Integrand size = 20, antiderivative size = 150 \[ \int x^2 (A+B x) \left (a+b x^2\right )^{5/2} \, dx=-\frac {5 a^3 A x \sqrt {a+b x^2}}{128 b}-\frac {5 a^2 A x \left (a+b x^2\right )^{3/2}}{192 b}-\frac {a A x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {(16 a B-63 A b x) \left (a+b x^2\right )^{7/2}}{504 b^2}-\frac {5 a^4 A \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}} \]
-5/192*a^2*A*x*(b*x^2+a)^(3/2)/b-1/48*a*A*x*(b*x^2+a)^(5/2)/b+1/9*B*x^2*(b *x^2+a)^(7/2)/b-1/504*(-63*A*b*x+16*B*a)*(b*x^2+a)^(7/2)/b^2-5/128*a^4*A*a rctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(3/2)-5/128*a^3*A*x*(b*x^2+a)^(1/2)/b
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83 \[ \int x^2 (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\frac {\sqrt {a+b x^2} \left (-256 a^4 B+112 b^4 x^7 (9 A+8 B x)+a^3 b x (315 A+128 B x)+8 a b^3 x^5 (357 A+304 B x)+6 a^2 b^2 x^3 (413 A+320 B x)\right )+315 a^4 A \sqrt {b} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8064 b^2} \]
(Sqrt[a + b*x^2]*(-256*a^4*B + 112*b^4*x^7*(9*A + 8*B*x) + a^3*b*x*(315*A + 128*B*x) + 8*a*b^3*x^5*(357*A + 304*B*x) + 6*a^2*b^2*x^3*(413*A + 320*B* x)) + 315*a^4*A*Sqrt[b]*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(8064*b^2)
Time = 0.26 (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.
| \(\displaystyle \int x^2 \left (a+b x^2\right )^{5/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\int x (2 a B-9 A b x) \left (b x^2+a\right )^{5/2}dx}{9 b}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {-\frac {\int -a b (9 A+16 B x) \left (b x^2+a\right )^{5/2}dx}{8 b}-\frac {9}{8} A x \left (a+b x^2\right )^{7/2}}{9 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\frac {\int a b (9 A+16 B x) \left (b x^2+a\right )^{5/2}dx}{8 b}-\frac {9}{8} A x \left (a+b x^2\right )^{7/2}}{9 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\frac {1}{8} a \int (9 A+16 B x) \left (b x^2+a\right )^{5/2}dx-\frac {9}{8} A x \left (a+b x^2\right )^{7/2}}{9 b}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\frac {1}{8} a \left (9 A \int \left (b x^2+a\right )^{5/2}dx+\frac {16 B \left (a+b x^2\right )^{7/2}}{7 b}\right )-\frac {9}{8} A x \left (a+b x^2\right )^{7/2}}{9 b}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {16 B \left (a+b x^2\right )^{7/2}}{7 b}\right )-\frac {9}{8} A x \left (a+b x^2\right )^{7/2}}{9 b}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {16 B \left (a+b x^2\right )^{7/2}}{7 b}\right )-\frac {9}{8} A x \left (a+b x^2\right )^{7/2}}{9 b}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {16 B \left (a+b x^2\right )^{7/2}}{7 b}\right )-\frac {9}{8} A x \left (a+b x^2\right )^{7/2}}{9 b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {16 B \left (a+b x^2\right )^{7/2}}{7 b}\right )-\frac {9}{8} A x \left (a+b x^2\right )^{7/2}}{9 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B x^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\frac {1}{8} a \left (9 A \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {16 B \left (a+b x^2\right )^{7/2}}{7 b}\right )-\frac {9}{8} A x \left (a+b x^2\right )^{7/2}}{9 b}\) |
(B*x^2*(a + b*x^2)^(7/2))/(9*b) - ((-9*A*x*(a + b*x^2)^(7/2))/8 + (a*((16* B*(a + b*x^2)^(7/2))/(7*b) + 9*A*((x*(a + b*x^2)^(5/2))/6 + (5*a*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqr t[a + b*x^2]])/(2*Sqrt[b])))/4))/6)))/8)/(9*b)
3.1.16.3.1 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 = 3.42 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85
| method | result | size |
| default | \(B \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )+A \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )\) | \(128\) |
| risch | \(\frac {\left (896 B \,x^{8} b^{4}+1008 A \,b^{4} x^{7}+2432 B \,x^{6} a \,b^{3}+2856 A a \,b^{3} x^{5}+1920 a^{2} B \,b^{2} x^{4}+2478 A \,a^{2} b^{2} x^{3}+128 B \,a^{3} b \,x^{2}+315 A \,a^{3} b x -256 B \,a^{4}\right ) \sqrt {b \,x^{2}+a}}{8064 b^{2}}-\frac {5 a^{4} A \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}\) | \(128\) |
B*(1/9*x^2*(b*x^2+a)^(7/2)/b-2/63*a/b^2*(b*x^2+a)^(7/2))+A*(1/8*x*(b*x^2+a )^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(1/4*x*(b*x^2+a)^(3/2)+3/4* a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))))
Time = 0.29 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.81 \[ \int x^2 (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\left [\frac {315 \, A a^{4} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (896 \, B b^{4} x^{8} + 1008 \, A b^{4} x^{7} + 2432 \, B a b^{3} x^{6} + 2856 \, A a b^{3} x^{5} + 1920 \, B a^{2} b^{2} x^{4} + 2478 \, A a^{2} b^{2} x^{3} + 128 \, B a^{3} b x^{2} + 315 \, A a^{3} b x - 256 \, B a^{4}\right )} \sqrt {b x^{2} + a}}{16128 \, b^{2}}, \frac {315 \, A a^{4} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (896 \, B b^{4} x^{8} + 1008 \, A b^{4} x^{7} + 2432 \, B a b^{3} x^{6} + 2856 \, A a b^{3} x^{5} + 1920 \, B a^{2} b^{2} x^{4} + 2478 \, A a^{2} b^{2} x^{3} + 128 \, B a^{3} b x^{2} + 315 \, A a^{3} b x - 256 \, B a^{4}\right )} \sqrt {b x^{2} + a}}{8064 \, b^{2}}\right ] \]
[1/16128*(315*A*a^4*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a ) + 2*(896*B*b^4*x^8 + 1008*A*b^4*x^7 + 2432*B*a*b^3*x^6 + 2856*A*a*b^3*x^ 5 + 1920*B*a^2*b^2*x^4 + 2478*A*a^2*b^2*x^3 + 128*B*a^3*b*x^2 + 315*A*a^3* b*x - 256*B*a^4)*sqrt(b*x^2 + a))/b^2, 1/8064*(315*A*a^4*sqrt(-b)*arctan(s qrt(-b)*x/sqrt(b*x^2 + a)) + (896*B*b^4*x^8 + 1008*A*b^4*x^7 + 2432*B*a*b^ 3*x^6 + 2856*A*a*b^3*x^5 + 1920*B*a^2*b^2*x^4 + 2478*A*a^2*b^2*x^3 + 128*B *a^3*b*x^2 + 315*A*a^3*b*x - 256*B*a^4)*sqrt(b*x^2 + a))/b^2]
Time = 0.59 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.23 \[ \int x^2 (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\begin {cases} - \frac {5 A a^{4} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{128 b} + \sqrt {a + b x^{2}} \cdot \left (\frac {5 A a^{3} x}{128 b} + \frac {59 A a^{2} x^{3}}{192} + \frac {17 A a b x^{5}}{48} + \frac {A b^{2} x^{7}}{8} - \frac {2 B a^{4}}{63 b^{2}} + \frac {B a^{3} x^{2}}{63 b} + \frac {5 B a^{2} x^{4}}{21} + \frac {19 B a b x^{6}}{63} + \frac {B b^{2} x^{8}}{9}\right ) & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{3}}{3} + \frac {B x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-5*A*a**4*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sq rt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2), True))/(128*b) + sqrt(a + b*x**2 )*(5*A*a**3*x/(128*b) + 59*A*a**2*x**3/192 + 17*A*a*b*x**5/48 + A*b**2*x** 7/8 - 2*B*a**4/(63*b**2) + B*a**3*x**2/(63*b) + 5*B*a**2*x**4/21 + 19*B*a* b*x**6/63 + B*b**2*x**8/9), Ne(b, 0)), (a**(5/2)*(A*x**3/3 + B*x**4/4), Tr ue))
Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.83 \[ \int x^2 (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x^{2}}{9 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A a x}{48 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{2} x}{192 \, b} - \frac {5 \, \sqrt {b x^{2} + a} A a^{3} x}{128 \, b} - \frac {5 \, A a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a}{63 \, b^{2}} \]
1/9*(b*x^2 + a)^(7/2)*B*x^2/b + 1/8*(b*x^2 + a)^(7/2)*A*x/b - 1/48*(b*x^2 + a)^(5/2)*A*a*x/b - 5/192*(b*x^2 + a)^(3/2)*A*a^2*x/b - 5/128*sqrt(b*x^2 + a)*A*a^3*x/b - 5/128*A*a^4*arcsinh(b*x/sqrt(a*b))/b^(3/2) - 2/63*(b*x^2 + a)^(7/2)*B*a/b^2
Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85 \[ \int x^2 (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\frac {5 \, A a^{4} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {3}{2}}} - \frac {1}{8064} \, {\left (\frac {256 \, B a^{4}}{b^{2}} - {\left (\frac {315 \, A a^{3}}{b} + 2 \, {\left (\frac {64 \, B a^{3}}{b} + {\left (1239 \, A a^{2} + 4 \, {\left (240 \, B a^{2} + {\left (357 \, A a b + 2 \, {\left (152 \, B a b + 7 \, {\left (8 \, B b^{2} x + 9 \, A b^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {b x^{2} + a} \]
5/128*A*a^4*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) - 1/8064*(256*B *a^4/b^2 - (315*A*a^3/b + 2*(64*B*a^3/b + (1239*A*a^2 + 4*(240*B*a^2 + (35 7*A*a*b + 2*(152*B*a*b + 7*(8*B*b^2*x + 9*A*b^2)*x)*x)*x)*x)*x)*x)*x)*sqrt (b*x^2 + a)
Timed out. \[ \int x^2 (A+B x) \left (a+b x^2\right )^{5/2} \, dx=\int x^2\,{\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]