Integrand size = 24, antiderivative size = 205 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x} \, dx=\frac {2 a^4 (b c-a d) \left (a x^2+b x^3\right )^{7/2}}{7 b^6 x^7}-\frac {2 a^3 (4 b c-5 a d) \left (a x^2+b x^3\right )^{9/2}}{9 b^6 x^9}+\frac {4 a^2 (3 b c-5 a d) \left (a x^2+b x^3\right )^{11/2}}{11 b^6 x^{11}}-\frac {4 a (2 b c-5 a d) \left (a x^2+b x^3\right )^{13/2}}{13 b^6 x^{13}}+\frac {2 (b c-5 a d) \left (a x^2+b x^3\right )^{15/2}}{15 b^6 x^{15}}+\frac {2 d \left (a x^2+b x^3\right )^{17/2}}{17 b^6 x^{17}} \] Output:
2/7*a^4*(-a*d+b*c)*(b*x^3+a*x^2)^(7/2)/b^6/x^7-2/9*a^3*(-5*a*d+4*b*c)*(b*x ^3+a*x^2)^(9/2)/b^6/x^9+4/11*a^2*(-5*a*d+3*b*c)*(b*x^3+a*x^2)^(11/2)/b^6/x ^11-4/13*a*(-5*a*d+2*b*c)*(b*x^3+a*x^2)^(13/2)/b^6/x^13+2/15*(-5*a*d+b*c)* (b*x^3+a*x^2)^(15/2)/b^6/x^15+2/17*d*(b*x^3+a*x^2)^(17/2)/b^6/x^17
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.58 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x} \, dx=\frac {2 x (a+b x)^4 \left (-1280 a^5 d+3003 b^5 x^4 (17 c+15 d x)+128 a^4 b (17 c+35 d x)-224 a^3 b^2 x (34 c+45 d x)+336 a^2 b^3 x^2 (51 c+55 d x)-462 a b^4 x^3 (68 c+65 d x)\right )}{765765 b^6 \sqrt {x^2 (a+b x)}} \] Input:
Integrate[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/x,x]
Output:
(2*x*(a + b*x)^4*(-1280*a^5*d + 3003*b^5*x^4*(17*c + 15*d*x) + 128*a^4*b*( 17*c + 35*d*x) - 224*a^3*b^2*x*(34*c + 45*d*x) + 336*a^2*b^3*x^2*(51*c + 5 5*d*x) - 462*a*b^4*x^3*(68*c + 65*d*x)))/(765765*b^6*Sqrt[x^2*(a + b*x)])
Time = 0.72 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1945, 1922, 1922, 1922, 1922, 1920}
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.
| \(\displaystyle \int \frac {\left (a x^2+b x^3\right )^{5/2} (c+d x)}{x} \, dx\) |
| \(\Big \downarrow \) 1945 |
| \(\displaystyle \frac {(17 b c-10 a d) \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x}dx}{17 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(17 b c-10 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^2}dx}{15 b}\right )}{17 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(17 b c-10 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^3}dx}{13 b}\right )}{15 b}\right )}{17 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(17 b c-10 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^4}dx}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {(17 b c-10 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{9 b x^6}-\frac {2 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^5}dx}{9 b}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {\left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{9 b x^6}-\frac {4 a \left (a x^2+b x^3\right )^{7/2}}{63 b^2 x^7}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right ) (17 b c-10 a d)}{17 b}+\frac {2 d \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}\) |
Input:
Int[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/x,x]
Output:
(2*d*(a*x^2 + b*x^3)^(7/2))/(17*b*x^2) + ((17*b*c - 10*a*d)*((2*(a*x^2 + b *x^3)^(7/2))/(15*b*x^3) - (8*a*((2*(a*x^2 + b*x^3)^(7/2))/(13*b*x^4) - (6* a*((2*(a*x^2 + b*x^3)^(7/2))/(11*b*x^5) - (4*a*((-4*a*(a*x^2 + b*x^3)^(7/2 ))/(63*b^2*x^7) + (2*(a*x^2 + b*x^3)^(7/2))/(9*b*x^6)))/(11*b)))/(13*b)))/ (15*b)))/(17*b)
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[ n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))) I nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1) /(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)) Int[(e* x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p *(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
Result contains higher order function than in optimal. Order 3 vs. order 2.
