Integrand size = 26, antiderivative size = 376 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {11 c^2 \left (5 b^2 c^4+48 a b c^2 d^2+640 a^2 d^4\right ) \sqrt {e x} \sqrt {c+d x}}{5120 d^4 e}+\frac {13 c \left (5 b^2 c^4+48 a b c^2 d^2+640 a^2 d^4\right ) (e x)^{3/2} \sqrt {c+d x}}{7680 d^3 e^2}+\frac {\left (5 b^2 c^4+48 a b c^2 d^2+640 a^2 d^4\right ) (e x)^{5/2} \sqrt {c+d x}}{1920 d^2 e^3}-\frac {b c \left (5 b c^2+48 a d^2\right ) \sqrt {e x} (c+d x)^{7/2}}{320 d^4 e}+\frac {b \left (5 b c^2+48 a d^2\right ) (e x)^{3/2} (c+d x)^{7/2}}{120 d^3 e^2}-\frac {b^2 c (e x)^{5/2} (c+d x)^{7/2}}{12 d^2 e^3}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}+\frac {c^3 \left (5 b^2 c^4+48 a b c^2 d^2+640 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{1024 d^{9/2} \sqrt {e}} \] Output:
11/5120*c^2*(640*a^2*d^4+48*a*b*c^2*d^2+5*b^2*c^4)*(e*x)^(1/2)*(d*x+c)^(1/ 2)/d^4/e+13/7680*c*(640*a^2*d^4+48*a*b*c^2*d^2+5*b^2*c^4)*(e*x)^(3/2)*(d*x +c)^(1/2)/d^3/e^2+1/1920*(640*a^2*d^4+48*a*b*c^2*d^2+5*b^2*c^4)*(e*x)^(5/2 )*(d*x+c)^(1/2)/d^2/e^3-1/320*b*c*(48*a*d^2+5*b*c^2)*(e*x)^(1/2)*(d*x+c)^( 7/2)/d^4/e+1/120*b*(48*a*d^2+5*b*c^2)*(e*x)^(3/2)*(d*x+c)^(7/2)/d^3/e^2-1/ 12*b^2*c*(e*x)^(5/2)*(d*x+c)^(7/2)/d^2/e^3+1/7*b^2*(e*x)^(7/2)*(d*x+c)^(7/ 2)/d/e^4+1/1024*c^3*(640*a^2*d^4+48*a*b*c^2*d^2+5*b^2*c^4)*arctanh(d^(1/2) *(e*x)^(1/2)/e^(1/2)/(d*x+c)^(1/2))/d^(9/2)/e^(1/2)
Time = 0.47 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.63 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {\sqrt {d} x \sqrt {c+d x} \left (4480 a^2 d^4 \left (33 c^2+26 c d x+8 d^2 x^2\right )+336 a b d^2 \left (-15 c^4+10 c^3 d x+248 c^2 d^2 x^2+336 c d^3 x^3+128 d^4 x^4\right )-5 b^2 \left (105 c^6-70 c^5 d x+56 c^4 d^2 x^2-48 c^3 d^3 x^3-4736 c^2 d^4 x^4-7424 c d^5 x^5-3072 d^6 x^6\right )\right )-105 c^3 \left (5 b^2 c^4+48 a b c^2 d^2+640 a^2 d^4\right ) \sqrt {x} \log \left (-\sqrt {d} \sqrt {x}+\sqrt {c+d x}\right )}{107520 d^{9/2} \sqrt {e x}} \] Input:
Integrate[((c + d*x)^(5/2)*(a + b*x^2)^2)/Sqrt[e*x],x]
Output:
(Sqrt[d]*x*Sqrt[c + d*x]*(4480*a^2*d^4*(33*c^2 + 26*c*d*x + 8*d^2*x^2) + 3 36*a*b*d^2*(-15*c^4 + 10*c^3*d*x + 248*c^2*d^2*x^2 + 336*c*d^3*x^3 + 128*d ^4*x^4) - 5*b^2*(105*c^6 - 70*c^5*d*x + 56*c^4*d^2*x^2 - 48*c^3*d^3*x^3 - 4736*c^2*d^4*x^4 - 7424*c*d^5*x^5 - 3072*d^6*x^6)) - 105*c^3*(5*b^2*c^4 + 48*a*b*c^2*d^2 + 640*a^2*d^4)*Sqrt[x]*Log[-(Sqrt[d]*Sqrt[x]) + Sqrt[c + d* x]])/(107520*d^(9/2)*Sqrt[e*x])
Time = 0.92 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.86, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {521, 27, 2125, 27, 521, 27, 90, 60, 60, 60, 65, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)^{5/2}}{\sqrt {e x}} \, dx\) |
| \(\Big \downarrow \) 521 |
| \(\displaystyle \frac {\int \frac {7 (c+d x)^{5/2} \left (-b^2 c x^3 e^4+4 a b d x^2 e^4+2 a^2 d e^4\right )}{2 \sqrt {e x}}dx}{7 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{5/2} \left (-b^2 c x^3 e^4+4 a b d x^2 e^4+2 a^2 d e^4\right )}{\sqrt {e x}}dx}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 2125 |
