3.689 \(\int \frac {(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx\)
Optimal. Leaf size=317 \[ -\frac {\sqrt {a+b x} \left (315 a^2 d^2-322 a b c d+15 b^2 c^2\right ) (b c-a d)}{192 a c^4 x \sqrt {c+d x}}+\frac {5 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{11/2}}-\frac {d \sqrt {a+b x} \left (-945 a^3 d^3+1785 a^2 b c d^2-839 a b^2 c^2 d+15 b^3 c^3\right )}{192 a c^5 \sqrt {c+d x}}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{96 c^3 x^2 \sqrt {c+d x}}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{24 c^2 x^3 \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}} \]
[Out]
5/64*(-a*d+b*c)^2*(-63*a^2*d^2+14*a*b*c*d+b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(3/2
)/c^(11/2)-1/4*a*(b*x+a)^(3/2)/c/x^4/(d*x+c)^(1/2)-1/192*d*(-945*a^3*d^3+1785*a^2*b*c*d^2-839*a*b^2*c^2*d+15*b
^3*c^3)*(b*x+a)^(1/2)/a/c^5/(d*x+c)^(1/2)-1/24*a*(-9*a*d+11*b*c)*(b*x+a)^(1/2)/c^2/x^3/(d*x+c)^(1/2)-1/96*(-63
*a*d+59*b*c)*(-a*d+b*c)*(b*x+a)^(1/2)/c^3/x^2/(d*x+c)^(1/2)-1/192*(-a*d+b*c)*(315*a^2*d^2-322*a*b*c*d+15*b^2*c
^2)*(b*x+a)^(1/2)/a/c^4/x/(d*x+c)^(1/2)
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Rubi [A] time = 0.35, antiderivative size = 317, normalized size of antiderivative = 1.00,
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 =
{98, 149, 151, 152, 12, 93, 208} \[ -\frac {\sqrt {a+b x} \left (315 a^2 d^2-322 a b c d+15 b^2 c^2\right ) (b c-a d)}{192 a c^4 x \sqrt {c+d x}}-\frac {d \sqrt {a+b x} \left (1785 a^2 b c d^2-945 a^3 d^3-839 a b^2 c^2 d+15 b^3 c^3\right )}{192 a c^5 \sqrt {c+d x}}+\frac {5 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{11/2}}-\frac {\sqrt {a+b x} (59 b c-63 a d) (b c-a d)}{96 c^3 x^2 \sqrt {c+d x}}-\frac {a \sqrt {a+b x} (11 b c-9 a d)}{24 c^2 x^3 \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)),x]
[Out]
-(d*(15*b^3*c^3 - 839*a*b^2*c^2*d + 1785*a^2*b*c*d^2 - 945*a^3*d^3)*Sqrt[a + b*x])/(192*a*c^5*Sqrt[c + d*x]) -
(a*(11*b*c - 9*a*d)*Sqrt[a + b*x])/(24*c^2*x^3*Sqrt[c + d*x]) - ((59*b*c - 63*a*d)*(b*c - a*d)*Sqrt[a + b*x])
/(96*c^3*x^2*Sqrt[c + d*x]) - ((b*c - a*d)*(15*b^2*c^2 - 322*a*b*c*d + 315*a^2*d^2)*Sqrt[a + b*x])/(192*a*c^4*
x*Sqrt[c + d*x]) - (a*(a + b*x)^(3/2))/(4*c*x^4*Sqrt[c + d*x]) + (5*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63*a
^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(11/2))
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)^{3/2}} \, dx &=-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\int \frac {\sqrt {a+b x} \left (-\frac {1}{2} a (11 b c-9 a d)-b (4 b c-3 a d) x\right )}{x^4 (c+d x)^{3/2}} \, dx}{4 c}\\ &=-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\int \frac {-\frac {1}{4} a (59 b c-63 a d) (b c-a d)-\frac {3}{2} b (8 b c-9 a d) (b c-a d) x}{x^3 \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{12 c^2}\\ &=-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}+\frac {\int \frac {\frac {1}{8} a (b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right )-\frac {1}{2} a b d (59 b c-63 a d) (b c-a d) x}{x^2 \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{24 a c^3}\\ &=-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\int \frac {\frac {15}{16} a (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )+\frac {1}{8} a b d (b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{24 a^2 c^4}\\ &=-\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}+\frac {\int -\frac {15 a (b c-a d)^3 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 a^2 c^5 (b c-a d)}\\ &=-\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a c^5}\\ &=-\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}-\frac {\left (5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\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^5}\\ &=-\frac {d \left (15 b^3 c^3-839 a b^2 c^2 d+1785 a^2 b c d^2-945 a^3 d^3\right ) \sqrt {a+b x}}{192 a c^5 \sqrt {c+d x}}-\frac {a (11 b c-9 a d) \sqrt {a+b x}}{24 c^2 x^3 \sqrt {c+d x}}-\frac {(59 b c-63 a d) (b c-a d) \sqrt {a+b x}}{96 c^3 x^2 \sqrt {c+d x}}-\frac {(b c-a d) \left (15 b^2 c^2-322 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{192 a c^4 x \sqrt {c+d x}}-\frac {a (a+b x)^{3/2}}{4 c x^4 \sqrt {c+d x}}+\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 206, normalized size = 0.65 \[ \frac {x^2 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \left (2 c^{5/2} (a+b x)^{5/2}+5 x (b c-a d) \left (\sqrt {c} \sqrt {a+b x} (a (c+3 d x)-2 b c x)+3 \sqrt {a} x \sqrt {c+d x} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )+8 c^{7/2} x (a+b x)^{7/2} (9 a d+b c)-48 a c^{9/2} (a+b x)^{7/2}}{192 a^2 c^{11/2} x^4 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)),x]
