Integrand size = 24, antiderivative size = 244 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{12}} \, dx=-\frac {b (5 b c+52 a d) \sqrt {a x^2+b x^3}}{160 x^5}-\frac {b^2 (5 b c+372 a d) \sqrt {a x^2+b x^3}}{960 a x^4}+\frac {b^3 (5 b c-12 a d) \sqrt {a x^2+b x^3}}{768 a^2 x^3}-\frac {b^4 (5 b c-12 a d) \sqrt {a x^2+b x^3}}{512 a^3 x^2}-\frac {(5 b c+12 a d) \left (a x^2+b x^3\right )^{3/2}}{60 x^8}-\frac {c \left (a x^2+b x^3\right )^{5/2}}{6 x^{11}}+\frac {b^5 (5 b c-12 a d) \text {arctanh}\left (\frac {\sqrt {a x^2+b x^3}}{\sqrt {a} x}\right )}{512 a^{7/2}} \] Output:
-1/160*b*(52*a*d+5*b*c)*(b*x^3+a*x^2)^(1/2)/x^5-1/960*b^2*(372*a*d+5*b*c)* (b*x^3+a*x^2)^(1/2)/a/x^4+1/768*b^3*(-12*a*d+5*b*c)*(b*x^3+a*x^2)^(1/2)/a^ 2/x^3-1/512*b^4*(-12*a*d+5*b*c)*(b*x^3+a*x^2)^(1/2)/a^3/x^2-1/60*(12*a*d+5 *b*c)*(b*x^3+a*x^2)^(3/2)/x^8-1/6*c*(b*x^3+a*x^2)^(5/2)/x^11+1/512*b^5*(-1 2*a*d+5*b*c)*arctanh((b*x^3+a*x^2)^(1/2)/a^(1/2)/x)/a^(7/2)
Time = 0.39 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.72 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{12}} \, dx=\frac {\sqrt {x^2 (a+b x)} \left (-\sqrt {a} \sqrt {a+b x} \left (75 b^5 c x^5+40 a^2 b^3 x^3 (c+3 d x)+256 a^5 (5 c+6 d x)-10 a b^4 x^4 (5 c+18 d x)+48 a^3 b^2 x^2 (45 c+62 d x)+64 a^4 b x (50 c+63 d x)\right )+15 b^5 (5 b c-12 a d) x^6 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{7680 a^{7/2} x^7 \sqrt {a+b x}} \] Input:
Integrate[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/x^12,x]
Output:
(Sqrt[x^2*(a + b*x)]*(-(Sqrt[a]*Sqrt[a + b*x]*(75*b^5*c*x^5 + 40*a^2*b^3*x ^3*(c + 3*d*x) + 256*a^5*(5*c + 6*d*x) - 10*a*b^4*x^4*(5*c + 18*d*x) + 48* a^3*b^2*x^2*(45*c + 62*d*x) + 64*a^4*b*x*(50*c + 63*d*x))) + 15*b^5*(5*b*c - 12*a*d)*x^6*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]))/(7680*a^(7/2)*x^7*Sqrt[a + b*x])
Time = 0.80 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1944, 1926, 1926, 1926, 1931, 1931, 1914, 219}
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.