| \(\displaystyle \frac {\frac {\int \frac {e^7 (c+d x)^{5/2} \left (24 a^2 d^2+b \left (5 b c^2+48 a d^2\right ) x^2\right )}{2 \sqrt {e x}}dx}{6 d e^3}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e^4 \int \frac {(c+d x)^{5/2} \left (24 a^2 d^2+b \left (5 b c^2+48 a d^2\right ) x^2\right )}{\sqrt {e x}}dx}{12 d}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 521 |
| \(\displaystyle \frac {\frac {e^4 \left (\frac {\int \frac {3 e^2 (c+d x)^{5/2} \left (80 a^2 d^3-b c \left (5 b c^2+48 a d^2\right ) x\right )}{2 \sqrt {e x}}dx}{5 d e^2}+\frac {b (e x)^{3/2} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{5 d e^2}\right )}{12 d}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e^4 \left (\frac {3 \int \frac {(c+d x)^{5/2} \left (80 a^2 d^3-b c \left (5 b c^2+48 a d^2\right ) x\right )}{\sqrt {e x}}dx}{10 d}+\frac {b (e x)^{3/2} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{5 d e^2}\right )}{12 d}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\frac {e^4 \left (\frac {3 \left (\frac {\left (640 a^2 d^4+48 a b c^2 d^2+5 b^2 c^4\right ) \int \frac {(c+d x)^{5/2}}{\sqrt {e x}}dx}{8 d}-\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{3/2} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{5 d e^2}\right )}{12 d}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {e^4 \left (\frac {3 \left (\frac {\left (640 a^2 d^4+48 a b c^2 d^2+5 b^2 c^4\right ) \left (\frac {5}{6} c \int \frac {(c+d x)^{3/2}}{\sqrt {e x}}dx+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}-\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{3/2} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{5 d e^2}\right )}{12 d}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {e^4 \left (\frac {3 \left (\frac {\left (640 a^2 d^4+48 a b c^2 d^2+5 b^2 c^4\right ) \left (\frac {5}{6} c \left (\frac {3}{4} c \int \frac {\sqrt {c+d x}}{\sqrt {e x}}dx+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}-\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{3/2} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{5 d e^2}\right )}{12 d}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {e^4 \left (\frac {3 \left (\frac {\left (640 a^2 d^4+48 a b c^2 d^2+5 b^2 c^4\right ) \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}-\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{3/2} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{5 d e^2}\right )}{12 d}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {\frac {e^4 \left (\frac {3 \left (\frac {\left (640 a^2 d^4+48 a b c^2 d^2+5 b^2 c^4\right ) \left (\frac {5}{6} c \left (\frac {3}{4} c \left (c \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}-\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{3/2} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{5 d e^2}\right )}{12 d}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {e^4 \left (\frac {3 \left (\frac {\left (640 a^2 d^4+48 a b c^2 d^2+5 b^2 c^4\right ) \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{\sqrt {d} \sqrt {e}}+\frac {\sqrt {e x} \sqrt {c+d x}}{e}\right )+\frac {\sqrt {e x} (c+d x)^{3/2}}{2 e}\right )+\frac {\sqrt {e x} (c+d x)^{5/2}}{3 e}\right )}{8 d}-\frac {b c \sqrt {e x} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{4 d e}\right )}{10 d}+\frac {b (e x)^{3/2} (c+d x)^{7/2} \left (48 a d^2+5 b c^2\right )}{5 d e^2}\right )}{12 d}-\frac {b^2 c e (e x)^{5/2} (c+d x)^{7/2}}{6 d}}{2 d e^4}+\frac {b^2 (e x)^{7/2} (c+d x)^{7/2}}{7 d e^4}\) |