[Out]
(-48*a*c^(9/2)*(a + b*x)^(7/2) + 8*c^(7/2)*(b*c + 9*a*d)*x*(a + b*x)^(7/2) + (b^2*c^2 + 14*a*b*c*d - 63*a^2*d^
2)*x^2*(2*c^(5/2)*(a + b*x)^(5/2) + 5*(b*c - a*d)*x*(Sqrt[c]*Sqrt[a + b*x]*(-2*b*c*x + a*(c + 3*d*x)) + 3*Sqrt
[a]*(b*c - a*d)*x*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])))/(192*a^2*c^(11/2)*
x^4*Sqrt[c + d*x])
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fricas [A] time = 16.27, size = 828, normalized size = 2.61 \[ \left [-\frac {15 \, {\left ({\left (b^{4} c^{4} d + 12 \, a b^{3} c^{3} d^{2} - 90 \, a^{2} b^{2} c^{2} d^{3} + 140 \, a^{3} b c d^{4} - 63 \, a^{4} d^{5}\right )} x^{5} + {\left (b^{4} c^{5} + 12 \, a b^{3} c^{4} d - 90 \, a^{2} b^{2} c^{3} d^{2} + 140 \, a^{3} b c^{2} d^{3} - 63 \, a^{4} c 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^{5} + {\left (15 \, a b^{3} c^{4} d - 839 \, a^{2} b^{2} c^{3} d^{2} + 1785 \, a^{3} b c^{2} d^{3} - 945 \, a^{4} c d^{4}\right )} x^{4} + {\left (15 \, a b^{3} c^{5} - 337 \, a^{2} b^{2} c^{4} d + 637 \, a^{3} b c^{3} d^{2} - 315 \, a^{4} c^{2} d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{5} - 122 \, a^{3} b c^{4} d + 63 \, a^{4} c^{3} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{5} - 9 \, a^{4} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (a^{2} c^{6} d x^{5} + a^{2} c^{7} x^{4}\right )}}, -\frac {15 \, {\left ({\left (b^{4} c^{4} d + 12 \, a b^{3} c^{3} d^{2} - 90 \, a^{2} b^{2} c^{2} d^{3} + 140 \, a^{3} b c d^{4} - 63 \, a^{4} d^{5}\right )} x^{5} + {\left (b^{4} c^{5} + 12 \, a b^{3} c^{4} d - 90 \, a^{2} b^{2} c^{3} d^{2} + 140 \, a^{3} b c^{2} d^{3} - 63 \, a^{4} c 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^{5} + {\left (15 \, a b^{3} c^{4} d - 839 \, a^{2} b^{2} c^{3} d^{2} + 1785 \, a^{3} b c^{2} d^{3} - 945 \, a^{4} c d^{4}\right )} x^{4} + {\left (15 \, a b^{3} c^{5} - 337 \, a^{2} b^{2} c^{4} d + 637 \, a^{3} b c^{3} d^{2} - 315 \, a^{4} c^{2} d^{3}\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{2} c^{5} - 122 \, a^{3} b c^{4} d + 63 \, a^{4} c^{3} d^{2}\right )} x^{2} + 8 \, {\left (17 \, a^{3} b c^{5} - 9 \, a^{4} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (a^{2} c^{6} d x^{5} + a^{2} c^{7} x^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),x, algorithm="fricas")
[Out]
[-1/768*(15*((b^4*c^4*d + 12*a*b^3*c^3*d^2 - 90*a^2*b^2*c^2*d^3 + 140*a^3*b*c*d^4 - 63*a^4*d^5)*x^5 + (b^4*c^5
+ 12*a*b^3*c^4*d - 90*a^2*b^2*c^3*d^2 + 140*a^3*b*c^2*d^3 - 63*a^4*c*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^5 + (15*a*b^3*c^4*d - 839*a^2*b^2*c^3*d^2 + 1785*a^3*b*c^2*d^3 - 945*a^4*c*
d^4)*x^4 + (15*a*b^3*c^5 - 337*a^2*b^2*c^4*d + 637*a^3*b*c^3*d^2 - 315*a^4*c^2*d^3)*x^3 + 2*(59*a^2*b^2*c^5 -
122*a^3*b*c^4*d + 63*a^4*c^3*d^2)*x^2 + 8*(17*a^3*b*c^5 - 9*a^4*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^
6*d*x^5 + a^2*c^7*x^4), -1/384*(15*((b^4*c^4*d + 12*a*b^3*c^3*d^2 - 90*a^2*b^2*c^2*d^3 + 140*a^3*b*c*d^4 - 63*
a^4*d^5)*x^5 + (b^4*c^5 + 12*a*b^3*c^4*d - 90*a^2*b^2*c^3*d^2 + 140*a^3*b*c^2*d^3 - 63*a^4*c*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^4*c^5 + (15*a*b^3*c^4*d - 839*a^2*b^2*c^3*d^2 + 1785*a^3*b*c^2*d^3 - 945*a^4*c*d^4)
*x^4 + (15*a*b^3*c^5 - 337*a^2*b^2*c^4*d + 637*a^3*b*c^3*d^2 - 315*a^4*c^2*d^3)*x^3 + 2*(59*a^2*b^2*c^5 - 122*
a^3*b*c^4*d + 63*a^4*c^3*d^2)*x^2 + 8*(17*a^3*b*c^5 - 9*a^4*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^6*d*
x^5 + a^2*c^7*x^4)]
________________________________________________________________________________________
giac [B] time = 171.59, size = 3762, normalized size = 11.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),x, algorithm="giac")
[Out]
