3.7.73 \(\int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx\)
Optimal. Leaf size=388 \[ -\frac {d \sqrt {a+b x} (b c-a d) \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{64 a c^5 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (b c-a d) \left (231 a^2 d^2-156 a b c d+5 b^2 c^2\right )}{64 a c^4 x (c+d x)^{3/2}}-\frac {d \sqrt {a+b x} \left (-1155 a^3 d^3+1715 a^2 b c d^2-581 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^6 \sqrt {c+d x}}+\frac {5 (b c-a d) \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{13/2}}-\frac {\sqrt {a+b x} (59 b c-99 a d) (b c-a d)}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {11 a \sqrt {a+b x} (b c-a d)}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.51, antiderivative size = 388, normalized size of antiderivative = 1.00,
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} \begin {gather*} -\frac {d \sqrt {a+b x} \left (1715 a^2 b c d^2-1155 a^3 d^3-581 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^6 \sqrt {c+d x}}-\frac {d \sqrt {a+b x} (b c-a d) \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{64 a c^5 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (b c-a d) \left (231 a^2 d^2-156 a b c d+5 b^2 c^2\right )}{64 a c^4 x (c+d x)^{3/2}}+\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 {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{13/2}}-\frac {\sqrt {a+b x} (59 b c-99 a d) (b c-a d)}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {11 a \sqrt {a+b x} (b c-a d)}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^(5/2)/(x^5*(c + d*x)^(5/2)),x]
[Out]
-(d*(b*c - a*d)*(5*b^2*c^2 - 238*a*b*c*d + 385*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^5*(c + d*x)^(3/2)) - (11*a*(b*c
- a*d)*Sqrt[a + b*x])/(24*c^2*x^3*(c + d*x)^(3/2)) - ((59*b*c - 99*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(96*c^3*x^
2*(c + d*x)^(3/2)) - ((b*c - a*d)*(5*b^2*c^2 - 156*a*b*c*d + 231*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^4*x*(c + d*x)
^(3/2)) - (a*(a + b*x)^(3/2))/(4*c*x^4*(c + d*x)^(3/2)) - (d*(5*b^3*c^3 - 581*a*b^2*c^2*d + 1715*a^2*b*c*d^2 -
1155*a^3*d^3)*Sqrt[a + b*x])/(64*a*c^6*Sqrt[c + d*x]) + (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)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(13/2))
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 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 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 {(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx &=-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac {\int \frac {\sqrt {a+b x} \left (-\frac {11}{2} a (b c-a d)-4 b (b c-a d) x\right )}{x^4 (c+d x)^{5/2}} \, dx}{4 c}\\ &=-\frac {11 a (b c-a d) \sqrt {a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac {\int \frac {-\frac {1}{4} a (59 b c-99 a d) (b c-a d)-2 b (6 b c-11 a d) (b c-a d) x}{x^3 \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{12 c^2}\\ &=-\frac {11 a (b c-a d) \sqrt {a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {(59 b c-99 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}+\frac {\int \frac {\frac {3}{8} a (b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right )-\frac {3}{4} a b d (59 b c-99 a d) (b c-a d) x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{24 a c^3}\\ &=-\frac {11 a (b c-a d) \sqrt {a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {(59 b c-99 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac {\int \frac {\frac {15}{16} a (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 )+\frac {3}{4} a b d (b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{24 a^2 c^4}\\ &=-\frac {d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac {11 a (b c-a d) \sqrt {a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {(59 b c-99 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}+\frac {\int \frac {-\frac {45}{32} a (b c-a d)^2 \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 )-\frac {9}{16} a b d (b c-a d)^2 \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{36 a^2 c^5 (b c-a d)}\\ &=-\frac {d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac {11 a (b c-a d) \sqrt {a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {(59 b c-99 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac {d \left (5 b^3 c^3-581 a b^2 c^2 d+1715 a^2 b c d^2-1155 a^3 d^3\right ) \sqrt {a+b x}}{64 a c^6 \sqrt {c+d x}}-\frac {\int \frac {45 a (b c-a d)^3 \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 )}{64 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{18 a^2 c^6 (b c-a d)^2}\\ &=-\frac {d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac {11 a (b c-a d) \sqrt {a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {(59 b c-99 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac {d \left (5 b^3 c^3-581 a b^2 c^2 d+1715 a^2 b c d^2-1155 a^3 d^3\right ) \sqrt {a+b x}}{64 a c^6 \sqrt {c+d x}}-\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}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a c^6}\\ &=-\frac {d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac {11 a (b c-a d) \sqrt {a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {(59 b c-99 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac {d \left (5 b^3 c^3-581 a b^2 c^2 d+1715 a^2 b c d^2-1155 a^3 d^3\right ) \sqrt {a+b x}}{64 a c^6 \sqrt {c+d x}}-\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}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 a c^6}\\ &=-\frac {d (b c-a d) \left (5 b^2 c^2-238 a b c d+385 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^5 (c+d x)^{3/2}}-\frac {11 a (b c-a d) \sqrt {a+b x}}{24 c^2 x^3 (c+d x)^{3/2}}-\frac {(59 b c-99 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 (c+d x)^{3/2}}-\frac {(b c-a d) \left (5 b^2 c^2-156 a b c d+231 a^2 d^2\right ) \sqrt {a+b x}}{64 a c^4 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}}-\frac {d \left (5 b^3 c^3-581 a b^2 c^2 d+1715 a^2 b c d^2-1155 a^3 d^3\right ) \sqrt {a+b x}}{64 a c^6 \sqrt {c+d x}}+\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 {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{13/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.71, size = 255, normalized size = 0.66 \begin {gather*} \frac {-48 a^2 c^{11/2} (a+b x)^{7/2}+x^2 \left (2 c^{7/2} (a+b x)^{7/2} \left (-99 a^2 d^2+26 a b c d+b^2 c^2\right )+x \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \left (3 c^{5/2} (a+b x)^{5/2}-5 x (b c-a d) \left (\sqrt {c} \sqrt {a+b x} (4 a c+3 a d x+b c x)-3 a^{3/2} (c+d x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )\right )+8 a c^{9/2} x (a+b x)^{7/2} (11 a d+b c)}{192 a^3 c^{13/2} x^4 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^(5/2)/(x^5*(c + d*x)^(5/2)),x]
[Out]
(-48*a^2*c^(11/2)*(a + b*x)^(7/2) + 8*a*c^(9/2)*(b*c + 11*a*d)*x*(a + b*x)^(7/2) + x^2*(2*c^(7/2)*(b^2*c^2 + 2
6*a*b*c*d - 99*a^2*d^2)*(a + b*x)^(7/2) + (b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*x*(3*c^(5
/2)*(a + b*x)^(5/2) - 5*(b*c - a*d)*x*(Sqrt[c]*Sqrt[a + b*x]*(4*a*c + b*c*x + 3*a*d*x) - 3*a^(3/2)*(c + d*x)^(
3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])]))))/(192*a^3*c^(13/2)*x^4*(c + d*x)^(3/2))
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.02, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(a + b*x)^(5/2)/(x^5*(c + d*x)^(5/2)),x]
[Out]
$Aborted
________________________________________________________________________________________
fricas [A] time = 50.44, size = 1100, normalized size = 2.84 \begin {gather*} \left [-\frac {15 \, {\left ({\left (b^{4} c^{4} d^{2} + 20 \, a b^{3} c^{3} d^{3} - 210 \, a^{2} b^{2} c^{2} d^{4} + 420 \, a^{3} b c d^{5} - 231 \, a^{4} d^{6}\right )} x^{6} + 2 \, {\left (b^{4} c^{5} d + 20 \, a b^{3} c^{4} d^{2} - 210 \, a^{2} b^{2} c^{3} d^{3} + 420 \, a^{3} b c^{2} d^{4} - 231 \, a^{4} c d^{5}\right )} x^{5} + {\left (b^{4} c^{6} + 20 \, a b^{3} c^{5} d - 210 \, a^{2} b^{2} c^{4} d^{2} + 420 \, a^{3} b c^{3} d^{3} - 231 \, a^{4} c^{2} d^{4}\right )} x^{4}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (48 \, a^{4} c^{6} + 3 \, {\left (5 \, a b^{3} c^{4} d^{2} - 581 \, a^{2} b^{2} c^{3} d^{3} + 1715 \, a^{3} b c^{2} d^{4} - 1155 \, a^{4} c d^{5}\right )} x^{5} + 6 \, {\left (5 \, a b^{3} c^{5} d - 412 \, a^{2} b^{2} c^{4} d^{2} + 1169 \, a^{3} b c^{3} d^{3} - 770 \, a^{4} c^{2} d^{4}\right )} x^{4} + 3 \, {\left (5 \, a b^{3} c^{6} - 161 \, a^{2} b^{2} c^{5} d + 387 \, a^{3} b c^{4} d^{2} - 231 \, a^{4} c^{3} d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{6} - 158 \, a^{3} b c^{5} d + 99 \, a^{4} c^{4} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{6} - 11 \, a^{4} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (a^{2} c^{7} d^{2} x^{6} + 2 \, a^{2} c^{8} d x^{5} + a^{2} c^{9} x^{4}\right )}}, -\frac {15 \, {\left ({\left (b^{4} c^{4} d^{2} + 20 \, a b^{3} c^{3} d^{3} - 210 \, a^{2} b^{2} c^{2} d^{4} + 420 \, a^{3} b c d^{5} - 231 \, a^{4} d^{6}\right )} x^{6} + 2 \, {\left (b^{4} c^{5} d + 20 \, a b^{3} c^{4} d^{2} - 210 \, a^{2} b^{2} c^{3} d^{3} + 420 \, a^{3} b c^{2} d^{4} - 231 \, a^{4} c d^{5}\right )} x^{5} + {\left (b^{4} c^{6} + 20 \, a b^{3} c^{5} d - 210 \, a^{2} b^{2} c^{4} d^{2} + 420 \, a^{3} b c^{3} d^{3} - 231 \, a^{4} c^{2} d^{4}\right )} x^{4}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (48 \, a^{4} c^{6} + 3 \, {\left (5 \, a b^{3} c^{4} d^{2} - 581 \, a^{2} b^{2} c^{3} d^{3} + 1715 \, a^{3} b c^{2} d^{4} - 1155 \, a^{4} c d^{5}\right )} x^{5} + 6 \, {\left (5 \, a b^{3} c^{5} d - 412 \, a^{2} b^{2} c^{4} d^{2} + 1169 \, a^{3} b c^{3} d^{3} - 770 \, a^{4} c^{2} d^{4}\right )} x^{4} + 3 \, {\left (5 \, a b^{3} c^{6} - 161 \, a^{2} b^{2} c^{5} d + 387 \, a^{3} b c^{4} d^{2} - 231 \, a^{4} c^{3} d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{6} - 158 \, a^{3} b c^{5} d + 99 \, a^{4} c^{4} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{6} - 11 \, a^{4} c^{5} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (a^{2} c^{7} d^{2} x^{6} + 2 \, a^{2} c^{8} d x^{5} + a^{2} c^{9} x^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(5/2),x, algorithm="fricas")
[Out]
[-1/768*(15*((b^4*c^4*d^2 + 20*a*b^3*c^3*d^3 - 210*a^2*b^2*c^2*d^4 + 420*a^3*b*c*d^5 - 231*a^4*d^6)*x^6 + 2*(b
^4*c^5*d + 20*a*b^3*c^4*d^2 - 210*a^2*b^2*c^3*d^3 + 420*a^3*b*c^2*d^4 - 231*a^4*c*d^5)*x^5 + (b^4*c^6 + 20*a*b
^3*c^5*d - 210*a^2*b^2*c^4*d^2 + 420*a^3*b*c^3*d^3 - 231*a^4*c^2*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^4*c^6 + 3*(5*a*b^3*c^4*d^2 - 581*a^2*b^2*c^3*d^3 + 1715*a^3*b*c^2*d^4 - 1155*a^4*c*d
^5)*x^5 + 6*(5*a*b^3*c^5*d - 412*a^2*b^2*c^4*d^2 + 1169*a^3*b*c^3*d^3 - 770*a^4*c^2*d^4)*x^4 + 3*(5*a*b^3*c^6
- 161*a^2*b^2*c^5*d + 387*a^3*b*c^4*d^2 - 231*a^4*c^3*d^3)*x^3 + 2*(59*a^2*b^2*c^6 - 158*a^3*b*c^5*d + 99*a^4*
c^4*d^2)*x^2 + 8*(17*a^3*b*c^6 - 11*a^4*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^7*d^2*x^6 + 2*a^2*c^8*d*
x^5 + a^2*c^9*x^4), -1/384*(15*((b^4*c^4*d^2 + 20*a*b^3*c^3*d^3 - 210*a^2*b^2*c^2*d^4 + 420*a^3*b*c*d^5 - 231*
a^4*d^6)*x^6 + 2*(b^4*c^5*d + 20*a*b^3*c^4*d^2 - 210*a^2*b^2*c^3*d^3 + 420*a^3*b*c^2*d^4 - 231*a^4*c*d^5)*x^5
+ (b^4*c^6 + 20*a*b^3*c^5*d - 210*a^2*b^2*c^4*d^2 + 420*a^3*b*c^3*d^3 - 231*a^4*c^2*d^4)*x^4)*sqrt(-a*c)*arcta
n(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^4*c^6 + 3*(5*a*b^3*c^4*d^2 - 581*a^2*b^2*c^3*d^3 + 1715*a^3*b*c^2*d^4 - 1155*a^4*c*d^5)*x^5
+ 6*(5*a*b^3*c^5*d - 412*a^2*b^2*c^4*d^2 + 1169*a^3*b*c^3*d^3 - 770*a^4*c^2*d^4)*x^4 + 3*(5*a*b^3*c^6 - 161*a^
2*b^2*c^5*d + 387*a^3*b*c^4*d^2 - 231*a^4*c^3*d^3)*x^3 + 2*(59*a^2*b^2*c^6 - 158*a^3*b*c^5*d + 99*a^4*c^4*d^2)
*x^2 + 8*(17*a^3*b*c^6 - 11*a^4*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^7*d^2*x^6 + 2*a^2*c^8*d*x^5 + a^
2*c^9*x^4)]
________________________________________________________________________________________
giac [B] time = 33.95, size = 3915, normalized size = 10.09
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(5/2),x, algorithm="giac")
[Out]
2/3*sqrt(b*x + a)*((8*b^6*c^9*d^4*abs(b) - 31*a*b^5*c^8*d^5*abs(b) + 38*a^2*b^4*c^7*d^6*abs(b) - 15*a^3*b^3*c^
6*d^7*abs(b))*(b*x + a)/(b^3*c^13*d - a*b^2*c^12*d^2) + 3*(3*b^7*c^10*d^3*abs(b) - 14*a*b^6*c^9*d^4*abs(b) + 2
4*a^2*b^5*c^8*d^5*abs(b) - 18*a^3*b^4*c^7*d^6*abs(b) + 5*a^4*b^3*c^6*d^7*abs(b))/(b^3*c^13*d - a*b^2*c^12*d^2)
)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + 5/64*(sqrt(b*d)*b^6*c^4 + 20*sqrt(b*d)*a*b^5*c^3*d - 210*sqrt(b*d)*a
^2*b^4*c^2*d^2 + 420*sqrt(b*d)*a^3*b^3*c*d^3 - 231*sqrt(b*d)*a^4*b^2*d^4)*arctan(-1/2*(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)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a*b*c^6*abs(b)
