-238 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^5 c (a+b x)^{3/2}}+\frac{11 c (b c-a d) \sqrt{c+d x}}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{(99 b c-59 a d) (b c-a d) \sqrt{c+d x}}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{(b c-a d) \left (231 b^2 c^2-156 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^4 c x (a+b x)^{3/2}}+\frac{b \left (1155 b^3 c^3-1715 a b^2 c^2 d+581 a^2 b c d^2-5 a^3 d^3\right ) \sqrt{c+d x}}{64 a^6 c \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}-\frac{\int -\frac{45 c (b c-a d)^3 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )}{64 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{18 a^6 c^2 (b c-a d)^2}\\ &=\frac{b (b c-a d) \left (385 b^2 c^2-238 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^5 c (a+b x)^{3/2}}+\frac{11 c (b c-a d) \sqrt{c+d x}}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{(99 b c-59 a d) (b c-a d) \sqrt{c+d x}}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{(b c-a d) \left (231 b^2 c^2-156 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^4 c x (a+b x)^{3/2}}+\frac{b \left (1155 b^3 c^3-1715 a b^2 c^2 d+581 a^2 b c d^2-5 a^3 d^3\right ) \sqrt{c+d x}}{64 a^6 c \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}+\frac{\left (5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a^6 c}\\ &=\frac{b (b c-a d) \left (385 b^2 c^2-238 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^5 c (a+b x)^{3/2}}+\frac{11 c (b c-a d) \sqrt{c+d x}}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{(99 b c-59 a d) (b c-a d) \sqrt{c+d x}}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{(b c-a d) \left (231 b^2 c^2-156 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^4 c x (a+b x)^{3/2}}+\frac{b \left (1155 b^3 c^3-1715 a b^2 c^2 d+581 a^2 b c d^2-5 a^3 d^3\right ) \sqrt{c+d x}}{64 a^6 c \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}+\frac{\left (5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{64 a^6 c}\\ &=\frac{b (b c-a d) \left (385 b^2 c^2-238 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^5 c (a+b x)^{3/2}}+\frac{11 c (b c-a d) \sqrt{c+d x}}{24 a^2 x^3 (a+b x)^{3/2}}-\frac{(99 b c-59 a d) (b c-a d) \sqrt{c+d x}}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{(b c-a d) \left (231 b^2 c^2-156 a b c d+5 a^2 d^2\right ) \sqrt{c+d x}}{64 a^4 c x (a+b x)^{3/2}}+\frac{b \left (1155 b^3 c^3-1715 a b^2 c^2 d+581 a^2 b c d^2-5 a^3 d^3\right ) \sqrt{c+d x}}{64 a^6 c \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}}-\frac{5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{13/2} c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.489806, size = 256, normalized size = 0.66 \[ \frac{-2 a^{7/2} x^2 (c+d x)^{7/2} \left (-a^2 d^2-26 a b c d+99 b^2 c^2\right )+x^3 \left (21 a^2 b c d^2+a^3 d^3-189 a b^2 c^2 d+231 b^3 c^3\right ) \left (3 a^{5/2} (c+d x)^{5/2}+5 x (b c-a d) \left (\sqrt{a} \sqrt{c+d x} (4 a c+a d x+3 b c x)-3 c^{3/2} (a+b x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )\right )+8 a^{9/2} c x (c+d x)^{7/2} (a d+11 b c)-48 a^{11/2} c^2 (c+d x)^{7/2}}{192 a^{13/2} c^3 x^4 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-48*a^(11/2)*c^2*(c + d*x)^(7/2) + 8*a^(9/2)*c*(11*b*c + a*d)*x*(c + d*x)^(7/2) - 2*a^(7/2)*(99*b^2*c^2 - 26*
a*b*c*d - a^2*d^2)*x^2*(c + d*x)^(7/2) + (231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2*b*c*d^2 + a^3*d^3)*x^3*(3*a^(
5/2)*(c + d*x)^(5/2) + 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c + d*x]*(4*a*c + 3*b*c*x + a*d*x) - 3*c^(3/2)*(a + b*x)^
(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])))/(192*a^(13/2)*c^3*x^4*(a + b*x)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.033, size = 1377, 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((d*x+c)^(5/2)/x^5/(b*x+a)^(5/2),x)

[Out]

1/384*(d*x+c)^(1/2)*(12600*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^2*b^4*c^3*d+3
00*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^5*b*c*d^3-3150*ln((a*d*x+b*c*x+2*(a*c