Time = 0.49 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.40
| method | result | size |
| pseudoelliptic | \(\frac {-2 a^{\frac {5}{2}} b c \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\frac {2 \sqrt {b x +a}\, \left (\left (d \,x^{3}+\frac {7}{5} c \,x^{2}\right ) b^{3}+\frac {77 x a \left (\frac {45 d x}{77}+c \right ) b^{2}}{15}+a^{2} \left (3 d x +\frac {161 c}{15}\right ) b +a^{3} d \right )}{7}}{b}\) | \(82\) |
| gosper | \(-\frac {2 \left (b x +a \right ) \left (-45045 d \,x^{5} b^{5}+30030 a \,b^{4} d \,x^{4}-51051 b^{5} c \,x^{4}-18480 a^{2} b^{3} d \,x^{3}+31416 a \,b^{4} c \,x^{3}+10080 a^{3} b^{2} d \,x^{2}-17136 a^{2} b^{3} c \,x^{2}-4480 a^{4} b d x +7616 a^{3} b^{2} c x +1280 a^{5} d -2176 a^{4} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{765765 b^{6} x^{5}}\) | \(133\) |
| default | \(-\frac {2 \left (b x +a \right ) \left (-45045 d \,x^{5} b^{5}+30030 a \,b^{4} d \,x^{4}-51051 b^{5} c \,x^{4}-18480 a^{2} b^{3} d \,x^{3}+31416 a \,b^{4} c \,x^{3}+10080 a^{3} b^{2} d \,x^{2}-17136 a^{2} b^{3} c \,x^{2}-4480 a^{4} b d x +7616 a^{3} b^{2} c x +1280 a^{5} d -2176 a^{4} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{765765 b^{6} x^{5}}\) | \(133\) |
| orering | \(-\frac {2 \left (b x +a \right ) \left (-45045 d \,x^{5} b^{5}+30030 a \,b^{4} d \,x^{4}-51051 b^{5} c \,x^{4}-18480 a^{2} b^{3} d \,x^{3}+31416 a \,b^{4} c \,x^{3}+10080 a^{3} b^{2} d \,x^{2}-17136 a^{2} b^{3} c \,x^{2}-4480 a^{4} b d x +7616 a^{3} b^{2} c x +1280 a^{5} d -2176 a^{4} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{765765 b^{6} x^{5}}\) | \(133\) |
| risch | \(-\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (-45045 d \,x^{8} b^{8}-105105 a \,b^{7} d \,x^{7}-51051 b^{8} c \,x^{7}-63525 a^{2} b^{6} d \,x^{6}-121737 a \,b^{7} c \,x^{6}-315 a^{3} b^{5} d \,x^{5}-76041 a^{2} b^{6} c \,x^{5}+350 a^{4} b^{4} d \,x^{4}-595 a^{3} b^{5} c \,x^{4}-400 a^{5} b^{3} d \,x^{3}+680 a^{4} b^{4} c \,x^{3}+480 a^{6} b^{2} d \,x^{2}-816 a^{5} b^{3} c \,x^{2}-640 a^{7} b d x +1088 a^{6} b^{2} c x +1280 a^{8} d -2176 a^{7} b c \right )}{765765 x \,b^{6}}\) | \(198\) |
| trager | \(-\frac {2 \left (-45045 d \,x^{8} b^{8}-105105 a \,b^{7} d \,x^{7}-51051 b^{8} c \,x^{7}-63525 a^{2} b^{6} d \,x^{6}-121737 a \,b^{7} c \,x^{6}-315 a^{3} b^{5} d \,x^{5}-76041 a^{2} b^{6} c \,x^{5}+350 a^{4} b^{4} d \,x^{4}-595 a^{3} b^{5} c \,x^{4}-400 a^{5} b^{3} d \,x^{3}+680 a^{4} b^{4} c \,x^{3}+480 a^{6} b^{2} d \,x^{2}-816 a^{5} b^{3} c \,x^{2}-640 a^{7} b d x +1088 a^{6} b^{2} c x +1280 a^{8} d -2176 a^{7} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{765765 b^{6} x}\) | \(200\) |
Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2)/x,x,method=_RETURNVERBOSE)