) - 1/96*(15*sqrt(b*d)*b^20*c^11 - 839*sqrt(b*d)*a*b^19*c^10*d + 8373*sqrt(b*d)*a^2*b^18*c^9*d^2 - 40125*sqrt(
b*d)*a^3*b^17*c^8*d^3 + 115302*sqrt(b*d)*a^4*b^16*c^7*d^4 - 217686*sqrt(b*d)*a^5*b^15*c^6*d^5 + 281274*sqrt(b*
d)*a^6*b^14*c^5*d^6 - 251658*sqrt(b*d)*a^7*b^13*c^4*d^7 + 153915*sqrt(b*d)*a^8*b^12*c^3*d^8 - 61587*sqrt(b*d)*
a^9*b^11*c^2*d^9 + 14561*sqrt(b*d)*a^10*b^10*c*d^10 - 1545*sqrt(b*d)*a^11*b^9*d^11 - 105*sqrt(b*d)*(sqrt(b*d)*
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^18*c^10 + 5794*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^17*c^9*d - 44109*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^16*c^8*d^2 + 145304*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^15*c^7*d^3 - 245954*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^14*c^6*d^4 + 190860*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^13*c^5*d^5 + 29006*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^12
*c^4*d^6 - 195176*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^11*c^3*d^7
+ 172011*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^8*b^10*c^2*d^8 - 68446
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^9*b^9*c*d^9 + 10815*sqrt(b*d)*(
sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^10*b^8*d^10 + 315*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^16*c^9 - 16853*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(
b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^15*c^8*d + 94076*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^14*c^7*d^2 - 196500*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^13*c^6*d^3 + 167002*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^12*c^5*d^4 - 44758*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^11*c^4*d^5 + 63660*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*b^10*c^3*
d^6 - 160004*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^7*b^9*c^2*d^7 + 125
507*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^8*b^8*c*d^8 - 32445*sqrt(b*d
)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^9*b^7*d^9 - 525*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^14*c^8 + 26720*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt
(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^13*c^7*d - 103860*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^6*a^2*b^12*c^6*d^2 + 118128*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^6*a^3*b^11*c^5*d^3 - 30006*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*
d))^6*a^4*b^10*c^4*d^4 - 11712*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^5
*b^9*c^3*d^5 + 57660*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^8*c^2*d
^6 - 110480*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^7*b^7*c*d^7 + 54075*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^8*b^6*d^8 + 525*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^12*c^7 - 24865*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^11*c^6*d + 62925*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*
c + (b*x + a)*b*d - a*b*d))^8*a^2*b^10*c^5*d^2 - 22697*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^8*a^3*b^9*c^4*d^3 - 5137*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^8*a^4*b^8*c^3*d^4 - 8955*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a
^5*b^7*c^2*d^5 + 46135*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^6*b^6*c*d
^6 - 54075*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^7*b^5*d^7 - 315*sqrt(
b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^10*c^6 + 13514*sqrt(b*d)*(sqrt(b*d)*
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a*b^9*c^5*d - 20573*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^2*b^8*c^4*d^2 - 9492*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^
2*c + (b*x + a)*b*d - a*b*d))^10*a^3*b^7*c^3*d^3 - 5893*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^10*a^4*b^6*c^2*d^4 - 9686*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^10*a^5*b^5*c*d^5 + 32445*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^1
