3.8.77 \(\int \frac {(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx\)
Optimal. Leaf size=388 \[ -\frac {\sqrt {c+d x} (99 b c-59 a d) (b c-a d)}{96 a^3 x^2 (a+b x)^{3/2}}+\frac {11 c \sqrt {c+d x} (b c-a d)}{24 a^2 x^3 (a+b x)^{3/2}}+\frac {b \sqrt {c+d x} (b c-a d) \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{64 a^5 c (a+b x)^{3/2}}+\frac {\sqrt {c+d x} (b c-a d) \left (5 a^2 d^2-156 a b c d+231 b^2 c^2\right )}{64 a^4 c x (a+b x)^{3/2}}-\frac {5 (b c-a d) \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{13/2} c^{3/2}}+\frac {b \sqrt {c+d x} \left (-5 a^3 d^3+581 a^2 b c d^2-1715 a b^2 c^2 d+1155 b^3 c^3\right )}{64 a^6 c \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}} \]
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Rubi [A] time = 0.53, 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 {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}} \end {gather*}
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 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 {(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*}
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Mathematica [A] time = 0.52, size = 256, normalized size = 0.66 \begin {gather*} \frac {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}-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 (a^3 d^3+21 a^2 b c d^2-189 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 )}{192 a^{13/2} c^3 x^4 (a+b x)^{3/2}} \end {gather*}
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))
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IntegrateAlgebraic [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(c + d*x)^(5/2)/(x^5*(a + b*x)^(5/2)),x]
[Out]
$Aborted
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fricas [A] time = 35.88, size = 1106, normalized size = 2.85 \begin {gather*} \left [-\frac {15 \, {\left ({\left (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}\right )} x^{6} + 2 \, {\left (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}\right )} x^{5} + {\left (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}\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^{6} c^{4} - 3 \, {\left (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}\right )} x^{5} - 6 \, {\left (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}\right )} x^{4} - 3 \, {\left (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}\right )} x^{3} + 2 \, {\left (99 \, a^{4} b^{2} c^{4} - 158 \, a^{5} b c^{3} d + 59 \, a^{6} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (11 \, a^{5} b c^{4} - 17 \, a^{6} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (a^{7} b^{2} c^{2} x^{6} + 2 \, a^{8} b c^{2} x^{5} + a^{9} c^{2} x^{4}\right )}}, \frac {15 \, {\left ({\left (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}\right )} x^{6} + 2 \, {\left (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}\right )} x^{5} + {\left (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}\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^{6} c^{4} - 3 \, {\left (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}\right )} x^{5} - 6 \, {\left (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}\right )} x^{4} - 3 \, {\left (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}\right )} x^{3} + 2 \, {\left (99 \, a^{4} b^{2} c^{4} - 158 \, a^{5} b c^{3} d + 59 \, a^{6} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (11 \, a^{5} b c^{4} - 17 \, a^{6} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (a^{7} b^{2} c^{2} x^{6} + 2 \, a^{8} b c^{2} x^{5} + a^{9} c^{2} x^{4}\right )}}\right ] \end {gather*}
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)]
________________________________________________________________________________________
giac [B] time = 41.26, size = 4500, normalized size = 11.60
result too large to display
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]
-5/64*(231*sqrt(b*d)*b^4*c^4*abs(b) - 420*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)
- 20*sqrt(b*d)*a^3*b*c*d^3*abs(b) - sqrt(b*d)*a^4*d^4*abs(b))*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^6*b*c) + 4/3*(15*sqrt(b*
d)*b^8*c^5*abs(b) - 68*sqrt(b*d)*a*b^7*c^4*d*abs(b) + 122*sqrt(b*d)*a^2*b^6*c^3*d^2*abs(b) - 108*sqrt(b*d)*a^3
*b^5*c^2*d^3*abs(b) + 47*sqrt(b*d)*a^4*b^4*c*d^4*abs(b) - 8*sqrt(b*d)*a^5*b^3*d^5*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*b^6*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*b^5*c^3*d*abs(b) - 144*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^2*b^4*c^2*d^2*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^3*c*d^3*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^4*b^2*d^4*abs(b) + 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^4*b^4*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*b^3*c^2*d*abs(b) + 27*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^2*c*
d^2*abs(b) - 6*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*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*a^6) + 1/96*(1545*sqrt(b
*d)*b^18*c^11*abs(b) - 14561*sqrt(b*d)*a*b^17*c^10*d*abs(b) + 61587*sqrt(b*d)*a^2*b^16*c^9*d^2*abs(b) - 153915
*sqrt(b*d)*a^3*b^15*c^8*d^3*abs(b) + 251658*sqrt(b*d)*a^4*b^14*c^7*d^4*abs(b) - 281274*sqrt(b*d)*a^5*b^13*c^6*
d^5*abs(b) + 217686*sqrt(b*d)*a^6*b^12*c^5*d^6*abs(b) - 115302*sqrt(b*d)*a^7*b^11*c^4*d^7*abs(b) + 40125*sqrt(
b*d)*a^8*b^10*c^3*d^8*abs(b) - 8373*sqrt(b*d)*a^9*b^9*c^2*d^9*abs(b) + 839*sqrt(b*d)*a^10*b^8*c*d^10*abs(b) -
15*sqrt(b*d)*a^11*b^7*d^11*abs(b) - 10815*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^2*b^16*c^10*abs(b) + 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
*b^15*c^9*d*abs(b) - 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^2*b^
14*c^8*d^2*abs(b) + 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^3*b^1
3*c^7*d^3*abs(b) - 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^4*b^12*
c^6*d^4*abs(b) - 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^11*c
^5*d^5*abs(b) + 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^6*b^10*c^
4*d^6*abs(b) - 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^7*b^9*c^3*
d^7*abs(b) + 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^8*b^8*c^2*d^8
*abs(b) - 5794*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^7*c*d^9*abs(b
) + 105*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^6*d^10*abs(b) + 324