Input:
Int[((c + d*x)^(5/2)*(a + b*x^2)^2)/Sqrt[e*x],x]
Output:
(b^2*(e*x)^(7/2)*(c + d*x)^(7/2))/(7*d*e^4) + (-1/6*(b^2*c*e*(e*x)^(5/2)*( c + d*x)^(7/2))/d + (e^4*((b*(5*b*c^2 + 48*a*d^2)*(e*x)^(3/2)*(c + d*x)^(7 /2))/(5*d*e^2) + (3*(-1/4*(b*c*(5*b*c^2 + 48*a*d^2)*Sqrt[e*x]*(c + d*x)^(7 /2))/(d*e) + ((5*b^2*c^4 + 48*a*b*c^2*d^2 + 640*a^2*d^4)*((Sqrt[e*x]*(c + d*x)^(5/2))/(3*e) + (5*c*((Sqrt[e*x]*(c + d*x)^(3/2))/(2*e) + (3*c*((Sqrt[ e*x]*Sqrt[c + d*x])/e + (c*ArcTanh[(Sqrt[d]*Sqrt[e*x])/(Sqrt[e]*Sqrt[c + d *x])])/(Sqrt[d]*Sqrt[e])))/4))/6))/(8*d)))/(10*d)))/(12*d))/(2*d*e^4)
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[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[b^p*(e*x)^(m + 2*p)*((c + d*x)^(n + 1)/(d*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(d*e^(2*p)*(m + n + 2*p + 1)) Int[(e*x)^m*(c + d*x)^n*ExpandToSum[d*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x^2)^p - b^p*(e*x)^ (2*p)) - b^p*(e*c)*(m + 2*p)*(e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] && !IntegerQ[m] && !I ntegerQ[n]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x )^(m + q)*((c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1))), x] + Simp[1/(d*b^q*( m + n + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q + 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x) ^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x]
Time = 0.28 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(\frac {\left (15360 b^{2} d^{6} x^{6}+37120 b^{2} c \,d^{5} x^{5}+43008 x^{4} a b \,d^{6}+23680 b^{2} c^{2} d^{4} x^{4}+112896 x^{3} a b c \,d^{5}+240 b^{2} c^{3} d^{3} x^{3}+35840 x^{2} a^{2} d^{6}+83328 x^{2} a b \,c^{2} d^{4}-280 b^{2} c^{4} d^{2} x^{2}+116480 x \,a^{2} c \,d^{5}+3360 x a b \,c^{3} d^{3}+350 b^{2} c^{5} d x +147840 a^{2} c^{2} d^{4}-5040 a b \,c^{4} d^{2}-525 c^{6} b^{2}\right ) x \sqrt {d x +c}}{107520 d^{4} \sqrt {e x}}+\frac {c^{3} \left (640 a^{2} d^{4}+48 b \,c^{2} d^{2} a +5 b^{2} c^{4}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) \sqrt {\left (d x +c \right ) e x}}{2048 d^{4} \sqrt {d e}\, \sqrt {e x}\, \sqrt {d x +c}}\) | \(283\) |
| default | \(\frac {\sqrt {d x +c}\, x \left (30720 b^{2} d^{6} x^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+74240 b^{2} c \,d^{5} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+86016 a b \,d^{6} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+47360 b^{2} c^{2} d^{4} x^{4} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+225792 a b c \,d^{5} x^{3} \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}+480 b^{2} c^{3} d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+71680 a^{2} d^{6} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+166656 a b \,c^{2} d^{4} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-560 b^{2} c^{4} d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+67200 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} c^{3} d^{4} e +5040 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{5} d^{2} e +525 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{7} e +232960 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c \,d^{5} x +6720 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{3} d^{3} x +700 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{5} d x +295680 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c^{2} d^{4}-10080 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, a b \,c^{4} d^{2}-1050 \sqrt {d e}\, \sqrt {\left (d x +c \right ) e x}\, b^{2} c^{6}\right )}{215040 d^{4} \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\) | \(564\) |
Input:
int((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/107520/d^4*(15360*b^2*d^6*x^6+37120*b^2*c*d^5*x^5+43008*a*b*d^6*x^4+2368 0*b^2*c^2*d^4*x^4+112896*a*b*c*d^5*x^3+240*b^2*c^3*d^3*x^3+35840*a^2*d^6*x ^2+83328*a*b*c^2*d^4*x^2-280*b^2*c^4*d^2*x^2+116480*a^2*c*d^5*x+3360*a*b*c ^3*d^3*x+350*b^2*c^5*d*x+147840*a^2*c^2*d^4-5040*a*b*c^4*d^2-525*b^2*c^6)* x*(d*x+c)^(1/2)/(e*x)^(1/2)+1/2048/d^4*c^3*(640*a^2*d^4+48*a*b*c^2*d^2+5*b ^2*c^4)*ln((1/2*c*e+d*e*x)/(d*e)^(1/2)+(d*e*x^2+c*e*x)^(1/2))/(d*e)^(1/2)* ((d*x+c)*e*x)^(1/2)/(e*x)^(1/2)/(d*x+c)^(1/2)
Time = 0.14 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.40 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\left [\frac {105 \, {\left (5 \, b^{2} c^{7} + 48 \, a b c^{5} d^{2} + 640 \, a^{2} c^{3} d^{4}\right )} \sqrt {d e} \log \left (2 \, d e x + c e + 2 \, \sqrt {d e} \sqrt {d x + c} \sqrt {e x}\right ) + 2 \, {\left (15360 \, b^{2} d^{7} x^{6} + 37120 \, b^{2} c d^{6} x^{5} - 525 \, b^{2} c^{6} d - 5040 \, a b c^{4} d^{3} + 147840 \, a^{2} c^{2} d^{5} + 128 \, {\left (185 \, b^{2} c^{2} d^{5} + 336 \, a b d^{7}\right )} x^{4} + 48 \, {\left (5 \, b^{2} c^{3} d^{4} + 2352 \, a b c d^{6}\right )} x^{3} - 56 \, {\left (5 \, b^{2} c^{4} d^{3} - 1488 \, a b c^{2} d^{5} - 640 \, a^{2} d^{7}\right )} x^{2} + 70 \, {\left (5 \, b^{2} c^{5} d^{2} + 48 \, a b c^{3} d^{4} + 1664 \, a^{2} c d^{6}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{215040 \, d^{5} e}, -\frac {105 \, {\left (5 \, b^{2} c^{7} + 48 \, a b c^{5} d^{2} + 640 \, a^{2} c^{3} d^{4}\right )} \sqrt {-d e} \arctan \left (\frac {\sqrt {-d e} \sqrt {d x + c} \sqrt {e x}}{d e x + c e}\right ) - {\left (15360 \, b^{2} d^{7} x^{6} + 37120 \, b^{2} c d^{6} x^{5} - 525 \, b^{2} c^{6} d - 5040 \, a b c^{4} d^{3} + 147840 \, a^{2} c^{2} d^{5} + 128 \, {\left (185 \, b^{2} c^{2} d^{5} + 336 \, a b d^{7}\right )} x^{4} + 48 \, {\left (5 \, b^{2} c^{3} d^{4} + 2352 \, a b c d^{6}\right )} x^{3} - 56 \, {\left (5 \, b^{2} c^{4} d^{3} - 1488 \, a b c^{2} d^{5} - 640 \, a^{2} d^{7}\right )} x^{2} + 70 \, {\left (5 \, b^{2} c^{5} d^{2} + 48 \, a b c^{3} d^{4} + 1664 \, a^{2} c d^{6}\right )} x\right )} \sqrt {d x + c} \sqrt {e x}}{107520 \, d^{5} e}\right ] \] Input:
integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(1/2),x, algorithm="fricas")
Output:
[1/215040*(105*(5*b^2*c^7 + 48*a*b*c^5*d^2 + 640*a^2*c^3*d^4)*sqrt(d*e)*lo g(2*d*e*x + c*e + 2*sqrt(d*e)*sqrt(d*x + c)*sqrt(e*x)) + 2*(15360*b^2*d^7* x^6 + 37120*b^2*c*d^6*x^5 - 525*b^2*c^6*d - 5040*a*b*c^4*d^3 + 147840*a^2* c^2*d^5 + 128*(185*b^2*c^2*d^5 + 336*a*b*d^7)*x^4 + 48*(5*b^2*c^3*d^4 + 23 52*a*b*c*d^6)*x^3 - 56*(5*b^2*c^4*d^3 - 1488*a*b*c^2*d^5 - 640*a^2*d^7)*x^ 2 + 70*(5*b^2*c^5*d^2 + 48*a*b*c^3*d^4 + 1664*a^2*c*d^6)*x)*sqrt(d*x + c)* sqrt(e*x))/(d^5*e), -1/107520*(105*(5*b^2*c^7 + 48*a*b*c^5*d^2 + 640*a^2*c ^3*d^4)*sqrt(-d*e)*arctan(sqrt(-d*e)*sqrt(d*x + c)*sqrt(e*x)/(d*e*x + c*e) ) - (15360*b^2*d^7*x^6 + 37120*b^2*c*d^6*x^5 - 525*b^2*c^6*d - 5040*a*b*c^ 4*d^3 + 147840*a^2*c^2*d^5 + 128*(185*b^2*c^2*d^5 + 336*a*b*d^7)*x^4 + 48* (5*b^2*c^3*d^4 + 2352*a*b*c*d^6)*x^3 - 56*(5*b^2*c^4*d^3 - 1488*a*b*c^2*d^ 5 - 640*a^2*d^7)*x^2 + 70*(5*b^2*c^5*d^2 + 48*a*b*c^3*d^4 + 1664*a^2*c*d^6 )*x)*sqrt(d*x + c)*sqrt(e*x))/(d^5*e)]
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(5/2)*(b*x**2+a)**2/(e*x)**(1/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(1/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.21 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {{\left (\sqrt {{\left (d x + c\right )} d e - c d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (d x + c\right )} {\left (10 \, {\left (d x + c\right )} {\left (\frac {12 \, {\left (d x + c\right )} b^{2}}{d^{5} e} - \frac {43 \, b^{2} c}{d^{5} e}\right )} + \frac {535 \, b^{2} c^{2} d^{20} e^{4} + 336 \, a b d^{22} e^{4}}{d^{25} e^{5}}\right )} - \frac {3 \, {\left (635 \, b^{2} c^{3} d^{20} e^{4} + 1232 \, a b c d^{22} e^{4}\right )}}{d^{25} e^{5}}\right )} {\left (d x + c\right )} + \frac {7 \, {\left (5 \, b^{2} c^{4} d^{20} e^{4} + 48 \, a b c^{2} d^{22} e^{4} + 640 \, a^{2} d^{24} e^{4}\right )}}{d^{25} e^{5}}\right )} {\left (d x + c\right )} + \frac {35 \, {\left (5 \, b^{2} c^{5} d^{20} e^{4} + 48 \, a b c^{3} d^{22} e^{4} + 640 \, a^{2} c d^{24} e^{4}\right )}}{d^{25} e^{5}}\right )} {\left (d x + c\right )} + \frac {105 \, {\left (5 \, b^{2} c^{6} d^{20} e^{4} + 48 \, a b c^{4} d^{22} e^{4} + 640 \, a^{2} c^{2} d^{24} e^{4}\right )}}{d^{25} e^{5}}\right )} \sqrt {d x + c} - \frac {105 \, {\left (5 \, b^{2} c^{7} + 48 \, a b c^{5} d^{2} + 640 \, a^{2} c^{3} d^{4}\right )} \log \left ({\left | -\sqrt {d e} \sqrt {d x + c} + \sqrt {{\left (d x + c\right )} d e - c d e} \right |}\right )}{\sqrt {d e} d^{4}}\right )} d}{107520 \, {\left | d \right |}} \] Input:
integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(1/2),x, algorithm="giac")
Output:
1/107520*(sqrt((d*x + c)*d*e - c*d*e)*(2*(4*(2*(8*(d*x + c)*(10*(d*x + c)* (12*(d*x + c)*b^2/(d^5*e) - 43*b^2*c/(d^5*e)) + (535*b^2*c^2*d^20*e^4 + 33 6*a*b*d^22*e^4)/(d^25*e^5)) - 3*(635*b^2*c^3*d^20*e^4 + 1232*a*b*c*d^22*e^ 4)/(d^25*e^5))*(d*x + c) + 7*(5*b^2*c^4*d^20*e^4 + 48*a*b*c^2*d^22*e^4 + 6 40*a^2*d^24*e^4)/(d^25*e^5))*(d*x + c) + 35*(5*b^2*c^5*d^20*e^4 + 48*a*b*c ^3*d^22*e^4 + 640*a^2*c*d^24*e^4)/(d^25*e^5))*(d*x + c) + 105*(5*b^2*c^6*d ^20*e^4 + 48*a*b*c^4*d^22*e^4 + 640*a^2*c^2*d^24*e^4)/(d^25*e^5))*sqrt(d*x + c) - 105*(5*b^2*c^7 + 48*a*b*c^5*d^2 + 640*a^2*c^3*d^4)*log(abs(-sqrt(d *e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/(sqrt(d*e)*d^4))*d/abs(d )
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (c+d\,x\right )}^{5/2}}{\sqrt {e\,x}} \,d x \] Input:
int(((a + b*x^2)^2*(c + d*x)^(5/2))/(e*x)^(1/2),x)
Output:
int(((a + b*x^2)^2*(c + d*x)^(5/2))/(e*x)^(1/2), x)
Time = 0.32 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {e x}} \, dx=\frac {\sqrt {e}\, \left (147840 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{5}+116480 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{6} x +35840 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{7} x^{2}-5040 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{3}+3360 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{4} x +83328 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{5} x^{2}+112896 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{6} x^{3}+43008 \sqrt {x}\, \sqrt {d x +c}\, a b \,d^{7} x^{4}-525 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} d +350 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d^{2} x -280 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{3} x^{2}+240 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d^{4} x^{3}+23680 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{5} x^{4}+37120 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{6} x^{5}+15360 \sqrt {x}\, \sqrt {d x +c}\, b^{2} d^{7} x^{6}+67200 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} c^{3} d^{4}+5040 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{5} d^{2}+525 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{7}\right )}{107520 d^{5} e} \] Input:
int((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(1/2),x)
Output:
(sqrt(e)*(147840*sqrt(x)*sqrt(c + d*x)*a**2*c**2*d**5 + 116480*sqrt(x)*sqr t(c + d*x)*a**2*c*d**6*x + 35840*sqrt(x)*sqrt(c + d*x)*a**2*d**7*x**2 - 50 40*sqrt(x)*sqrt(c + d*x)*a*b*c**4*d**3 + 3360*sqrt(x)*sqrt(c + d*x)*a*b*c* *3*d**4*x + 83328*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**5*x**2 + 112896*sqrt(x )*sqrt(c + d*x)*a*b*c*d**6*x**3 + 43008*sqrt(x)*sqrt(c + d*x)*a*b*d**7*x** 4 - 525*sqrt(x)*sqrt(c + d*x)*b**2*c**6*d + 350*sqrt(x)*sqrt(c + d*x)*b**2 *c**5*d**2*x - 280*sqrt(x)*sqrt(c + d*x)*b**2*c**4*d**3*x**2 + 240*sqrt(x) *sqrt(c + d*x)*b**2*c**3*d**4*x**3 + 23680*sqrt(x)*sqrt(c + d*x)*b**2*c**2 *d**5*x**4 + 37120*sqrt(x)*sqrt(c + d*x)*b**2*c*d**6*x**5 + 15360*sqrt(x)* sqrt(c + d*x)*b**2*d**7*x**6 + 67200*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)* sqrt(d))/sqrt(c))*a**2*c**3*d**4 + 5040*sqrt(d)*log((sqrt(c + d*x) + sqrt( x)*sqrt(d))/sqrt(c))*a*b*c**5*d**2 + 525*sqrt(d)*log((sqrt(c + d*x) + sqrt (x)*sqrt(d))/sqrt(c))*b**2*c**7))/(107520*d**5*e)