Output:
2/7*(-7*a^(5/2)*b*c*arctanh((b*x+a)^(1/2)/a^(1/2))+(b*x+a)^(1/2)*((d*x^3+7 /5*c*x^2)*b^3+77/15*x*a*(45/77*d*x+c)*b^2+a^2*(3*d*x+161/15*c)*b+a^3*d))/b
Time = 0.15 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.98 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x} \, dx=\frac {2 \, {\left (45045 \, b^{8} d x^{8} + 2176 \, a^{7} b c - 1280 \, a^{8} d + 3003 \, {\left (17 \, b^{8} c + 35 \, a b^{7} d\right )} x^{7} + 231 \, {\left (527 \, a b^{7} c + 275 \, a^{2} b^{6} d\right )} x^{6} + 63 \, {\left (1207 \, a^{2} b^{6} c + 5 \, a^{3} b^{5} d\right )} x^{5} + 35 \, {\left (17 \, a^{3} b^{5} c - 10 \, a^{4} b^{4} d\right )} x^{4} - 40 \, {\left (17 \, a^{4} b^{4} c - 10 \, a^{5} b^{3} d\right )} x^{3} + 48 \, {\left (17 \, a^{5} b^{3} c - 10 \, a^{6} b^{2} d\right )} x^{2} - 64 \, {\left (17 \, a^{6} b^{2} c - 10 \, a^{7} b d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{765765 \, b^{6} x} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x,x, algorithm="fricas")
Output:
2/765765*(45045*b^8*d*x^8 + 2176*a^7*b*c - 1280*a^8*d + 3003*(17*b^8*c + 3 5*a*b^7*d)*x^7 + 231*(527*a*b^7*c + 275*a^2*b^6*d)*x^6 + 63*(1207*a^2*b^6* c + 5*a^3*b^5*d)*x^5 + 35*(17*a^3*b^5*c - 10*a^4*b^4*d)*x^4 - 40*(17*a^4*b ^4*c - 10*a^5*b^3*d)*x^3 + 48*(17*a^5*b^3*c - 10*a^6*b^2*d)*x^2 - 64*(17*a ^6*b^2*c - 10*a^7*b*d)*x)*sqrt(b*x^3 + a*x^2)/(b^6*x)
\[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {5}{2}} \left (c + d x\right )}{x}\, dx \] Input:
integrate((d*x+c)*(b*x**3+a*x**2)**(5/2)/x,x)
Output:
Integral((x**2*(a + b*x))**(5/2)*(c + d*x)/x, x)
Time = 0.05 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.91 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x} \, dx=\frac {2 \, {\left (3003 \, b^{7} x^{7} + 7161 \, a b^{6} x^{6} + 4473 \, a^{2} b^{5} x^{5} + 35 \, a^{3} b^{4} x^{4} - 40 \, a^{4} b^{3} x^{3} + 48 \, a^{5} b^{2} x^{2} - 64 \, a^{6} b x + 128 \, a^{7}\right )} \sqrt {b x + a} c}{45045 \, b^{5}} + \frac {2 \, {\left (9009 \, b^{8} x^{8} + 21021 \, a b^{7} x^{7} + 12705 \, a^{2} b^{6} x^{6} + 63 \, a^{3} b^{5} x^{5} - 70 \, a^{4} b^{4} x^{4} + 80 \, a^{5} b^{3} x^{3} - 96 \, a^{6} b^{2} x^{2} + 128 \, a^{7} b x - 256 \, a^{8}\right )} \sqrt {b x + a} d}{153153 \, b^{6}} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x,x, algorithm="maxima")
Output:
2/45045*(3003*b^7*x^7 + 7161*a*b^6*x^6 + 4473*a^2*b^5*x^5 + 35*a^3*b^4*x^4 - 40*a^4*b^3*x^3 + 48*a^5*b^2*x^2 - 64*a^6*b*x + 128*a^7)*sqrt(b*x + a)*c /b^5 + 2/153153*(9009*b^8*x^8 + 21021*a*b^7*x^7 + 12705*a^2*b^6*x^6 + 63*a ^3*b^5*x^5 - 70*a^4*b^4*x^4 + 80*a^5*b^3*x^3 - 96*a^6*b^2*x^2 + 128*a^7*b* x - 256*a^8)*sqrt(b*x + a)*d/b^6
Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (181) = 362\).
Time = 0.13 (sec) , antiderivative size = 746, normalized size of antiderivative = 3.64 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x,x, algorithm="giac")
Output:
2/765765*(2431*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a) ^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*a^3*c*sgn(x) /b^4 + 3315*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^( 7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt( b*x + a)*a^5)*a^2*c*sgn(x)/b^4 + 1105*(63*(b*x + a)^(11/2) - 385*(b*x + a) ^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*a^3*d*sgn(x)/b^5 + 765*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b* x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3 003*sqrt(b*x + a)*a^6)*a*c*sgn(x)/b^4 + 765*(231*(b*x + a)^(13/2) - 1638*( b*x + a)^(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a ^6)*a^2*d*sgn(x)/b^5 + 119*(429*(b*x + a)^(15/2) - 3465*(b*x + a)^(13/2)*a + 12285*(b*x + a)^(11/2)*a^2 - 25025*(b*x + a)^(9/2)*a^3 + 32175*(b*x + a )^(7/2)*a^4 - 27027*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6 - 6435 *sqrt(b*x + a)*a^7)*c*sgn(x)/b^4 + 357*(429*(b*x + a)^(15/2) - 3465*(b*x + a)^(13/2)*a + 12285*(b*x + a)^(11/2)*a^2 - 25025*(b*x + a)^(9/2)*a^3 + 32 175*(b*x + a)^(7/2)*a^4 - 27027*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2 )*a^6 - 6435*sqrt(b*x + a)*a^7)*a*d*sgn(x)/b^5 + 7*(6435*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2)*a + 235620*(b*x + a)^(13/2)*a^2 - 556920*(b*x ...
Time = 9.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x} \, dx=\frac {\sqrt {b\,x^3+a\,x^2}\,\left (\frac {2\,a\,x^6\,\left (275\,a\,d+527\,b\,c\right )}{3315}+\frac {2\,b\,x^7\,\left (35\,a\,d+17\,b\,c\right )}{255}+\frac {2\,b^2\,d\,x^8}{17}-\frac {256\,a^7\,\left (10\,a\,d-17\,b\,c\right )}{765765\,b^6}+\frac {128\,a^6\,x\,\left (10\,a\,d-17\,b\,c\right )}{765765\,b^5}-\frac {2\,a^3\,x^4\,\left (10\,a\,d-17\,b\,c\right )}{21879\,b^2}+\frac {16\,a^4\,x^3\,\left (10\,a\,d-17\,b\,c\right )}{153153\,b^3}-\frac {32\,a^5\,x^2\,\left (10\,a\,d-17\,b\,c\right )}{255255\,b^4}+\frac {2\,a^2\,x^5\,\left (5\,a\,d+1207\,b\,c\right )}{12155\,b}\right )}{x} \] Input:
int(((a*x^2 + b*x^3)^(5/2)*(c + d*x))/x,x)
Output:
((a*x^2 + b*x^3)^(1/2)*((2*a*x^6*(275*a*d + 527*b*c))/3315 + (2*b*x^7*(35* a*d + 17*b*c))/255 + (2*b^2*d*x^8)/17 - (256*a^7*(10*a*d - 17*b*c))/(76576 5*b^6) + (128*a^6*x*(10*a*d - 17*b*c))/(765765*b^5) - (2*a^3*x^4*(10*a*d - 17*b*c))/(21879*b^2) + (16*a^4*x^3*(10*a*d - 17*b*c))/(153153*b^3) - (32* a^5*x^2*(10*a*d - 17*b*c))/(255255*b^4) + (2*a^2*x^5*(5*a*d + 1207*b*c))/( 12155*b)))/x
Time = 0.20 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x} \, dx=\frac {2 \sqrt {b x +a}\, \left (45045 b^{8} d \,x^{8}+105105 a \,b^{7} d \,x^{7}+51051 b^{8} c \,x^{7}+63525 a^{2} b^{6} d \,x^{6}+121737 a \,b^{7} c \,x^{6}+315 a^{3} b^{5} d \,x^{5}+76041 a^{2} b^{6} c \,x^{5}-350 a^{4} b^{4} d \,x^{4}+595 a^{3} b^{5} c \,x^{4}+400 a^{5} b^{3} d \,x^{3}-680 a^{4} b^{4} c \,x^{3}-480 a^{6} b^{2} d \,x^{2}+816 a^{5} b^{3} c \,x^{2}+640 a^{7} b d x -1088 a^{6} b^{2} c x -1280 a^{8} d +2176 a^{7} b c \right )}{765765 b^{6}} \] Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2)/x,x)
Output:
(2*sqrt(a + b*x)*( - 1280*a**8*d + 2176*a**7*b*c + 640*a**7*b*d*x - 1088*a **6*b**2*c*x - 480*a**6*b**2*d*x**2 + 816*a**5*b**3*c*x**2 + 400*a**5*b**3 *d*x**3 - 680*a**4*b**4*c*x**3 - 350*a**4*b**4*d*x**4 + 595*a**3*b**5*c*x* *4 + 315*a**3*b**5*d*x**5 + 76041*a**2*b**6*c*x**5 + 63525*a**2*b**6*d*x** 6 + 121737*a*b**7*c*x**6 + 105105*a*b**7*d*x**7 + 51051*b**8*c*x**7 + 4504 5*b**8*d*x**8))/(765765*b**6)