Integrand size = 20, antiderivative size = 127 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=-\frac {a^2 A x \sqrt {a+c x^2}}{16 c}-\frac {a A x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {(12 a B-35 A c x) \left (a+c x^2\right )^{5/2}}{210 c^2}-\frac {a^3 A \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}} \]
-1/24*a*A*x*(c*x^2+a)^(3/2)/c+1/7*B*x^2*(c*x^2+a)^(5/2)/c-1/210*(-35*A*c*x +12*B*a)*(c*x^2+a)^(5/2)/c^2-1/16*a^3*A*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2)) /c^(3/2)-1/16*a^2*A*x*(c*x^2+a)^(1/2)/c
Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {\sqrt {a+c x^2} \left (-96 a^3 B+40 c^3 x^5 (7 A+6 B x)+3 a^2 c x (35 A+16 B x)+2 a c^2 x^3 (245 A+192 B x)\right )+105 a^3 A \sqrt {c} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{1680 c^2} \]
(Sqrt[a + c*x^2]*(-96*a^3*B + 40*c^3*x^5*(7*A + 6*B*x) + 3*a^2*c*x*(35*A + 16*B*x) + 2*a*c^2*x^3*(245*A + 192*B*x)) + 105*a^3*A*Sqrt[c]*Log[-(Sqrt[c ]*x) + Sqrt[a + c*x^2]])/(1680*c^2)
Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {533, 533, 25, 27, 455, 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.
\(\displaystyle \int x^2 \left (a+c x^2\right )^{3/2} (A+B x) \, dx\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\int x (2 a B-7 A c x) \left (c x^2+a\right )^{3/2}dx}{7 c}\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {-\frac {\int -a c (7 A+12 B x) \left (c x^2+a\right )^{3/2}dx}{6 c}-\frac {7}{6} A x \left (a+c x^2\right )^{5/2}}{7 c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {\int a c (7 A+12 B x) \left (c x^2+a\right )^{3/2}dx}{6 c}-\frac {7}{6} A x \left (a+c x^2\right )^{5/2}}{7 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{6} a \int (7 A+12 B x) \left (c x^2+a\right )^{3/2}dx-\frac {7}{6} A x \left (a+c x^2\right )^{5/2}}{7 c}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{6} a \left (7 A \int \left (c x^2+a\right )^{3/2}dx+\frac {12 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {7}{6} A x \left (a+c x^2\right )^{5/2}}{7 c}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{6} a \left (7 A \left (\frac {3}{4} a \int \sqrt {c x^2+a}dx+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {12 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {7}{6} A x \left (a+c x^2\right )^{5/2}}{7 c}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{6} a \left (7 A \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {c x^2+a}}dx+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {12 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {7}{6} A x \left (a+c x^2\right )^{5/2}}{7 c}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{6} a \left (7 A \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {12 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {7}{6} A x \left (a+c x^2\right )^{5/2}}{7 c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{6} a \left (7 A \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {12 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {7}{6} A x \left (a+c x^2\right )^{5/2}}{7 c}\) |
(B*x^2*(a + c*x^2)^(5/2))/(7*c) - ((-7*A*x*(a + c*x^2)^(5/2))/6 + (a*((12* B*(a + c*x^2)^(5/2))/(5*c) + 7*A*((x*(a + c*x^2)^(3/2))/4 + (3*a*((x*Sqrt[ a + c*x^2])/2 + (a*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*Sqrt[c])))/4)) )/6)/(7*c)
3.4.28.3.1 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 = 104, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {\left (240 B \,c^{3} x^{6}+280 A \,c^{3} x^{5}+384 a B \,c^{2} x^{4}+490 a A \,c^{2} x^{3}+48 a^{2} B c \,x^{2}+105 a^{2} A c x -96 B \,a^{3}\right ) \sqrt {c \,x^{2}+a}}{1680 c^{2}}-\frac {a^{3} A \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}\) | \(104\) |