2*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*sqrt(b*x + a)/(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*c^5*abs(b)) +
5/64*(sqrt(b*d)*b^6*c^4 + 12*sqrt(b*d)*a*b^5*c^3*d - 90*sqrt(b*d)*a^2*b^4*c^2*d^2 + 140*sqrt(b*d)*a^3*b^3*c*d
^3 - 63*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^5*abs(b)) - 1/96*(15*sqrt(b*d)*b^20*c^11 - 575*sqrt
(b*d)*a*b^19*c^10*d + 5077*sqrt(b*d)*a^2*b^18*c^9*d^2 - 22277*sqrt(b*d)*a^3*b^17*c^8*d^3 + 59494*sqrt(b*d)*a^4
*b^16*c^7*d^4 - 105350*sqrt(b*d)*a^5*b^15*c^6*d^5 + 128506*sqrt(b*d)*a^6*b^14*c^5*d^6 - 109082*sqrt(b*d)*a^7*b
^13*c^4*d^7 + 63547*sqrt(b*d)*a^8*b^12*c^3*d^8 - 24299*sqrt(b*d)*a^9*b^11*c^2*d^9 + 5505*sqrt(b*d)*a^10*b^10*c
*d^10 - 561*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 + 3946*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^17*c^
9*d - 26165*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 + 773
04*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 - 118178*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 + 80188*sqrt(b*d)*(sqr
t(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 + 22494*sqrt(b*d)*(sqrt(b*d)*sq
rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^12*c^4*d^6 - 87560*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 + 70411*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 - 26262*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 + 3927*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 - 11309*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 + 532
92*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 - 97092*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 + 73162*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 - 21174*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^5*b^11*c^4*d^5 + 36540*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 - 71636*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 + 49683*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 - 11781*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)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^14
*c^8 + 17480*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 - 53740*
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 + 51320*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 - 12238*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 - 6664*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 + 31892*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^6*b^8*c^2*d^6 - 47160*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 + 19635*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 - 15625*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 + 26
765*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 - 7185*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 - 1489*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 - 5683*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 + 24375*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s
qrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a^6*b^6*c*d^6 - 19635*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 + 7970*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 - 4917*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 - 40
68*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 - 3829*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 - 8670*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 + 11781*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^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 - 2091*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 - 510*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^6*c^3*d^2 + 2946*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^5*c
^2*d^3 + 3477*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 - 392
7*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)*(sqr
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^14*b^6*c^4 + 204*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 + 198*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 - 948*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 + 561*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^5*abs(b))