45*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^14*c^9*abs(b) - 125507*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^13*c^8*d*abs(b) + 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^2*b^12*c^7*d^2*abs(b) - 63660*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^11*c^6*d^3*abs(b) + 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^4*b^10*c^5*d^4*abs(b) - 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^5*b^9*c^4*d^5*abs(b) + 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^6*b^8*c^3*d^6*abs(b) - 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^7*b^7*c^2*d^7*abs(b) + 16853*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^8*b^6*c*d^8*abs(b) - 315*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^5*d^9*abs(b) - 54075*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^12*c^8*abs(b) + 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*b^11*c^7*d*abs(b) - 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^2*b^10*c^6*d^2*abs(b) + 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^3*b^9*c^5*d^3*abs(b) + 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^8*c^4*d^4*abs(b) - 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^5*b^7*c^3*d^5*abs(b) + 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^6*b^6*c^2*d^6*abs(b) - 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^7*b^5*c*d^7*abs(b) + 525*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^4*d^8*abs(b) + 54075*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^1
0*c^7*abs(b) - 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*b^9*c^6*d*a
bs(b) + 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^2*b^8*c^5*d^2*abs(b
) + 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^3*b^7*c^4*d^3*abs(b) +
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^4*b^6*c^3*d^4*abs(b) - 629
25*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^5*c^2*d^5*abs(b) + 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^6*b^4*c*d^6*abs(b) - 525*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^3*d^7*abs(b) - 32445*sqrt(b*d)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^10*b^8*c^6*abs(b) + 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*b^7*c^5*d*abs(b) + 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^2*b^6*c^4*d^2*abs(b) + 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^5*c^3*d^3*abs(b) + 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^4*b^4*c^2*d^4*abs(b) - 13514*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^10*a^5*b^3*c*d^5*abs(b) + 315*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2
*c + (b*x + a)*b*d - a*b*d))^10*a^6*b^2*d^6*abs(b) + 10815*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (
b*x + a)*b*d - a*b*d))^12*b^6*c^5*abs(b) - 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*b^5*c^4*d*abs(b) - 7074*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b
*d))^12*a^2*b^4*c^3*d^2*abs(b) - 3474*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^12*a^3*b^3*c^2*d^3*abs(b) + 3939*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^1
2*a^4*b^2*c*d^4*abs(b) - 105*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*d^5*abs(b) - 1545*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*b^4*c^4*abs(b
) + 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*b^3*c^3*d*abs(b) + 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^2*c^2*d^2*abs(b) - 468*sqr
t(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*c*d^3*abs(b) + 15*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*a^4*d^4*abs(b))/((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^6*c)
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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((d*x+c)^(5/2)/x^5/(b*x+a)^(5/2),x)
[Out]
1/384*(d*x+c)^(1/2)*(-30*x^5*a^3*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-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*b^5*c^4-3465*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^2*b^4*c^4+6930*x^5*b^5*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-30*x^3*a^5*d^3*((b*x+a)*(d*x+c))
^(1/2)*(a*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*a^4*b^2*d^4+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*a^5*b*d^4-96*a^5*c^3*((b*x+a)*(d*x+c))^(1/2)*(a
*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*b^6*c^4+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*a^6*d^4+3486*x^5*a^2*b^3*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*
c)^(1/2)-10290*x^5*a*b^4*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+4944*x^4*a^3*b^2*c*d^2*((b*x+a)*(d*x+c))^(1
/2)*(a*c)^(1/2)-14028*x^4*a^2*b^3*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+966*x^3*a^4*b*c*d^2*((b*x+a)*(d*x+
c))^(1/2)*(a*c)^(1/2)-2322*x^3*a^3*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+632*x^2*a^4*b*c^2*d*((b*x+a)*
(d*x+c))^(1/2)*(a*c)^(1/2)-60*x^4*a^4*b*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+9240*x^4*a*b^4*c^3*((b*x+a)*(d
*x+c))^(1/2)*(a*c)^(1/2)+1386*x^3*a^2*b^3*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-236*x^2*a^5*c*d^2*((b*x+a)*(
d*x+c))^(1/2)*(a*c)^(1/2)-396*x^2*a^3*b^2*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-272*x*a^5*c^2*d*((b*x+a)*(d*
x+c))^(1/2)*(a*c)^(1/2)+176*x*a^4*b*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+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^6*a^3*b^3*c*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^6*a^2*b^4*c^2*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^6*a*b^5*
c^3*d+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^4*b^2*c*d^3-6300*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^3*b^3*c^2*d^2+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^2*b^4*c^3*d+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^5*b*c*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^4*b^2*c^2*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^3*c^3*d)/c/a^6/((b*x+a)*(d*x+
c))^(1/2)/x^4/(a*c)^(1/2)/(b*x+a)^(3/2)
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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((d*x+c)^(5/2)/x^5/(b*x+a)^(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?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^5\,{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^(5/2)/(x^5*(a + b*x)^(5/2)),x)
[Out]
int((c + d*x)^(5/2)/(x^5*(a + b*x)^(5/2)), x)
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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((d*x+c)**(5/2)/x**5/(b*x+a)**(5/2),x)
[Out]
Timed out
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