Integrand size = 20, antiderivative size = 209 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=-\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}+\frac {2 a^3 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}} \]
-2/3*a^3*(-a*d+b*c)*(d*x+c)^(3/2)/b^5-2/5*a^3*(d*x+c)^(5/2)/b^4+2/7*(a^2*d ^2+a*b*c*d+b^2*c^2)*(d*x+c)^(7/2)/b^3/d^3-2/9*(a*d+2*b*c)*(d*x+c)^(9/2)/b^ 2/d^3+2/11*(d*x+c)^(11/2)/b/d^3+2*a^3*(-a*d+b*c)^(5/2)*arctanh(b^(1/2)*(d* x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(13/2)-2*a^3*(-a*d+b*c)^2*(d*x+c)^(1/2)/b^6
Time = 0.37 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\frac {2 \sqrt {c+d x} \left (-3465 a^5 d^5+495 a^2 b^3 d^2 (c+d x)^3+55 a b^4 d (2 c-7 d x) (c+d x)^3+1155 a^4 b d^4 (7 c+d x)-231 a^3 b^2 d^3 \left (23 c^2+11 c d x+3 d^2 x^2\right )+5 b^5 (c+d x)^3 \left (8 c^2-28 c d x+63 d^2 x^2\right )\right )}{3465 b^6 d^3}+\frac {2 a^3 (-b c+a d)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{13/2}} \]
(2*Sqrt[c + d*x]*(-3465*a^5*d^5 + 495*a^2*b^3*d^2*(c + d*x)^3 + 55*a*b^4*d *(2*c - 7*d*x)*(c + d*x)^3 + 1155*a^4*b*d^4*(7*c + d*x) - 231*a^3*b^2*d^3* (23*c^2 + 11*c*d*x + 3*d^2*x^2) + 5*b^5*(c + d*x)^3*(8*c^2 - 28*c*d*x + 63 *d^2*x^2)))/(3465*b^6*d^3) + (2*a^3*(-(b*c) + a*d)^(5/2)*ArcTan[(Sqrt[b]*S qrt[c + d*x])/Sqrt[-(b*c) + a*d]])/b^(13/2)
Time = 0.39 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {99, 2009}
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.
| \(\displaystyle \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^3 (c+d x)^{5/2}}{b^3 (a+b x)}+\frac {(c+d x)^{5/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^2}+\frac {(c+d x)^{7/2} (-a d-2 b c)}{b^2 d^2}+\frac {(c+d x)^{9/2}}{b d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^3 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}-\frac {2 a^3 \sqrt {c+d x} (b c-a d)^2}{b^6}-\frac {2 a^3 (c+d x)^{3/2} (b c-a d)}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 (c+d x)^{7/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{7 b^3 d^3}-\frac {2 (c+d x)^{9/2} (a d+2 b c)}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}\) |
(-2*a^3*(b*c - a*d)^2*Sqrt[c + d*x])/b^6 - (2*a^3*(b*c - a*d)*(c + d*x)^(3 /2))/(3*b^5) - (2*a^3*(c + d*x)^(5/2))/(5*b^4) + (2*(b^2*c^2 + a*b*c*d + a ^2*d^2)*(c + d*x)^(7/2))/(7*b^3*d^3) - (2*(2*b*c + a*d)*(c + d*x)^(9/2))/( 9*b^2*d^3) + (2*(c + d*x)^(11/2))/(11*b*d^3) + (2*a^3*(b*c - a*d)^(5/2)*Ar cTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(13/2)
3.5.55.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 2.01 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {8 \left (\frac {63}{8} d^{2} x^{2}-\frac {7}{2} c d x +c^{2}\right ) \left (d x +c \right )^{3} b^{5}}{693}-\frac {2 \left (-\frac {7 d x}{2}+c \right ) d \left (d x +c \right )^{3} a \,b^{4}}{63}-\frac {a^{2} d^{2} \left (d x +c \right )^{3} b^{3}}{7}+\frac {23 d^{3} \left (\frac {3}{23} d^{2} x^{2}+\frac {11}{23} c d x +c^{2}\right ) a^{3} b^{2}}{15}-\frac {7 \left (\frac {d x}{7}+c \right ) d^{4} a^{4} b}{3}+a^{5} d^{5}\right ) \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}-a^{3} d^{3} \left (a d -b c \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )\right )}{\sqrt {\left (a d -b c \right ) b}\, d^{3} b^{6}}\) | \(200\) |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{\frac {11}{2}} b^{5}}{11}+\frac {\left (a d \,b^{4}+2 b^{5} c \right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {\left (-2 a d \,b^{4} c -b \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {\left (a d \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )-b \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {\left (a d \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )-b \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+a d \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right ) \sqrt {d x +c}\right )}{b^{6}}+\frac {2 a^{3} d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\) | \(348\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (d x +c \right )^{\frac {11}{2}} b^{5}}{11}+\frac {\left (a d \,b^{4}+2 b^{5} c \right ) \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {\left (-2 a d \,b^{4} c -b \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {\left (a d \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )-b \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {\left (a d \left (-a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d \right )-b \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}+a d \left (a^{4} d^{4}-2 a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}\right ) \sqrt {d x +c}\right )}{b^{6}}+\frac {2 a^{3} d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\) | \(348\) |
| risch | \(-\frac {2 \left (-315 b^{5} d^{5} x^{5}+385 x^{4} a \,b^{4} d^{5}-805 x^{4} b^{5} c \,d^{4}-495 x^{3} a^{2} b^{3} d^{5}+1045 a \,b^{4} c \,d^{4} x^{3}-565 x^{3} b^{5} c^{2} d^{3}+693 x^{2} a^{3} b^{2} d^{5}-1485 a^{2} b^{3} c \,d^{4} x^{2}+825 a \,b^{4} c^{2} d^{3} x^{2}-15 x^{2} b^{5} c^{3} d^{2}-1155 x \,a^{4} b \,d^{5}+2541 a^{3} b^{2} c \,d^{4} x -1485 a^{2} b^{3} c^{2} d^{3} x +55 a \,b^{4} c^{3} d^{2} x +20 x \,b^{5} c^{4} d +3465 a^{5} d^{5}-8085 a^{4} b c \,d^{4}+5313 a^{3} b^{2} c^{2} d^{3}-495 a^{2} b^{3} c^{3} d^{2}-110 a \,b^{4} c^{4} d -40 b^{5} c^{5}\right ) \sqrt {d x +c}}{3465 d^{3} b^{6}}+\frac {2 a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}\) | \(355\) |
-2/((a*d-b*c)*b)^(1/2)*((-8/693*(63/8*d^2*x^2-7/2*c*d*x+c^2)*(d*x+c)^3*b^5 -2/63*(-7/2*d*x+c)*d*(d*x+c)^3*a*b^4-1/7*a^2*d^2*(d*x+c)^3*b^3+23/15*d^3*( 3/23*d^2*x^2+11/23*c*d*x+c^2)*a^3*b^2-7/3*(1/7*d*x+c)*d^4*a^4*b+a^5*d^5)*( (a*d-b*c)*b)^(1/2)*(d*x+c)^(1/2)-a^3*d^3*(a*d-b*c)^3*arctan(b*(d*x+c)^(1/2 )/((a*d-b*c)*b)^(1/2)))/d^3/b^6
Time = 0.23 (sec) , antiderivative size = 713, normalized size of antiderivative = 3.41 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\left [\frac {3465 \, {\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \, {\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \, {\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} - {\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{3465 \, b^{6} d^{3}}, \frac {2 \, {\left (3465 \, {\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \, {\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \, {\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} - {\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}\right )}}{3465 \, b^{6} d^{3}}\right ] \]
[1/3465*(3465*(a^3*b^2*c^2*d^3 - 2*a^4*b*c*d^4 + a^5*d^5)*sqrt((b*c - a*d) /b)*log((b*d*x + 2*b*c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*(315*b^5*d^5*x^5 + 40*b^5*c^5 + 110*a*b^4*c^4*d + 495*a^2*b^3*c ^3*d^2 - 5313*a^3*b^2*c^2*d^3 + 8085*a^4*b*c*d^4 - 3465*a^5*d^5 + 35*(23*b ^5*c*d^4 - 11*a*b^4*d^5)*x^4 + 5*(113*b^5*c^2*d^3 - 209*a*b^4*c*d^4 + 99*a ^2*b^3*d^5)*x^3 + 3*(5*b^5*c^3*d^2 - 275*a*b^4*c^2*d^3 + 495*a^2*b^3*c*d^4 - 231*a^3*b^2*d^5)*x^2 - (20*b^5*c^4*d + 55*a*b^4*c^3*d^2 - 1485*a^2*b^3* c^2*d^3 + 2541*a^3*b^2*c*d^4 - 1155*a^4*b*d^5)*x)*sqrt(d*x + c))/(b^6*d^3) , 2/3465*(3465*(a^3*b^2*c^2*d^3 - 2*a^4*b*c*d^4 + a^5*d^5)*sqrt(-(b*c - a* d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (315*b^5 *d^5*x^5 + 40*b^5*c^5 + 110*a*b^4*c^4*d + 495*a^2*b^3*c^3*d^2 - 5313*a^3*b ^2*c^2*d^3 + 8085*a^4*b*c*d^4 - 3465*a^5*d^5 + 35*(23*b^5*c*d^4 - 11*a*b^4 *d^5)*x^4 + 5*(113*b^5*c^2*d^3 - 209*a*b^4*c*d^4 + 99*a^2*b^3*d^5)*x^3 + 3 *(5*b^5*c^3*d^2 - 275*a*b^4*c^2*d^3 + 495*a^2*b^3*c*d^4 - 231*a^3*b^2*d^5) *x^2 - (20*b^5*c^4*d + 55*a*b^4*c^3*d^2 - 1485*a^2*b^3*c^2*d^3 + 2541*a^3* b^2*c*d^4 - 1155*a^4*b*d^5)*x)*sqrt(d*x + c))/(b^6*d^3)]
Time = 2.86 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.38 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\begin {cases} \frac {2 \left (- \frac {a^{3} d^{4} \left (c + d x\right )^{\frac {5}{2}}}{5 b^{4}} + \frac {a^{3} d^{4} \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{7} \sqrt {\frac {a d - b c}{b}}} + \frac {d \left (c + d x\right )^{\frac {11}{2}}}{11 b} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (- a d^{2} - 2 b c d\right )}{9 b^{2}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (a^{2} d^{3} + a b c d^{2} + b^{2} c^{2} d\right )}{7 b^{3}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{4} d^{5} - a^{3} b c d^{4}\right )}{3 b^{5}} + \frac {\sqrt {c + d x} \left (- a^{5} d^{6} + 2 a^{4} b c d^{5} - a^{3} b^{2} c^{2} d^{4}\right )}{b^{6}}\right )}{d^{4}} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (- \frac {a^{3} \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{3}} + \frac {a^{2} x}{b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{3}}{3 b}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(-a**3*d**4*(c + d*x)**(5/2)/(5*b**4) + a**3*d**4*(a*d - b*c) **3*atan(sqrt(c + d*x)/sqrt((a*d - b*c)/b))/(b**7*sqrt((a*d - b*c)/b)) + d *(c + d*x)**(11/2)/(11*b) + (c + d*x)**(9/2)*(-a*d**2 - 2*b*c*d)/(9*b**2) + (c + d*x)**(7/2)*(a**2*d**3 + a*b*c*d**2 + b**2*c**2*d)/(7*b**3) + (c + d*x)**(3/2)*(a**4*d**5 - a**3*b*c*d**4)/(3*b**5) + sqrt(c + d*x)*(-a**5*d* *6 + 2*a**4*b*c*d**5 - a**3*b**2*c**2*d**4)/b**6)/d**4, Ne(d, 0)), (c**(5/ 2)*(-a**3*Piecewise((x/a, Eq(b, 0)), (log(a + b*x)/b, True))/b**3 + a**2*x /b**3 - a*x**2/(2*b**2) + x**3/(3*b)), True))
Exception generated. \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.46 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=-\frac {2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{6}} + \frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{10} d^{30} - 770 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{10} c d^{30} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{10} c^{2} d^{30} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} a b^{9} d^{31} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{9} c d^{31} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} a^{2} b^{8} d^{32} - 693 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{3} b^{7} d^{33} - 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{7} c d^{33} - 3465 \, \sqrt {d x + c} a^{3} b^{7} c^{2} d^{33} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{4} b^{6} d^{34} + 6930 \, \sqrt {d x + c} a^{4} b^{6} c d^{34} - 3465 \, \sqrt {d x + c} a^{5} b^{5} d^{35}\right )}}{3465 \, b^{11} d^{33}} \]
-2*(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*arctan(sqrt(d *x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^6) + 2/3465*(315*( d*x + c)^(11/2)*b^10*d^30 - 770*(d*x + c)^(9/2)*b^10*c*d^30 + 495*(d*x + c )^(7/2)*b^10*c^2*d^30 - 385*(d*x + c)^(9/2)*a*b^9*d^31 + 495*(d*x + c)^(7/ 2)*a*b^9*c*d^31 + 495*(d*x + c)^(7/2)*a^2*b^8*d^32 - 693*(d*x + c)^(5/2)*a ^3*b^7*d^33 - 1155*(d*x + c)^(3/2)*a^3*b^7*c*d^33 - 3465*sqrt(d*x + c)*a^3 *b^7*c^2*d^33 + 1155*(d*x + c)^(3/2)*a^4*b^6*d^34 + 6930*sqrt(d*x + c)*a^4 *b^6*c*d^34 - 3465*sqrt(d*x + c)*a^5*b^5*d^35)/(b^11*d^33)
Time = 0.11 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.71 \[ \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx=\left (\frac {6\,c^2}{7\,b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{7\,b\,d^3}\right )\,{\left (c+d\,x\right )}^{7/2}-\left (\frac {2\,c^3}{5\,b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{5\,b\,d^3}\right )\,{\left (c+d\,x\right )}^{5/2}-\left (\frac {2\,c}{3\,b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{9\,b^2\,d^6}\right )\,{\left (c+d\,x\right )}^{9/2}+\frac {2\,{\left (c+d\,x\right )}^{11/2}}{11\,b\,d^3}+\frac {2\,a^3\,\mathrm {atan}\left (\frac {a^3\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}\,\sqrt {c+d\,x}}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{13/2}}-\frac {\left (\frac {2\,c^3}{b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,{\left (a\,d^4-b\,c\,d^3\right )}^2\,\sqrt {c+d\,x}}{b^2\,d^6}+\frac {\left (\frac {2\,c^3}{b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,b\,d^3} \]
((6*c^2)/(7*b*d^3) + (((6*c)/(b*d^3) + (2*(a*d^4 - b*c*d^3))/(b^2*d^6))*(a *d^4 - b*c*d^3))/(7*b*d^3))*(c + d*x)^(7/2) - ((2*c^3)/(5*b*d^3) + (((6*c^ 2)/(b*d^3) + (((6*c)/(b*d^3) + (2*(a*d^4 - b*c*d^3))/(b^2*d^6))*(a*d^4 - b *c*d^3))/(b*d^3))*(a*d^4 - b*c*d^3))/(5*b*d^3))*(c + d*x)^(5/2) - ((2*c)/( 3*b*d^3) + (2*(a*d^4 - b*c*d^3))/(9*b^2*d^6))*(c + d*x)^(9/2) + (2*(c + d* x)^(11/2))/(11*b*d^3) + (2*a^3*atan((a^3*b^(1/2)*(a*d - b*c)^(5/2)*(c + d* x)^(1/2))/(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2))*(a*d - b*c)^(5/2))/b^(13/2) - (((2*c^3)/(b*d^3) + (((6*c^2)/(b*d^3) + (((6*c)/( b*d^3) + (2*(a*d^4 - b*c*d^3))/(b^2*d^6))*(a*d^4 - b*c*d^3))/(b*d^3))*(a*d ^4 - b*c*d^3))/(b*d^3))*(a*d^4 - b*c*d^3)^2*(c + d*x)^(1/2))/(b^2*d^6) + ( ((2*c^3)/(b*d^3) + (((6*c^2)/(b*d^3) + (((6*c)/(b*d^3) + (2*(a*d^4 - b*c*d ^3))/(b^2*d^6))*(a*d^4 - b*c*d^3))/(b*d^3))*(a*d^4 - b*c*d^3))/(b*d^3))*(a *d^4 - b*c*d^3)*(c + d*x)^(3/2))/(3*b*d^3)