0*a^6*b^4*d^6 + 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*b^8*c^5 - 393
9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a*b^7*c^4*d + 3474*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^2*b^6*c^3*d^2 + 7074*sqrt(b*d)*(sqrt(b*d)*s
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^3*b^5*c^2*d^3 + 4101*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x +
a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^12*a^4*b^4*c*d^4 - 10815*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b
^2*c + (b*x + a)*b*d - a*b*d))^12*a^5*b^3*d^5 - 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)
*b*d - a*b*d))^14*b^6*c^4 + 468*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a
*b^5*c^3*d - 306*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^2*b^4*c^2*d^2
- 1692*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^3*b^3*c*d^3 + 1545*sqrt(
b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^4*b^2*d^4)/((b^4*c^2 - 2*a*b^3*c*d +
a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d
- a*b*d))^4)^4*a*c^6*abs(b))
________________________________________________________________________________________
maple [B] time = 0.04, size = 1377, normalized size = 3.55
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^(5/2)/x^5/(d*x+c)^(5/2),x)
[Out]
-1/384*(b*x+a)^(1/2)*(-15*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*b^4*c^6+96*a^3*c
^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+3465*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^
6*a^4*d^6-6930*x^5*a^3*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-1386*x^3*a^3*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+
c))^(1/2)+396*x^2*a^3*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+236*x^2*a*b^2*c^5*(a*c)^(1/2)*((b*x+a)*(d*x+
c))^(1/2)-176*x*a^3*c^4*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+272*x*a^2*b*c^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)-6300*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^6*a^3*b*c*d^5+3150*ln((a*d*x+b*c*x+
2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^6*a^2*b^2*c^2*d^4-300*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((
b*x+a)*(d*x+c))^(1/2))/x)*x^6*a*b^3*c^3*d^3-12600*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))
/x)*x^5*a^3*b*c^2*d^4+6300*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^5*a^2*b^2*c^3*d^3
-600*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^5*a*b^3*c^4*d^2-6300*ln((a*d*x+b*c*x+2*
a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4*a^3*b*c^3*d^3+3150*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x
+a)*(d*x+c))^(1/2))/x)*x^4*a^2*b^2*c^4*d^2-300*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)
*x^4*a*b^3*c^5*d+30*x^5*b^3*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-9240*x^4*a^3*c*d^4*(a*c)^(1/2)*((b*x+a
)*(d*x+c))^(1/2)+60*x^4*b^3*c^4*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+30*x^3*b^3*c^5*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)-15*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^6*b^4*c^4*d^2+6930*ln((a*d*x+b
*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^5*a^4*c*d^5-30*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2))/x)*x^5*b^4*c^5*d+3465*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^4
*a^4*c^2*d^4+14028*x^4*a^2*b*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-4944*x^4*a*b^2*c^3*d^2*(a*c)^(1/2)*((
b*x+a)*(d*x+c))^(1/2)+2322*x^3*a^2*b*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-966*x^3*a*b^2*c^4*d*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)-632*x^2*a^2*b*c^4*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+10290*x^5*a^2*b*c*d^4*(a*c)
^(1/2)*((b*x+a)*(d*x+c))^(1/2)-3486*x^5*a*b^2*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/c^6/a/((b*x+a)*(d*x
+c))^(1/2)/x^4/(a*c)^(1/2)/(d*x+c)^(3/2)
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{x^5\,{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^(5/2)/(x^5*(c + d*x)^(5/2)),x)
[Out]
int((a + b*x)^(5/2)/(x^5*(c + d*x)^(5/2)), x)
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**(5/2)/x**5/(d*x+c)**(5/2),x)
[Out]
Timed out
________________________________________________________________________________________