Integrand size = 22, antiderivative size = 319 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^5}+\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^4 (b c-a d)}+\frac {\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^3 (b c-a d)^2}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-5 a d) (c+d x)^{7/2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{11/2} \sqrt {d}} \]
-2/3*a^2*(d*x+c)^(7/2)/b^2/(-a*d+b*c)/(b*x+a)^(3/2)+5/8*(-a*d+b*c)*(21*a^2 *d^2-14*a*b*c*d+b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/ 2))/b^(11/2)/d^(1/2)+4/3*a*(-5*a*d+3*b*c)*(d*x+c)^(7/2)/b^2/(-a*d+b*c)^2/( b*x+a)^(1/2)+5/12*(21*a^2*d^2-14*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*(b*x+a)^(1 /2)/b^4/(-a*d+b*c)+1/3*(21*a^2*d^2-14*a*b*c*d+b^2*c^2)*(d*x+c)^(5/2)*(b*x+ a)^(1/2)/b^3/(-a*d+b*c)^2+5/8*(21*a^2*d^2-14*a*b*c*d+b^2*c^2)*(b*x+a)^(1/2 )*(d*x+c)^(1/2)/b^5
Time = 10.52 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.68 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {c+d x} \left (\frac {315 a^4 d^2+420 a^3 b d (-c+d x)-6 a b^3 x \left (-27 c^2+16 c d x+3 d^2 x^2\right )+b^4 x^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )+a^2 b^2 \left (113 c^2-574 c d x+63 d^2 x^2\right )}{(a+b x)^{3/2}}+\frac {15 \sqrt {b c-a d} \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{24 b^5} \]
(Sqrt[c + d*x]*((315*a^4*d^2 + 420*a^3*b*d*(-c + d*x) - 6*a*b^3*x*(-27*c^2 + 16*c*d*x + 3*d^2*x^2) + b^4*x^2*(33*c^2 + 26*c*d*x + 8*d^2*x^2) + a^2*b ^2*(113*c^2 - 574*c*d*x + 63*d^2*x^2))/(a + b*x)^(3/2) + (15*Sqrt[b*c - a* d]*(b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqr t[b*c - a*d]])/(Sqrt[d]*Sqrt[(b*(c + d*x))/(b*c - a*d)])))/(24*b^5)
Time = 0.38 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {100, 27, 87, 60, 60, 60, 66, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {2 \int -\frac {(c+d x)^{5/2} (a (3 b c-7 a d)-3 b (b c-a d) x)}{2 (a+b x)^{3/2}}dx}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {(c+d x)^{5/2} (a (3 b c-7 a d)-3 b (b c-a d) x)}{(a+b x)^{3/2}}dx}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}}dx}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}}dx}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {-\frac {3 \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{b c-a d}-\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{\sqrt {a+b x} (b c-a d)}}{3 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}\) |
(-2*a^2*(c + d*x)^(7/2))/(3*b^2*(b*c - a*d)*(a + b*x)^(3/2)) - ((-4*a*(3*b *c - 5*a*d)*(c + d*x)^(7/2))/((b*c - a*d)*Sqrt[a + b*x]) - (3*(b^2*c^2 - 1 4*a*b*c*d + 21*a^2*d^2)*((Sqrt[a + b*x]*(c + d*x)^(5/2))/(3*b) + (5*(b*c - a*d)*((Sqrt[a + b*x]*(c + d*x)^(3/2))/(2*b) + (3*(b*c - a*d)*((Sqrt[a + b *x]*Sqrt[c + d*x])/b + ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[ b]*Sqrt[c + d*x])])/(b^(3/2)*Sqrt[d])))/(4*b)))/(6*b)))/(b*c - a*d))/(3*b^ 2*(b*c - a*d))
3.8.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(1001\) vs. \(2(275)=550\).
Time = 0.60 (sec) , antiderivative size = 1002, normalized size of antiderivative = 3.14
-1/48*(d*x+c)^(1/2)*(-16*b^4*d^2*x^4*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+3 15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1 /2))*a^3*b^2*d^3*x^2-525*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^( 1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^3*c*d^2*x^2+225*ln(1/2*(2*b*d*x+2*((b*x+a )*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^4*c^2*d*x^2-15*ln(1 /2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^ 5*c^3*x^2+36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^3*d^2*x^3-52*((b*x+a) *(d*x+c))^(1/2)*(b*d)^(1/2)*b^4*c*d*x^3+630*ln(1/2*(2*b*d*x+2*((b*x+a)*(d* x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b*d^3*x-1050*ln(1/2*(2*b *d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^2*c *d^2*x+450*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/ (b*d)^(1/2))*a^2*b^3*c^2*d*x-30*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)* (b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^4*c^3*x-126*(b*d)^(1/2)*((b*x+a)*(d* x+c))^(1/2)*a^2*b^2*d^2*x^2+192*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^3* c*d*x^2-66*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^4*c^2*x^2+315*ln(1/2*(2*b *d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*d^3-5 25*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1 /2))*a^4*b*c*d^2+225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2) +a*d+b*c)/(b*d)^(1/2))*a^3*b^2*c^2*d-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c) )^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^3*c^3-840*(b*d)^(1/2)*(...
Time = 0.49 (sec) , antiderivative size = 804, normalized size of antiderivative = 2.52 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left (a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 35 \, a^{4} b c d^{2} - 21 \, a^{5} d^{3} + {\left (b^{5} c^{3} - 15 \, a b^{4} c^{2} d + 35 \, a^{2} b^{3} c d^{2} - 21 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} - 15 \, a^{2} b^{3} c^{2} d + 35 \, a^{3} b^{2} c d^{2} - 21 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{5} d^{3} x^{4} + 113 \, a^{2} b^{3} c^{2} d - 420 \, a^{3} b^{2} c d^{2} + 315 \, a^{4} b d^{3} + 2 \, {\left (13 \, b^{5} c d^{2} - 9 \, a b^{4} d^{3}\right )} x^{3} + 3 \, {\left (11 \, b^{5} c^{2} d - 32 \, a b^{4} c d^{2} + 21 \, a^{2} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (81 \, a b^{4} c^{2} d - 287 \, a^{2} b^{3} c d^{2} + 210 \, a^{3} b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}}, -\frac {15 \, {\left (a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 35 \, a^{4} b c d^{2} - 21 \, a^{5} d^{3} + {\left (b^{5} c^{3} - 15 \, a b^{4} c^{2} d + 35 \, a^{2} b^{3} c d^{2} - 21 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} - 15 \, a^{2} b^{3} c^{2} d + 35 \, a^{3} b^{2} c d^{2} - 21 \, a^{4} b d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (8 \, b^{5} d^{3} x^{4} + 113 \, a^{2} b^{3} c^{2} d - 420 \, a^{3} b^{2} c d^{2} + 315 \, a^{4} b d^{3} + 2 \, {\left (13 \, b^{5} c d^{2} - 9 \, a b^{4} d^{3}\right )} x^{3} + 3 \, {\left (11 \, b^{5} c^{2} d - 32 \, a b^{4} c d^{2} + 21 \, a^{2} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (81 \, a b^{4} c^{2} d - 287 \, a^{2} b^{3} c d^{2} + 210 \, a^{3} b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}}\right ] \]
[-1/96*(15*(a^2*b^3*c^3 - 15*a^3*b^2*c^2*d + 35*a^4*b*c*d^2 - 21*a^5*d^3 + (b^5*c^3 - 15*a*b^4*c^2*d + 35*a^2*b^3*c*d^2 - 21*a^3*b^2*d^3)*x^2 + 2*(a *b^4*c^3 - 15*a^2*b^3*c^2*d + 35*a^3*b^2*c*d^2 - 21*a^4*b*d^3)*x)*sqrt(b*d )*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c + a *d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*( 8*b^5*d^3*x^4 + 113*a^2*b^3*c^2*d - 420*a^3*b^2*c*d^2 + 315*a^4*b*d^3 + 2* (13*b^5*c*d^2 - 9*a*b^4*d^3)*x^3 + 3*(11*b^5*c^2*d - 32*a*b^4*c*d^2 + 21*a ^2*b^3*d^3)*x^2 + 2*(81*a*b^4*c^2*d - 287*a^2*b^3*c*d^2 + 210*a^3*b^2*d^3) *x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^8*d*x^2 + 2*a*b^7*d*x + a^2*b^6*d), -1 /48*(15*(a^2*b^3*c^3 - 15*a^3*b^2*c^2*d + 35*a^4*b*c*d^2 - 21*a^5*d^3 + (b ^5*c^3 - 15*a*b^4*c^2*d + 35*a^2*b^3*c*d^2 - 21*a^3*b^2*d^3)*x^2 + 2*(a*b^ 4*c^3 - 15*a^2*b^3*c^2*d + 35*a^3*b^2*c*d^2 - 21*a^4*b*d^3)*x)*sqrt(-b*d)* arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b ^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 2*(8*b^5*d^3*x^4 + 113*a^ 2*b^3*c^2*d - 420*a^3*b^2*c*d^2 + 315*a^4*b*d^3 + 2*(13*b^5*c*d^2 - 9*a*b^ 4*d^3)*x^3 + 3*(11*b^5*c^2*d - 32*a*b^4*c*d^2 + 21*a^2*b^3*d^3)*x^2 + 2*(8 1*a*b^4*c^2*d - 287*a^2*b^3*c*d^2 + 210*a^3*b^2*d^3)*x)*sqrt(b*x + a)*sqrt (d*x + c))/(b^8*d*x^2 + 2*a*b^7*d*x + a^2*b^6*d)]
\[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x^{2} \left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (275) = 550\).
Time = 0.57 (sec) , antiderivative size = 848, normalized size of antiderivative = 2.66 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {1}{24} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{7}} + \frac {13 \, b^{21} c d^{5} {\left | b \right |} - 25 \, a b^{20} d^{6} {\left | b \right |}}{b^{27} d^{4}}\right )} + \frac {3 \, {\left (11 \, b^{22} c^{2} d^{4} {\left | b \right |} - 58 \, a b^{21} c d^{5} {\left | b \right |} + 55 \, a^{2} b^{20} d^{6} {\left | b \right |}\right )}}{b^{27} d^{4}}\right )} - \frac {5 \, {\left (b^{3} c^{3} {\left | b \right |} - 15 \, a b^{2} c^{2} d {\left | b \right |} + 35 \, a^{2} b c d^{2} {\left | b \right |} - 21 \, a^{3} d^{3} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{16 \, \sqrt {b d} b^{6}} + \frac {4 \, {\left (6 \, a b^{7} c^{5} d {\left | b \right |} - 37 \, a^{2} b^{6} c^{4} d^{2} {\left | b \right |} + 88 \, a^{3} b^{5} c^{3} d^{3} {\left | b \right |} - 102 \, a^{4} b^{4} c^{2} d^{4} {\left | b \right |} + 58 \, a^{5} b^{3} c d^{5} {\left | b \right |} - 13 \, a^{6} b^{2} d^{6} {\left | b \right |} - 12 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{4} d {\left | b \right |} + 60 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c^{3} d^{2} {\left | b \right |} - 108 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} c^{2} d^{3} {\left | b \right |} + 84 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{2} c d^{4} {\left | b \right |} - 24 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{5} b d^{5} {\left | b \right |} + 6 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c^{3} d {\left | b \right |} - 27 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} c^{2} d^{2} {\left | b \right |} + 36 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b c d^{3} {\left | b \right |} - 15 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{4} d^{4} {\left | b \right |}\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} \sqrt {b d} b^{5}} \]
1/24*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b* x + a)*d^2*abs(b)/b^7 + (13*b^21*c*d^5*abs(b) - 25*a*b^20*d^6*abs(b))/(b^2 7*d^4)) + 3*(11*b^22*c^2*d^4*abs(b) - 58*a*b^21*c*d^5*abs(b) + 55*a^2*b^20 *d^6*abs(b))/(b^27*d^4)) - 5/16*(b^3*c^3*abs(b) - 15*a*b^2*c^2*d*abs(b) + 35*a^2*b*c*d^2*abs(b) - 21*a^3*d^3*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b^6) + 4/3*(6*a*b^7*c^5 *d*abs(b) - 37*a^2*b^6*c^4*d^2*abs(b) + 88*a^3*b^5*c^3*d^3*abs(b) - 102*a^ 4*b^4*c^2*d^4*abs(b) + 58*a^5*b^3*c*d^5*abs(b) - 13*a^6*b^2*d^6*abs(b) - 1 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^5* c^4*d*abs(b) + 60*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^4*c^3*d^2*abs(b) - 108*(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^2*d^3*abs(b) + 84*(sqrt(b*d)*sqrt (b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^2*c*d^4*abs(b) - 24*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b *d^5*abs(b) + 6*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a* b*d))^4*a*b^3*c^3*d*abs(b) - 27*(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^2*d^2*abs(b) + 36*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b*c*d^3*abs(b) - 15*(sqrt( b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*d^4*abs(b) )/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b...
Timed out. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x^2\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]