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maple [B] time = 0.03, size = 982, normalized size = 3.10 \[ -\frac {\sqrt {b x +a}\, \left (945 a^{4} d^{5} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-2100 a^{3} b c \,d^{4} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+1350 a^{2} b^{2} c^{2} d^{3} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-180 a \,b^{3} c^{3} d^{2} x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-15 b^{4} c^{4} d \,x^{5} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+945 a^{4} c \,d^{4} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-2100 a^{3} b \,c^{2} d^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+1350 a^{2} b^{2} c^{3} d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-180 a \,b^{3} c^{4} d \,x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-15 b^{4} c^{5} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-1890 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} d^{4} x^{4}+3570 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b c \,d^{3} x^{4}-1678 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{2} d^{2} x^{4}+30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{3} c^{3} d \,x^{4}-630 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c \,d^{3} x^{3}+1274 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{2} d^{2} x^{3}-674 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{3} d \,x^{3}+30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{3} c^{4} x^{3}+252 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{2} d^{2} x^{2}-488 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{3} d \,x^{2}+236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{4} x^{2}-144 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{3} d x +272 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b \,c^{4} x +96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} c^{4}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, \sqrt {d x +c}\, a \,c^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),x)
[Out]
-1/384*(b*x+a)^(1/2)*(945*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*d^5-2100*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*d^4+1350*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^2*d^3-180*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^3*d^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^5*b^4*c^4
*d+945*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*d^4-2100*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^2*d^3+1350*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^3*d^2-180*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^4*d-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^5-1890*x^4*a^3*d^4
*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+3570*x^4*a^2*b*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-1678*x^4*a*b^2*c
^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+30*x^4*b^3*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-630*x^3*a^3*c*
d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+1274*x^3*a^2*b*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-674*x^3*a*b
^2*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+30*x^3*b^3*c^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+252*x^2*a^3*c^
2*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-488*x^2*a^2*b*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+236*x^2*a*b^
2*c^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-144*x*a^3*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+272*x*a^2*b*c^4*
((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+96*a^3*c^4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2))/c^5/a/((b*x+a)*(d*x+c))^(1
/2)/x^4/(a*c)^(1/2)/(d*x+c)^(1/2)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),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 \[ \int \frac {{\left (a+b\,x\right )}^{5/2}}{x^5\,{\left (c+d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)),x)
[Out]
int((a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**(5/2)/x**5/(d*x+c)**(3/2),x)
[Out]
Timed out
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