| \(\displaystyle \int \frac {\left (a x^2+b x^3\right )^{5/2} (c+d x)}{x^{12}} \, dx\) |
| \(\Big \downarrow \) 1944 |
| \(\displaystyle -\frac {(5 b c-12 a d) \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^{11}}dx}{12 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{6 a x^{13}}\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle -\frac {(5 b c-12 a d) \left (\frac {1}{2} b \int \frac {\left (b x^3+a x^2\right )^{3/2}}{x^8}dx-\frac {\left (a x^2+b x^3\right )^{5/2}}{5 x^{10}}\right )}{12 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{6 a x^{13}}\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle -\frac {(5 b c-12 a d) \left (\frac {1}{2} b \left (\frac {3}{8} b \int \frac {\sqrt {b x^3+a x^2}}{x^5}dx-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{5 x^{10}}\right )}{12 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{6 a x^{13}}\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle -\frac {(5 b c-12 a d) \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \int \frac {1}{x^2 \sqrt {b x^3+a x^2}}dx-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{5 x^{10}}\right )}{12 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{6 a x^{13}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {(5 b c-12 a d) \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \int \frac {1}{x \sqrt {b x^3+a x^2}}dx}{4 a}-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}\right )-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{5 x^{10}}\right )}{12 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{6 a x^{13}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {(5 b c-12 a d) \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{\sqrt {b x^3+a x^2}}dx}{2 a}-\frac {\sqrt {a x^2+b x^3}}{a x^2}\right )}{4 a}-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}\right )-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{5 x^{10}}\right )}{12 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{6 a x^{13}}\) |
| \(\Big \downarrow \) 1914 |
| \(\displaystyle -\frac {(5 b c-12 a d) \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (\frac {b \int \frac {1}{1-\frac {a x^2}{b x^3+a x^2}}d\frac {x}{\sqrt {b x^3+a x^2}}}{a}-\frac {\sqrt {a x^2+b x^3}}{a x^2}\right )}{4 a}-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}\right )-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{5 x^{10}}\right )}{12 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{6 a x^{13}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{3/2}}-\frac {\sqrt {a x^2+b x^3}}{a x^2}\right )}{4 a}-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}\right )-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{5 x^{10}}\right ) (5 b c-12 a d)}{12 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{6 a x^{13}}\) |
Input:
Int[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/x^12,x]
Output:
-1/6*(c*(a*x^2 + b*x^3)^(7/2))/(a*x^13) - ((5*b*c - 12*a*d)*(-1/5*(a*x^2 + b*x^3)^(5/2)/x^10 + (b*(-1/4*(a*x^2 + b*x^3)^(3/2)/x^7 + (3*b*(-1/3*Sqrt[ a*x^2 + b*x^3]/x^4 + (b*(-1/2*Sqrt[a*x^2 + b*x^3]/(a*x^3) - (3*b*(-(Sqrt[a *x^2 + b*x^3]/(a*x^2)) + (b*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/a^(3 /2)))/(4*a)))/6))/8))/2))/(12*a)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[2/(2 - n) Subst[Int[1/(1 - a*x^2), x], x, x/Sqrt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]
Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + j*p + 1))), x] - Simp[b*p *((n - j)/(c^n*(m + j*p + 1))) Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Integer sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 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, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[ m + j*p + 1, 0]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] + Simp[(a*d*(m + j*p + 1) - b *c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1)) Int[(e*x)^(m + n)*(a*x^ j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m + j*p, -1 ] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, (-n)*p - 1])) && ( GtQ[e, 0] || IntegersQ[j, n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1 , 0]
Time = 0.90 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {\left (-180 a \,b^{4} d \,x^{5}+75 b^{5} c \,x^{5}+120 x^{4} a^{2} b^{3} d -50 x^{4} a \,b^{4} c +2976 a^{3} b^{2} d \,x^{3}+40 a^{2} b^{3} c \,x^{3}+4032 a^{4} b d \,x^{2}+2160 a^{3} b^{2} c \,x^{2}+1536 a^{5} d x +3200 a^{4} b c x +1280 c \,a^{5}\right ) \sqrt {x^{2} \left (b x +a \right )}}{7680 x^{7} a^{3}}-\frac {\left (12 a d -5 b c \right ) b^{5} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {x^{2} \left (b x +a \right )}}{512 a^{\frac {7}{2}} x \sqrt {b x +a}}\) | \(180\) |
| pseudoelliptic | \(\frac {\frac {143 b^{10} x^{11} \left (a d -\frac {15 b c}{22}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{131072}-\frac {39 \sqrt {b x +a}\, \left (\frac {80 x^{5} \left (\frac {121 d x}{75}+c \right ) b^{5} a^{\frac {11}{2}}}{99}-\frac {320 x^{4} \left (\frac {143 d x}{90}+c \right ) b^{4} a^{\frac {13}{2}}}{429}+\frac {896 x^{3} b^{3} \left (\frac {11 d x}{7}+c \right ) a^{\frac {15}{2}}}{1287}+\frac {444416 x^{2} b^{2} \left (\frac {4213 d x}{3720}+c \right ) a^{\frac {17}{2}}}{1287}+\frac {86016 x \left (\frac {451 d x}{405}+c \right ) b \,a^{\frac {19}{2}}}{143}+\left (\frac {57344 d x}{195}+\frac {114688 c}{429}\right ) a^{\frac {21}{2}}+x^{6} \left (\frac {35 x^{3} \left (\frac {11 d x}{5}+c \right ) b^{3} a^{\frac {3}{2}}}{24}-\frac {7 x^{2} b^{2} \left (\frac {11 d x}{6}+c \right ) a^{\frac {5}{2}}}{6}+b x \left (\frac {77 d x}{45}+c \right ) a^{\frac {7}{2}}+\left (-\frac {22 d x}{15}-\frac {8 c}{9}\right ) a^{\frac {9}{2}}-\frac {35 \sqrt {a}\, b^{4} c \,x^{4}}{16}\right ) b^{6}\right )}{114688}}{a^{\frac {17}{2}} x^{11}}\) | \(218\) |
| default | \(\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}} \left (180 \left (b x +a \right )^{\frac {11}{2}} a^{\frac {9}{2}} d -75 \left (b x +a \right )^{\frac {11}{2}} a^{\frac {7}{2}} b c -1020 \left (b x +a \right )^{\frac {9}{2}} a^{\frac {11}{2}} d +425 \left (b x +a \right )^{\frac {9}{2}} a^{\frac {9}{2}} b c -696 \left (b x +a \right )^{\frac {7}{2}} a^{\frac {13}{2}} d -990 \left (b x +a \right )^{\frac {7}{2}} a^{\frac {11}{2}} b c -180 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{4} b^{6} d \,x^{6}+75 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{3} b^{7} c \,x^{6}+2376 \left (b x +a \right )^{\frac {5}{2}} a^{\frac {15}{2}} d -990 \left (b x +a \right )^{\frac {5}{2}} a^{\frac {13}{2}} b c -1020 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {17}{2}} d +425 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {15}{2}} b c +180 \sqrt {b x +a}\, a^{\frac {19}{2}} d -75 \sqrt {b x +a}\, a^{\frac {17}{2}} b c \right )}{7680 b \,x^{11} \left (b x +a \right )^{\frac {5}{2}} a^{\frac {13}{2}}}\) | \(243\) |
Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^12,x,method=_RETURNVERBOSE)
Output:
-1/7680*(-180*a*b^4*d*x^5+75*b^5*c*x^5+120*a^2*b^3*d*x^4-50*a*b^4*c*x^4+29 76*a^3*b^2*d*x^3+40*a^2*b^3*c*x^3+4032*a^4*b*d*x^2+2160*a^3*b^2*c*x^2+1536 *a^5*d*x+3200*a^4*b*c*x+1280*a^5*c)/x^7/a^3*(x^2*(b*x+a))^(1/2)-1/512*(12* a*d-5*b*c)*b^5/a^(7/2)*arctanh((b*x+a)^(1/2)/a^(1/2))*(x^2*(b*x+a))^(1/2)/ x/(b*x+a)^(1/2)
Time = 0.16 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.60 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{12}} \, dx=\left [-\frac {15 \, {\left (5 \, b^{6} c - 12 \, a b^{5} d\right )} \sqrt {a} x^{7} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, {\left (1280 \, a^{6} c + 15 \, {\left (5 \, a b^{5} c - 12 \, a^{2} b^{4} d\right )} x^{5} - 10 \, {\left (5 \, a^{2} b^{4} c - 12 \, a^{3} b^{3} d\right )} x^{4} + 8 \, {\left (5 \, a^{3} b^{3} c + 372 \, a^{4} b^{2} d\right )} x^{3} + 144 \, {\left (15 \, a^{4} b^{2} c + 28 \, a^{5} b d\right )} x^{2} + 128 \, {\left (25 \, a^{5} b c + 12 \, a^{6} d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{15360 \, a^{4} x^{7}}, -\frac {15 \, {\left (5 \, b^{6} c - 12 \, a b^{5} d\right )} \sqrt {-a} x^{7} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{b x^{2} + a x}\right ) + {\left (1280 \, a^{6} c + 15 \, {\left (5 \, a b^{5} c - 12 \, a^{2} b^{4} d\right )} x^{5} - 10 \, {\left (5 \, a^{2} b^{4} c - 12 \, a^{3} b^{3} d\right )} x^{4} + 8 \, {\left (5 \, a^{3} b^{3} c + 372 \, a^{4} b^{2} d\right )} x^{3} + 144 \, {\left (15 \, a^{4} b^{2} c + 28 \, a^{5} b d\right )} x^{2} + 128 \, {\left (25 \, a^{5} b c + 12 \, a^{6} d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{7680 \, a^{4} x^{7}}\right ] \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^12,x, algorithm="fricas")
Output:
[-1/15360*(15*(5*b^6*c - 12*a*b^5*d)*sqrt(a)*x^7*log((b*x^2 + 2*a*x - 2*sq rt(b*x^3 + a*x^2)*sqrt(a))/x^2) + 2*(1280*a^6*c + 15*(5*a*b^5*c - 12*a^2*b ^4*d)*x^5 - 10*(5*a^2*b^4*c - 12*a^3*b^3*d)*x^4 + 8*(5*a^3*b^3*c + 372*a^4 *b^2*d)*x^3 + 144*(15*a^4*b^2*c + 28*a^5*b*d)*x^2 + 128*(25*a^5*b*c + 12*a ^6*d)*x)*sqrt(b*x^3 + a*x^2))/(a^4*x^7), -1/7680*(15*(5*b^6*c - 12*a*b^5*d )*sqrt(-a)*x^7*arctan(sqrt(b*x^3 + a*x^2)*sqrt(-a)/(b*x^2 + a*x)) + (1280* a^6*c + 15*(5*a*b^5*c - 12*a^2*b^4*d)*x^5 - 10*(5*a^2*b^4*c - 12*a^3*b^3*d )*x^4 + 8*(5*a^3*b^3*c + 372*a^4*b^2*d)*x^3 + 144*(15*a^4*b^2*c + 28*a^5*b *d)*x^2 + 128*(25*a^5*b*c + 12*a^6*d)*x)*sqrt(b*x^3 + a*x^2))/(a^4*x^7)]
\[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{12}} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {5}{2}} \left (c + d x\right )}{x^{12}}\, dx \] Input:
integrate((d*x+c)*(b*x**3+a*x**2)**(5/2)/x**12,x)
Output:
Integral((x**2*(a + b*x))**(5/2)*(c + d*x)/x**12, x)
\[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{12}} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {5}{2}} {\left (d x + c\right )}}{x^{12}} \,d x } \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^12,x, algorithm="maxima")
Output:
integrate((b*x^3 + a*x^2)^(5/2)*(d*x + c)/x^12, x)
Time = 0.20 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{12}} \, dx=-\frac {\frac {15 \, {\left (5 \, b^{7} c \mathrm {sgn}\left (x\right ) - 12 \, a b^{6} d \mathrm {sgn}\left (x\right )\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {75 \, {\left (b x + a\right )}^{\frac {11}{2}} b^{7} c \mathrm {sgn}\left (x\right ) - 425 \, {\left (b x + a\right )}^{\frac {9}{2}} a b^{7} c \mathrm {sgn}\left (x\right ) + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} b^{7} c \mathrm {sgn}\left (x\right ) + 990 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} b^{7} c \mathrm {sgn}\left (x\right ) - 425 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{7} c \mathrm {sgn}\left (x\right ) + 75 \, \sqrt {b x + a} a^{5} b^{7} c \mathrm {sgn}\left (x\right ) - 180 \, {\left (b x + a\right )}^{\frac {11}{2}} a b^{6} d \mathrm {sgn}\left (x\right ) + 1020 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} b^{6} d \mathrm {sgn}\left (x\right ) + 696 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} b^{6} d \mathrm {sgn}\left (x\right ) - 2376 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} b^{6} d \mathrm {sgn}\left (x\right ) + 1020 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} b^{6} d \mathrm {sgn}\left (x\right ) - 180 \, \sqrt {b x + a} a^{6} b^{6} d \mathrm {sgn}\left (x\right )}{a^{3} b^{6} x^{6}}}{7680 \, b} \] Input:
integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^12,x, algorithm="giac")
Output:
-1/7680*(15*(5*b^7*c*sgn(x) - 12*a*b^6*d*sgn(x))*arctan(sqrt(b*x + a)/sqrt (-a))/(sqrt(-a)*a^3) + (75*(b*x + a)^(11/2)*b^7*c*sgn(x) - 425*(b*x + a)^( 9/2)*a*b^7*c*sgn(x) + 990*(b*x + a)^(7/2)*a^2*b^7*c*sgn(x) + 990*(b*x + a) ^(5/2)*a^3*b^7*c*sgn(x) - 425*(b*x + a)^(3/2)*a^4*b^7*c*sgn(x) + 75*sqrt(b *x + a)*a^5*b^7*c*sgn(x) - 180*(b*x + a)^(11/2)*a*b^6*d*sgn(x) + 1020*(b*x + a)^(9/2)*a^2*b^6*d*sgn(x) + 696*(b*x + a)^(7/2)*a^3*b^6*d*sgn(x) - 2376 *(b*x + a)^(5/2)*a^4*b^6*d*sgn(x) + 1020*(b*x + a)^(3/2)*a^5*b^6*d*sgn(x) - 180*sqrt(b*x + a)*a^6*b^6*d*sgn(x))/(a^3*b^6*x^6))/b
Timed out. \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{12}} \, dx=\int \frac {{\left (b\,x^3+a\,x^2\right )}^{5/2}\,\left (c+d\,x\right )}{x^{12}} \,d x \] Input:
int(((a*x^2 + b*x^3)^(5/2)*(c + d*x))/x^12,x)
Output:
int(((a*x^2 + b*x^3)^(5/2)*(c + d*x))/x^12, x)
Time = 0.22 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.14 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{12}} \, dx=\frac {-2560 \sqrt {b x +a}\, a^{6} c -3072 \sqrt {b x +a}\, a^{6} d x -6400 \sqrt {b x +a}\, a^{5} b c x -8064 \sqrt {b x +a}\, a^{5} b d \,x^{2}-4320 \sqrt {b x +a}\, a^{4} b^{2} c \,x^{2}-5952 \sqrt {b x +a}\, a^{4} b^{2} d \,x^{3}-80 \sqrt {b x +a}\, a^{3} b^{3} c \,x^{3}-240 \sqrt {b x +a}\, a^{3} b^{3} d \,x^{4}+100 \sqrt {b x +a}\, a^{2} b^{4} c \,x^{4}+360 \sqrt {b x +a}\, a^{2} b^{4} d \,x^{5}-150 \sqrt {b x +a}\, a \,b^{5} c \,x^{5}+180 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a \,b^{5} d \,x^{6}-75 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{6} c \,x^{6}-180 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a \,b^{5} d \,x^{6}+75 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{6} c \,x^{6}}{15360 a^{4} x^{6}} \] Input:
int((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^12,x)
Output:
( - 2560*sqrt(a + b*x)*a**6*c - 3072*sqrt(a + b*x)*a**6*d*x - 6400*sqrt(a + b*x)*a**5*b*c*x - 8064*sqrt(a + b*x)*a**5*b*d*x**2 - 4320*sqrt(a + b*x)* a**4*b**2*c*x**2 - 5952*sqrt(a + b*x)*a**4*b**2*d*x**3 - 80*sqrt(a + b*x)* a**3*b**3*c*x**3 - 240*sqrt(a + b*x)*a**3*b**3*d*x**4 + 100*sqrt(a + b*x)* a**2*b**4*c*x**4 + 360*sqrt(a + b*x)*a**2*b**4*d*x**5 - 150*sqrt(a + b*x)* a*b**5*c*x**5 + 180*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*a*b**5*d*x**6 - 7 5*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*b**6*c*x**6 - 180*sqrt(a)*log(sqrt( a + b*x) + sqrt(a))*a*b**5*d*x**6 + 75*sqrt(a)*log(sqrt(a + b*x) + sqrt(a) )*b**6*c*x**6)/(15360*a**4*x**6)