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^4*b^2*c^2*d^2+6300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)+2*a*c)/x)*x^4*a^3*b^3*c^3*d-30*x^5*a^3*b^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-60*x^4*a^4*b*d^3
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+9240*x^4*a*b^4*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1386*x^3*a^2*b^3*c
^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-236*x^2*a^5*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-396*x^2*a^3*b^2*c
^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-272*x*a^5*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-96*a^5*c^3*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)-3465*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^6*b^6*c^4+1
5*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^6*d^4-6300*ln((a*d*x+b*c*x+2*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^3*b^3*c^2*d^2+6930*x^5*b^5*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+
3486*x^5*a^2*b^3*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-10290*x^5*a*b^4*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)+4944*x^4*a^3*b^2*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-14028*x^4*a^2*b^3*c^2*d*(a*c)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)+966*x^3*a^4*b*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-2322*x^3*a^3*b^2*c^2*d*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+632*x^2*a^4*b*c^2*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-30*x^3*a^5*d^3*(a*c)^(1/2)*((b*x+
a)*(d*x+c))^(1/2)+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^6*a^4*b^2*d^4+30*ln((a*
d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^5*b*d^4-6930*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a*b^5*c^4-3465*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)
/x)*x^4*a^2*b^4*c^4+176*x*a^4*b*c^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^6*a^3*b^3*c*d^3-3150*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a
*c)/x)*x^6*a^2*b^4*c^2*d^2+6300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^6*a*b^5*c^3*
d+600*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^4*b^2*c*d^3)/c/a^6/((b*x+a)*(d*x+c
))^(1/2)/(a*c)^(1/2)/x^4/(b*x+a)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 150.116, size = 2395, normalized size = 6.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*((231*b^6*c^4 - 420*a*b^5*c^3*d + 210*a^2*b^4*c^2*d^2 - 20*a^3*b^3*c*d^3 - a^4*b^2*d^4)*x^6 + 2*(2
31*a*b^5*c^4 - 420*a^2*b^4*c^3*d + 210*a^3*b^3*c^2*d^2 - 20*a^4*b^2*c*d^3 - a^5*b*d^4)*x^5 + (231*a^2*b^4*c^4
- 420*a^3*b^3*c^3*d + 210*a^4*b^2*c^2*d^2 - 20*a^5*b*c*d^3 - a^6*d^4)*x^4)*sqrt(a*c)*log((8*a^2*c^2 + (b^2*c^2
+ 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a
^2*c*d)*x)/x^2) + 4*(48*a^6*c^4 - 3*(1155*a*b^5*c^4 - 1715*a^2*b^4*c^3*d + 581*a^3*b^3*c^2*d^2 - 5*a^4*b^2*c*d
^3)*x^5 - 6*(770*a^2*b^4*c^4 - 1169*a^3*b^3*c^3*d + 412*a^4*b^2*c^2*d^2 - 5*a^5*b*c*d^3)*x^4 - 3*(231*a^3*b^3*
c^4 - 387*a^4*b^2*c^3*d + 161*a^5*b*c^2*d^2 - 5*a^6*c*d^3)*x^3 + 2*(99*a^4*b^2*c^4 - 158*a^5*b*c^3*d + 59*a^6*
c^2*d^2)*x^2 - 8*(11*a^5*b*c^4 - 17*a^6*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^7*b^2*c^2*x^6 + 2*a^8*b*c^2*
x^5 + a^9*c^2*x^4), 1/384*(15*((231*b^6*c^4 - 420*a*b^5*c^3*d + 210*a^2*b^4*c^2*d^2 - 20*a^3*b^3*c*d^3 - a^4*b
^2*d^4)*x^6 + 2*(231*a*b^5*c^4 - 420*a^2*b^4*c^3*d + 210*a^3*b^3*c^2*d^2 - 20*a^4*b^2*c*d^3 - a^5*b*d^4)*x^5 +
(231*a^2*b^4*c^4 - 420*a^3*b^3*c^3*d + 210*a^4*b^2*c^2*d^2 - 20*a^5*b*c*d^3 - a^6*d^4)*x^4)*sqrt(-a*c)*arctan
(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*
d)*x)) - 2*(48*a^6*c^4 - 3*(1155*a*b^5*c^4 - 1715*a^2*b^4*c^3*d + 581*a^3*b^3*c^2*d^2 - 5*a^4*b^2*c*d^3)*x^5 -
6*(770*a^2*b^4*c^4 - 1169*a^3*b^3*c^3*d + 412*a^4*b^2*c^2*d^2 - 5*a^5*b*c*d^3)*x^4 - 3*(231*a^3*b^3*c^4 - 387
*a^4*b^2*c^3*d + 161*a^5*b*c^2*d^2 - 5*a^6*c*d^3)*x^3 + 2*(99*a^4*b^2*c^4 - 158*a^5*b*c^3*d + 59*a^6*c^2*d^2)*
x^2 - 8*(11*a^5*b*c^4 - 17*a^6*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^7*b^2*c^2*x^6 + 2*a^8*b*c^2*x^5 + a^9
*c^2*x^4)]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError