Integrand size = 20, antiderivative size = 150 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{5/2} \, dx=-\frac {5 a^3 A x \sqrt {a+c x^2}}{128 c}-\frac {5 a^2 A x \left (a+c x^2\right )^{3/2}}{192 c}-\frac {a A x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {(16 a B-63 A c x) \left (a+c x^2\right )^{7/2}}{504 c^2}-\frac {5 a^4 A \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}} \]
-5/192*a^2*A*x*(c*x^2+a)^(3/2)/c-1/48*a*A*x*(c*x^2+a)^(5/2)/c+1/9*B*x^2*(c *x^2+a)^(7/2)/c-1/504*(-63*A*c*x+16*B*a)*(c*x^2+a)^(7/2)/c^2-5/128*a^4*A*a rctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)-5/128*a^3*A*x*(c*x^2+a)^(1/2)/c
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\frac {\sqrt {a+c x^2} \left (-256 a^4 B+112 c^4 x^7 (9 A+8 B x)+a^3 c x (315 A+128 B x)+8 a c^3 x^5 (357 A+304 B x)+6 a^2 c^2 x^3 (413 A+320 B x)\right )+315 a^4 A \sqrt {c} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{8064 c^2} \]
(Sqrt[a + c*x^2]*(-256*a^4*B + 112*c^4*x^7*(9*A + 8*B*x) + a^3*c*x*(315*A + 128*B*x) + 8*a*c^3*x^5*(357*A + 304*B*x) + 6*a^2*c^2*x^3*(413*A + 320*B* x)) + 315*a^4*A*Sqrt[c]*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(8064*c^2)
Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {533, 533, 25, 27, 455, 211, 211, 211, 224, 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.
| \(\displaystyle \int x^2 \left (a+c x^2\right )^{5/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {\int x (2 a B-9 A c x) \left (c x^2+a\right )^{5/2}dx}{9 c}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {-\frac {\int -a c (9 A+16 B x) \left (c x^2+a\right )^{5/2}dx}{8 c}-\frac {9}{8} A x \left (a+c x^2\right )^{7/2}}{9 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {\frac {\int a c (9 A+16 B x) \left (c x^2+a\right )^{5/2}dx}{8 c}-\frac {9}{8} A x \left (a+c x^2\right )^{7/2}}{9 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {\frac {1}{8} a \int (9 A+16 B x) \left (c x^2+a\right )^{5/2}dx-\frac {9}{8} A x \left (a+c x^2\right )^{7/2}}{9 c}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {\frac {1}{8} a \left (9 A \int \left (c x^2+a\right )^{5/2}dx+\frac {16 B \left (a+c x^2\right )^{7/2}}{7 c}\right )-\frac {9}{8} A x \left (a+c x^2\right )^{7/2}}{9 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \int \left (c x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {16 B \left (a+c x^2\right )^{7/2}}{7 c}\right )-\frac {9}{8} A x \left (a+c x^2\right )^{7/2}}{9 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {c x^2+a}dx+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {16 B \left (a+c x^2\right )^{7/2}}{7 c}\right )-\frac {9}{8} A x \left (a+c x^2\right )^{7/2}}{9 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {c x^2+a}}dx+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {16 B \left (a+c x^2\right )^{7/2}}{7 c}\right )-\frac {9}{8} A x \left (a+c x^2\right )^{7/2}}{9 c}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {16 B \left (a+c x^2\right )^{7/2}}{7 c}\right )-\frac {9}{8} A x \left (a+c x^2\right )^{7/2}}{9 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{7/2}}{9 c}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {16 B \left (a+c x^2\right )^{7/2}}{7 c}\right )-\frac {9}{8} A x \left (a+c x^2\right )^{7/2}}{9 c}\) |
(B*x^2*(a + c*x^2)^(7/2))/(9*c) - ((-9*A*x*(a + c*x^2)^(7/2))/8 + (a*((16* B*(a + c*x^2)^(7/2))/(7*c) + 9*A*((x*(a + c*x^2)^(5/2))/6 + (5*a*((x*(a + c*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + c*x^2])/2 + (a*ArcTanh[(Sqrt[c]*x)/Sqr t[a + c*x^2]])/(2*Sqrt[c])))/4))/6)))/8)/(9*c)
3.4.41.3.1 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_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85
| method | result | size |
| default | \(B \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{9 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{63 c^{2}}\right )+A \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{8 c}\right )\) | \(128\) |
| risch | \(\frac {\left (896 B \,c^{4} x^{8}+1008 A \,c^{4} x^{7}+2432 a B \,c^{3} x^{6}+2856 a A \,c^{3} x^{5}+1920 a^{2} B \,c^{2} x^{4}+2478 a^{2} A \,c^{2} x^{3}+128 a^{3} B c \,x^{2}+315 a^{3} A c x -256 B \,a^{4}\right ) \sqrt {c \,x^{2}+a}}{8064 c^{2}}-\frac {5 A \,a^{4} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {3}{2}}}\) | \(128\) |
B*(1/9*x^2*(c*x^2+a)^(7/2)/c-2/63*a/c^2*(c*x^2+a)^(7/2))+A*(1/8*x*(c*x^2+a )^(7/2)/c-1/8*a/c*(1/6*x*(c*x^2+a)^(5/2)+5/6*a*(1/4*x*(c*x^2+a)^(3/2)+3/4* a*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))))))
Time = 0.41 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.81 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\left [\frac {315 \, A a^{4} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (896 \, B c^{4} x^{8} + 1008 \, A c^{4} x^{7} + 2432 \, B a c^{3} x^{6} + 2856 \, A a c^{3} x^{5} + 1920 \, B a^{2} c^{2} x^{4} + 2478 \, A a^{2} c^{2} x^{3} + 128 \, B a^{3} c x^{2} + 315 \, A a^{3} c x - 256 \, B a^{4}\right )} \sqrt {c x^{2} + a}}{16128 \, c^{2}}, \frac {315 \, A a^{4} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (896 \, B c^{4} x^{8} + 1008 \, A c^{4} x^{7} + 2432 \, B a c^{3} x^{6} + 2856 \, A a c^{3} x^{5} + 1920 \, B a^{2} c^{2} x^{4} + 2478 \, A a^{2} c^{2} x^{3} + 128 \, B a^{3} c x^{2} + 315 \, A a^{3} c x - 256 \, B a^{4}\right )} \sqrt {c x^{2} + a}}{8064 \, c^{2}}\right ] \]
[1/16128*(315*A*a^4*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a ) + 2*(896*B*c^4*x^8 + 1008*A*c^4*x^7 + 2432*B*a*c^3*x^6 + 2856*A*a*c^3*x^ 5 + 1920*B*a^2*c^2*x^4 + 2478*A*a^2*c^2*x^3 + 128*B*a^3*c*x^2 + 315*A*a^3* c*x - 256*B*a^4)*sqrt(c*x^2 + a))/c^2, 1/8064*(315*A*a^4*sqrt(-c)*arctan(s qrt(-c)*x/sqrt(c*x^2 + a)) + (896*B*c^4*x^8 + 1008*A*c^4*x^7 + 2432*B*a*c^ 3*x^6 + 2856*A*a*c^3*x^5 + 1920*B*a^2*c^2*x^4 + 2478*A*a^2*c^2*x^3 + 128*B *a^3*c*x^2 + 315*A*a^3*c*x - 256*B*a^4)*sqrt(c*x^2 + a))/c^2]
Time = 0.61 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.23 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\begin {cases} - \frac {5 A a^{4} \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right )}{128 c} + \sqrt {a + c x^{2}} \cdot \left (\frac {5 A a^{3} x}{128 c} + \frac {59 A a^{2} x^{3}}{192} + \frac {17 A a c x^{5}}{48} + \frac {A c^{2} x^{7}}{8} - \frac {2 B a^{4}}{63 c^{2}} + \frac {B a^{3} x^{2}}{63 c} + \frac {5 B a^{2} x^{4}}{21} + \frac {19 B a c x^{6}}{63} + \frac {B c^{2} x^{8}}{9}\right ) & \text {for}\: c \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{3}}{3} + \frac {B x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-5*A*a**4*Piecewise((log(2*sqrt(c)*sqrt(a + c*x**2) + 2*c*x)/sq rt(c), Ne(a, 0)), (x*log(x)/sqrt(c*x**2), True))/(128*c) + sqrt(a + c*x**2 )*(5*A*a**3*x/(128*c) + 59*A*a**2*x**3/192 + 17*A*a*c*x**5/48 + A*c**2*x** 7/8 - 2*B*a**4/(63*c**2) + B*a**3*x**2/(63*c) + 5*B*a**2*x**4/21 + 19*B*a* c*x**6/63 + B*c**2*x**8/9), Ne(c, 0)), (a**(5/2)*(A*x**3/3 + B*x**4/4), Tr ue))
Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.83 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B x^{2}}{9 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A x}{8 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A a x}{48 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A a^{2} x}{192 \, c} - \frac {5 \, \sqrt {c x^{2} + a} A a^{3} x}{128 \, c} - \frac {5 \, A a^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {3}{2}}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} B a}{63 \, c^{2}} \]
1/9*(c*x^2 + a)^(7/2)*B*x^2/c + 1/8*(c*x^2 + a)^(7/2)*A*x/c - 1/48*(c*x^2 + a)^(5/2)*A*a*x/c - 5/192*(c*x^2 + a)^(3/2)*A*a^2*x/c - 5/128*sqrt(c*x^2 + a)*A*a^3*x/c - 5/128*A*a^4*arcsinh(c*x/sqrt(a*c))/c^(3/2) - 2/63*(c*x^2 + a)^(7/2)*B*a/c^2
Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\frac {5 \, A a^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {3}{2}}} - \frac {1}{8064} \, {\left (\frac {256 \, B a^{4}}{c^{2}} - {\left (\frac {315 \, A a^{3}}{c} + 2 \, {\left (\frac {64 \, B a^{3}}{c} + {\left (1239 \, A a^{2} + 4 \, {\left (240 \, B a^{2} + {\left (357 \, A a c + 2 \, {\left (152 \, B a c + 7 \, {\left (8 \, B c^{2} x + 9 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} + a} \]
5/128*A*a^4*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2) - 1/8064*(256*B *a^4/c^2 - (315*A*a^3/c + 2*(64*B*a^3/c + (1239*A*a^2 + 4*(240*B*a^2 + (35 7*A*a*c + 2*(152*B*a*c + 7*(8*B*c^2*x + 9*A*c^2)*x)*x)*x)*x)*x)*x)*x)*sqrt (c*x^2 + a)
Timed out. \[ \int x^2 (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\int x^2\,{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]