Integrand size = 22, antiderivative size = 497 \[ \int \frac {x^4 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^7 d^2}+\frac {\left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \sqrt {a+b x} (c+d x)^{3/2}}{192 b^6 d^2 (b c-a d)}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac {2 (8 b c-13 a d) x^3 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt {a+b x}}+\frac {(93 b c-143 a d) x^2 \sqrt {a+b x} (c+d x)^{5/2}}{15 b^3 (b c-a d)}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (15 b^3 c^3+125 a b^2 c^2 d-2343 a^2 b c d^2+3003 a^3 d^3-2 b d \left (15 b^2 c^2-902 a b c d+1287 a^2 d^2\right ) x\right )}{240 b^5 d^2 (b c-a d)}+\frac {(b c-a d) \left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{15/2} d^{5/2}} \]
-2/3*x^4*(d*x+c)^(5/2)/b/(b*x+a)^(3/2)+1/128*(-a*d+b*c)*(3003*a^4*d^4-2772 *a^3*b*c*d^3+378*a^2*b^2*c^2*d^2+28*a*b^3*c^3*d+3*b^4*c^4)*arctanh(d^(1/2) *(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(15/2)/d^(5/2)-2/3*(-13*a*d+8*b*c) *x^3*(d*x+c)^(5/2)/b^2/(-a*d+b*c)/(b*x+a)^(1/2)+1/192*(3003*a^4*d^4-2772*a ^3*b*c*d^3+378*a^2*b^2*c^2*d^2+28*a*b^3*c^3*d+3*b^4*c^4)*(d*x+c)^(3/2)*(b* x+a)^(1/2)/b^6/d^2/(-a*d+b*c)+1/15*(-143*a*d+93*b*c)*x^2*(d*x+c)^(5/2)*(b* x+a)^(1/2)/b^3/(-a*d+b*c)-1/240*(d*x+c)^(5/2)*(15*b^3*c^3+125*a*b^2*c^2*d- 2343*a^2*b*c*d^2+3003*a^3*d^3-2*b*d*(1287*a^2*d^2-902*a*b*c*d+15*b^2*c^2)* x)*(b*x+a)^(1/2)/b^5/d^2/(-a*d+b*c)+1/128*(3003*a^4*d^4-2772*a^3*b*c*d^3+3 78*a^2*b^2*c^2*d^2+28*a*b^3*c^3*d+3*b^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b ^7/d^2
Time = 11.16 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.79 \[ \int \frac {x^4 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {c+d x} \left (\frac {\sqrt {d} \left (45045 a^6 d^4+2310 a^5 b d^3 (-31 c+26 d x)+21 a^4 b^2 d^2 \left (1304 c^2-4642 c d x+429 d^2 x^2\right )-6 a^3 b^3 d \left (65 c^3-6441 c^2 d x+2673 c d^2 x^2+429 d^3 x^3\right )+3 b^6 x^2 \left (-15 c^4+10 c^3 d x+248 c^2 d^2 x^2+336 c d^3 x^3+128 d^4 x^4\right )-2 a b^5 x \left (45 c^4+165 c^3 d x+917 c^2 d^2 x^2+944 c d^3 x^3+312 d^4 x^4\right )+a^2 b^4 \left (-45 c^4-750 c^3 d x+7404 c^2 d^2 x^2+4378 c d^3 x^3+1144 d^4 x^4\right )\right )}{(a+b x)^{3/2}}+\frac {15 \sqrt {b c-a d} \left (3 b^4 c^4+28 a b^3 c^3 d+378 a^2 b^2 c^2 d^2-2772 a^3 b c d^3+3003 a^4 d^4\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{1920 b^7 d^{5/2}} \]
(Sqrt[c + d*x]*((Sqrt[d]*(45045*a^6*d^4 + 2310*a^5*b*d^3*(-31*c + 26*d*x) + 21*a^4*b^2*d^2*(1304*c^2 - 4642*c*d*x + 429*d^2*x^2) - 6*a^3*b^3*d*(65*c ^3 - 6441*c^2*d*x + 2673*c*d^2*x^2 + 429*d^3*x^3) + 3*b^6*x^2*(-15*c^4 + 1 0*c^3*d*x + 248*c^2*d^2*x^2 + 336*c*d^3*x^3 + 128*d^4*x^4) - 2*a*b^5*x*(45 *c^4 + 165*c^3*d*x + 917*c^2*d^2*x^2 + 944*c*d^3*x^3 + 312*d^4*x^4) + a^2* b^4*(-45*c^4 - 750*c^3*d*x + 7404*c^2*d^2*x^2 + 4378*c*d^3*x^3 + 1144*d^4* x^4)))/(a + b*x)^(3/2) + (15*Sqrt[b*c - a*d]*(3*b^4*c^4 + 28*a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*ArcSinh[(Sqrt[d]*S qrt[a + b*x])/Sqrt[b*c - a*d]])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(1920*b^ 7*d^(5/2))
Time = 0.52 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.82, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {108, 27, 167, 27, 170, 27, 164, 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.
| \(\displaystyle \int \frac {x^4 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2 \int \frac {x^3 (c+d x)^{3/2} (8 c+13 d x)}{2 (a+b x)^{3/2}}dx}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^3 (c+d x)^{3/2} (8 c+13 d x)}{(a+b x)^{3/2}}dx}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {\frac {2 \int \frac {x^2 (c+d x)^{3/2} (6 c (8 b c-13 a d)+d (93 b c-143 a d) x)}{2 \sqrt {a+b x}}dx}{b (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 (c+d x)^{3/2} (6 c (8 b c-13 a d)+d (93 b c-143 a d) x)}{\sqrt {a+b x}}dx}{b (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {d x (c+d x)^{3/2} \left (4 a c (93 b c-143 a d)-\left (15 b^2 c^2-902 a b d c+1287 a^2 d^2\right ) x\right )}{2 \sqrt {a+b x}}dx}{5 b d}+\frac {x^2 \sqrt {a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{5 b}}{b (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{5 b}-\frac {\int \frac {x (c+d x)^{3/2} \left (4 a c (93 b c-143 a d)-\left (15 b^2 c^2-902 a b d c+1287 a^2 d^2\right ) x\right )}{\sqrt {a+b x}}dx}{10 b}}{b (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (3003 a^3 d^3-2 b d x \left (1287 a^2 d^2-902 a b c d+15 b^2 c^2\right )-2343 a^2 b c d^2+125 a b^2 c^2 d+15 b^3 c^3\right )}{8 b^2 d^2}-\frac {5 \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}}dx}{16 b^2 d^2}}{10 b}}{b (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (3003 a^3 d^3-2 b d x \left (1287 a^2 d^2-902 a b c d+15 b^2 c^2\right )-2343 a^2 b c d^2+125 a b^2 c^2 d+15 b^3 c^3\right )}{8 b^2 d^2}-\frac {5 \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right ) \left (\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d^2}}{10 b}}{b (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (3003 a^3 d^3-2 b d x \left (1287 a^2 d^2-902 a b c d+15 b^2 c^2\right )-2343 a^2 b c d^2+125 a b^2 c^2 d+15 b^3 c^3\right )}{8 b^2 d^2}-\frac {5 \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d^2}}{10 b}}{b (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (3003 a^3 d^3-2 b d x \left (1287 a^2 d^2-902 a b c d+15 b^2 c^2\right )-2343 a^2 b c d^2+125 a b^2 c^2 d+15 b^3 c^3\right )}{8 b^2 d^2}-\frac {5 \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d^2}}{10 b}}{b (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {x^2 \sqrt {a+b x} (c+d x)^{5/2} (93 b c-143 a d)}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (3003 a^3 d^3-2 b d x \left (1287 a^2 d^2-902 a b c d+15 b^2 c^2\right )-2343 a^2 b c d^2+125 a b^2 c^2 d+15 b^3 c^3\right )}{8 b^2 d^2}-\frac {5 \left (3003 a^4 d^4-2772 a^3 b c d^3+378 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+3 b^4 c^4\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d^2}}{10 b}}{b (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2} (8 b c-13 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^4 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
(-2*x^4*(c + d*x)^(5/2))/(3*b*(a + b*x)^(3/2)) + ((-2*(8*b*c - 13*a*d)*x^3 *(c + d*x)^(5/2))/(b*(b*c - a*d)*Sqrt[a + b*x]) + (((93*b*c - 143*a*d)*x^2 *Sqrt[a + b*x]*(c + d*x)^(5/2))/(5*b) - ((Sqrt[a + b*x]*(c + d*x)^(5/2)*(1 5*b^3*c^3 + 125*a*b^2*c^2*d - 2343*a^2*b*c*d^2 + 3003*a^3*d^3 - 2*b*d*(15* b^2*c^2 - 902*a*b*c*d + 1287*a^2*d^2)*x))/(8*b^2*d^2) - (5*(3*b^4*c^4 + 28 *a*b^3*c^3*d + 378*a^2*b^2*c^2*d^2 - 2772*a^3*b*c*d^3 + 3003*a^4*d^4)*((Sq rt[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)))/(16*b^2*d^2))/(10*b))/(b*(b*c - a*d) ))/(3*b)
3.8.93.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] - Simp[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] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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. \(1761\) vs. \(2(447)=894\).
Time = 0.60 (sec) , antiderivative size = 1762, normalized size of antiderivative = 3.55
-1/3840*(d*x+c)^(1/2)*(3776*a*b^5*c*d^3*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^ (1/2)-375*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^6*c^4*d*x^2+180*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^5* c^4*x+780*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^3*d-45*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^7*c^5*x ^2-45*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^5*c^5-90090*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^6*d^4+1500 *((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^4*c^3*d*x-90*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^6*c^5*x-375* 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^4*c^4*d+45045*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*b^2*d^5*x^2-750*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^5*c^4*d*x-768*b^6*d^4* x^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+90*((b*x+a)*(d*x+c))^(1/2)*(b*d)^( 1/2)*b^6*c^4*x^2+90*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^4*c^4+45045* 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^7*d^5-8756*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^4*c*d^3*x^3+3668* ((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^5*c^2*d^2*x^3+90090*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^6*b*d^5*x- 86625*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...
Time = 0.88 (sec) , antiderivative size = 1338, normalized size of antiderivative = 2.69 \[ \int \frac {x^4 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \]
[-1/7680*(15*(3*a^2*b^5*c^5 + 25*a^3*b^4*c^4*d + 350*a^4*b^3*c^3*d^2 - 315 0*a^5*b^2*c^2*d^3 + 5775*a^6*b*c*d^4 - 3003*a^7*d^5 + (3*b^7*c^5 + 25*a*b^ 6*c^4*d + 350*a^2*b^5*c^3*d^2 - 3150*a^3*b^4*c^2*d^3 + 5775*a^4*b^3*c*d^4 - 3003*a^5*b^2*d^5)*x^2 + 2*(3*a*b^6*c^5 + 25*a^2*b^5*c^4*d + 350*a^3*b^4* c^3*d^2 - 3150*a^4*b^3*c^2*d^3 + 5775*a^5*b^2*c*d^4 - 3003*a^6*b*d^5)*x)*s qrt(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*(384*b^7*d^5*x^6 - 45*a^2*b^5*c^4*d - 390*a^3*b^4*c^3*d^2 + 27384*a^ 4*b^3*c^2*d^3 - 71610*a^5*b^2*c*d^4 + 45045*a^6*b*d^5 + 48*(21*b^7*c*d^4 - 13*a*b^6*d^5)*x^5 + 8*(93*b^7*c^2*d^3 - 236*a*b^6*c*d^4 + 143*a^2*b^5*d^5 )*x^4 + 2*(15*b^7*c^3*d^2 - 917*a*b^6*c^2*d^3 + 2189*a^2*b^5*c*d^4 - 1287* a^3*b^4*d^5)*x^3 - 3*(15*b^7*c^4*d + 110*a*b^6*c^3*d^2 - 2468*a^2*b^5*c^2* d^3 + 5346*a^3*b^4*c*d^4 - 3003*a^4*b^3*d^5)*x^2 - 6*(15*a*b^6*c^4*d + 125 *a^2*b^5*c^3*d^2 - 6441*a^3*b^4*c^2*d^3 + 16247*a^4*b^3*c*d^4 - 10010*a^5* b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^10*d^3*x^2 + 2*a*b^9*d^3*x + a ^2*b^8*d^3), -1/3840*(15*(3*a^2*b^5*c^5 + 25*a^3*b^4*c^4*d + 350*a^4*b^3*c ^3*d^2 - 3150*a^5*b^2*c^2*d^3 + 5775*a^6*b*c*d^4 - 3003*a^7*d^5 + (3*b^7*c ^5 + 25*a*b^6*c^4*d + 350*a^2*b^5*c^3*d^2 - 3150*a^3*b^4*c^2*d^3 + 5775*a^ 4*b^3*c*d^4 - 3003*a^5*b^2*d^5)*x^2 + 2*(3*a*b^6*c^5 + 25*a^2*b^5*c^4*d + 350*a^3*b^4*c^3*d^2 - 3150*a^4*b^3*c^2*d^3 + 5775*a^5*b^2*c*d^4 - 3003*...
\[ \int \frac {x^4 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x^{4} \left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^4 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 1047 vs. \(2 (447) = 894\).
Time = 0.75 (sec) , antiderivative size = 1047, normalized size of antiderivative = 2.11 \[ \int \frac {x^4 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \]
1/1920*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8 *(b*x + a)*d^2*abs(b)/b^9 + (21*b^45*c*d^9*abs(b) - 61*a*b^44*d^10*abs(b)) /(b^53*d^8)) + (93*b^46*c^2*d^8*abs(b) - 866*a*b^45*c*d^9*abs(b) + 1253*a^ 2*b^44*d^10*abs(b))/(b^53*d^8)) + 5*(3*b^47*c^3*d^7*abs(b) - 481*a*b^46*c^ 2*d^8*abs(b) + 2201*a^2*b^45*c*d^9*abs(b) - 2107*a^3*b^44*d^10*abs(b))/(b^ 53*d^8))*(b*x + a) - 15*(3*b^48*c^4*d^6*abs(b) + 28*a*b^47*c^3*d^7*abs(b) - 1158*a^2*b^46*c^2*d^8*abs(b) + 3372*a^3*b^45*c*d^9*abs(b) - 2373*a^4*b^4 4*d^10*abs(b))/(b^53*d^8))*sqrt(b*x + a) - 1/256*(3*b^5*c^5*abs(b) + 25*a* b^4*c^4*d*abs(b) + 350*a^2*b^3*c^3*d^2*abs(b) - 3150*a^3*b^2*c^2*d^3*abs(b ) + 5775*a^4*b*c*d^4*abs(b) - 3003*a^5*d^5*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^8*d^2) + 4/3* (12*a^3*b^7*c^5*d*abs(b) - 67*a^4*b^6*c^4*d^2*abs(b) + 148*a^5*b^5*c^3*d^3 *abs(b) - 162*a^6*b^4*c^2*d^4*abs(b) + 88*a^7*b^3*c*d^5*abs(b) - 19*a^8*b^ 2*d^6*abs(b) - 24*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^5*c^4*d*abs(b) + 108*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^4*c^3*d^2*abs(b) - 180*(sqrt(b*d)*sqrt( b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^3*c^2*d^3*abs(b) + 132*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6 *b^2*c*d^4*abs(b) - 36*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b *d - a*b*d))^2*a^7*b*d^5*abs(b) + 12*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^...
Timed out. \[ \int \frac {x^4 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x^4\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]