default | \(B \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{7 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{35 c^{2}}\right )+A \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )\) | \(112\) |
1/1680*(240*B*c^3*x^6+280*A*c^3*x^5+384*B*a*c^2*x^4+490*A*a*c^2*x^3+48*B*a ^2*c*x^2+105*A*a^2*c*x-96*B*a^3)/c^2*(c*x^2+a)^(1/2)-1/16*a^3*A/c^(3/2)*ln (x*c^(1/2)+(c*x^2+a)^(1/2))
Time = 0.41 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.76 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\left [\frac {105 \, A a^{3} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 384 \, B a c^{2} x^{4} + 490 \, A a c^{2} x^{3} + 48 \, B a^{2} c x^{2} + 105 \, A a^{2} c x - 96 \, B a^{3}\right )} \sqrt {c x^{2} + a}}{3360 \, c^{2}}, \frac {105 \, A a^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 384 \, B a c^{2} x^{4} + 490 \, A a c^{2} x^{3} + 48 \, B a^{2} c x^{2} + 105 \, A a^{2} c x - 96 \, B a^{3}\right )} \sqrt {c x^{2} + a}}{1680 \, c^{2}}\right ] \]
[1/3360*(105*A*a^3*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(240*B*c^3*x^6 + 280*A*c^3*x^5 + 384*B*a*c^2*x^4 + 490*A*a*c^2*x^3 + 48*B*a^2*c*x^2 + 105*A*a^2*c*x - 96*B*a^3)*sqrt(c*x^2 + a))/c^2, 1/1680*(1 05*A*a^3*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (240*B*c^3*x^6 + 28 0*A*c^3*x^5 + 384*B*a*c^2*x^4 + 490*A*a*c^2*x^3 + 48*B*a^2*c*x^2 + 105*A*a ^2*c*x - 96*B*a^3)*sqrt(c*x^2 + a))/c^2]
Time = 0.59 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.18 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\begin {cases} - \frac {A a^{3} \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right )}{16 c} + \sqrt {a + c x^{2}} \left (\frac {A a^{2} x}{16 c} + \frac {7 A a x^{3}}{24} + \frac {A c x^{5}}{6} - \frac {2 B a^{3}}{35 c^{2}} + \frac {B a^{2} x^{2}}{35 c} + \frac {8 B a x^{4}}{35} + \frac {B c x^{6}}{7}\right ) & \text {for}\: c \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{3}}{3} + \frac {B x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-A*a**3*Piecewise((log(2*sqrt(c)*sqrt(a + c*x**2) + 2*c*x)/sqrt (c), Ne(a, 0)), (x*log(x)/sqrt(c*x**2), True))/(16*c) + sqrt(a + c*x**2)*( A*a**2*x/(16*c) + 7*A*a*x**3/24 + A*c*x**5/6 - 2*B*a**3/(35*c**2) + B*a**2 *x**2/(35*c) + 8*B*a*x**4/35 + B*c*x**6/7), Ne(c, 0)), (a**(3/2)*(A*x**3/3 + B*x**4/4), True))
Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B x^{2}}{7 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A x}{6 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A a x}{24 \, c} - \frac {\sqrt {c x^{2} + a} A a^{2} x}{16 \, c} - \frac {A a^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {3}{2}}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B a}{35 \, c^{2}} \]
1/7*(c*x^2 + a)^(5/2)*B*x^2/c + 1/6*(c*x^2 + a)^(5/2)*A*x/c - 1/24*(c*x^2 + a)^(3/2)*A*a*x/c - 1/16*sqrt(c*x^2 + a)*A*a^2*x/c - 1/16*A*a^3*arcsinh(c *x/sqrt(a*c))/c^(3/2) - 2/35*(c*x^2 + a)^(5/2)*B*a/c^2
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.81 \[ \int x^2 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {A a^{3} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {3}{2}}} - \frac {1}{1680} \, \sqrt {c x^{2} + a} {\left (\frac {96 \, B a^{3}}{c^{2}} - {\left (\frac {105 \, A a^{2}}{c} + 2 \, {\left (\frac {24 \, B a^{2}}{c} + {\left (245 \, A a + 4 \, {\left (48 \, B a + 5 \, {\left (6 \, B c x + 7 \, A c\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
1/16*A*a^3*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2) - 1/1680*sqrt(c* x^2 + a)*(96*B*a^3/c^2 - (105*A*a^2/c + 2*(24*B*a^2/c + (245*A*a + 4*(48*B *a + 5*(6*B*c*x + 7*A*c)*x)*x)*x)*x)*x)
Timed out. \[ \int x^2 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\int x^2\,{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \]