Integrand size = 22, antiderivative size = 168 \[ \int \left (a+c x^2\right )^{5/2} \left (A+B x+C x^2\right ) \, dx=\frac {5 a^2 (8 A c-a C) x \sqrt {a+c x^2}}{128 c}+\frac {5 a (8 A c-a C) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {(8 A c-a C) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {B \left (a+c x^2\right )^{7/2}}{7 c}+\frac {C x \left (a+c x^2\right )^{7/2}}{8 c}+\frac {5 a^3 (8 A c-a C) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}} \]
5/192*a*(8*A*c-C*a)*x*(c*x^2+a)^(3/2)/c+1/48*(8*A*c-C*a)*x*(c*x^2+a)^(5/2) /c+1/7*B*(c*x^2+a)^(7/2)/c+1/8*C*x*(c*x^2+a)^(7/2)/c+5/128*a^3*(8*A*c-C*a) *arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)+5/128*a^2*(8*A*c-C*a)*x*(c*x^2 +a)^(1/2)/c
Time = 0.50 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.85 \[ \int \left (a+c x^2\right )^{5/2} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {c} \sqrt {a+c x^2} \left (3 a^3 (128 B+35 C x)+16 c^3 x^5 (28 A+3 x (8 B+7 C x))+8 a c^2 x^3 (182 A+x (144 B+119 C x))+2 a^2 c x (924 A+x (576 B+413 C x))\right )+105 a^3 (-8 A c+a C) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{2688 c^{3/2}} \]
(Sqrt[c]*Sqrt[a + c*x^2]*(3*a^3*(128*B + 35*C*x) + 16*c^3*x^5*(28*A + 3*x* (8*B + 7*C*x)) + 8*a*c^2*x^3*(182*A + x*(144*B + 119*C*x)) + 2*a^2*c*x*(92 4*A + x*(576*B + 413*C*x))) + 105*a^3*(-8*A*c + a*C)*Log[-(Sqrt[c]*x) + Sq rt[a + c*x^2]])/(2688*c^(3/2))
Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.86, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2346, 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.
| \(\displaystyle \int \left (a+c x^2\right )^{5/2} \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {\int (8 A c+8 B x c-a C) \left (c x^2+a\right )^{5/2}dx}{8 c}+\frac {C x \left (a+c x^2\right )^{7/2}}{8 c}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {(8 A c-a C) \int \left (c x^2+a\right )^{5/2}dx+\frac {8}{7} B \left (a+c x^2\right )^{7/2}}{8 c}+\frac {C x \left (a+c x^2\right )^{7/2}}{8 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(8 A c-a C) \left (\frac {5}{6} a \int \left (c x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {8}{7} B \left (a+c x^2\right )^{7/2}}{8 c}+\frac {C x \left (a+c x^2\right )^{7/2}}{8 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(8 A c-a C) \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {c x^2+a}dx+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {8}{7} B \left (a+c x^2\right )^{7/2}}{8 c}+\frac {C x \left (a+c x^2\right )^{7/2}}{8 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(8 A c-a C) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {c x^2+a}}dx+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {8}{7} B \left (a+c x^2\right )^{7/2}}{8 c}+\frac {C x \left (a+c x^2\right )^{7/2}}{8 c}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {(8 A c-a C) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {8}{7} B \left (a+c x^2\right )^{7/2}}{8 c}+\frac {C x \left (a+c x^2\right )^{7/2}}{8 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(8 A c-a C) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {8}{7} B \left (a+c x^2\right )^{7/2}}{8 c}+\frac {C x \left (a+c x^2\right )^{7/2}}{8 c}\) |
(C*x*(a + c*x^2)^(7/2))/(8*c) + ((8*B*(a + c*x^2)^(7/2))/7 + (8*A*c - a*C) *((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)/Sqrt[a + c*x^2]])/(2*Sqrt[c])))/4)) /6))/(8*c)
3.1.100.3.1 Defintions of rubi rules used
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[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Time = 0.60 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(\frac {\left (336 c^{3} C \,x^{7}+384 B \,c^{3} x^{6}+448 A \,c^{3} x^{5}+952 a \,c^{2} C \,x^{5}+1152 a B \,c^{2} x^{4}+1456 a A \,c^{2} x^{3}+826 C \,a^{2} c \,x^{3}+1152 a^{2} B c \,x^{2}+1848 a^{2} A c x +105 a^{3} C x +384 B \,a^{3}\right ) \sqrt {c \,x^{2}+a}}{2688 c}+\frac {5 a^{3} \left (8 A c -C a \right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {3}{2}}}\) | \(148\) |
| default | \(A \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )+C \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{8 c}\right )+\frac {B \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 c}\) | \(177\) |
1/2688/c*(336*C*c^3*x^7+384*B*c^3*x^6+448*A*c^3*x^5+952*C*a*c^2*x^5+1152*B *a*c^2*x^4+1456*A*a*c^2*x^3+826*C*a^2*c*x^3+1152*B*a^2*c*x^2+1848*A*a^2*c* x+105*C*a^3*x+384*B*a^3)*(c*x^2+a)^(1/2)+5/128*a^3*(8*A*c-C*a)/c^(3/2)*ln( x*c^(1/2)+(c*x^2+a)^(1/2))
Time = 0.30 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.98 \[ \int \left (a+c x^2\right )^{5/2} \left (A+B x+C x^2\right ) \, dx=\left [-\frac {105 \, {\left (C a^{4} - 8 \, A a^{3} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (336 \, C c^{4} x^{7} + 384 \, B c^{4} x^{6} + 1152 \, B a c^{3} x^{4} + 1152 \, B a^{2} c^{2} x^{2} + 56 \, {\left (17 \, C a c^{3} + 8 \, A c^{4}\right )} x^{5} + 384 \, B a^{3} c + 14 \, {\left (59 \, C a^{2} c^{2} + 104 \, A a c^{3}\right )} x^{3} + 21 \, {\left (5 \, C a^{3} c + 88 \, A a^{2} c^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{5376 \, c^{2}}, \frac {105 \, {\left (C a^{4} - 8 \, A a^{3} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (336 \, C c^{4} x^{7} + 384 \, B c^{4} x^{6} + 1152 \, B a c^{3} x^{4} + 1152 \, B a^{2} c^{2} x^{2} + 56 \, {\left (17 \, C a c^{3} + 8 \, A c^{4}\right )} x^{5} + 384 \, B a^{3} c + 14 \, {\left (59 \, C a^{2} c^{2} + 104 \, A a c^{3}\right )} x^{3} + 21 \, {\left (5 \, C a^{3} c + 88 \, A a^{2} c^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2688 \, c^{2}}\right ] \]
[-1/5376*(105*(C*a^4 - 8*A*a^3*c)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a) *sqrt(c)*x - a) - 2*(336*C*c^4*x^7 + 384*B*c^4*x^6 + 1152*B*a*c^3*x^4 + 11 52*B*a^2*c^2*x^2 + 56*(17*C*a*c^3 + 8*A*c^4)*x^5 + 384*B*a^3*c + 14*(59*C* a^2*c^2 + 104*A*a*c^3)*x^3 + 21*(5*C*a^3*c + 88*A*a^2*c^2)*x)*sqrt(c*x^2 + a))/c^2, 1/2688*(105*(C*a^4 - 8*A*a^3*c)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt( c*x^2 + a)) + (336*C*c^4*x^7 + 384*B*c^4*x^6 + 1152*B*a*c^3*x^4 + 1152*B*a ^2*c^2*x^2 + 56*(17*C*a*c^3 + 8*A*c^4)*x^5 + 384*B*a^3*c + 14*(59*C*a^2*c^ 2 + 104*A*a*c^3)*x^3 + 21*(5*C*a^3*c + 88*A*a^2*c^2)*x)*sqrt(c*x^2 + a))/c ^2]
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (150) = 300\).
Time = 0.48 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.95 \[ \int \left (a+c x^2\right )^{5/2} \left (A+B x+C x^2\right ) \, dx=\begin {cases} \sqrt {a + c x^{2}} \left (\frac {B a^{3}}{7 c} + \frac {3 B a^{2} x^{2}}{7} + \frac {3 B a c x^{4}}{7} + \frac {B c^{2} x^{6}}{7} + \frac {C c^{2} x^{7}}{8} + \frac {x^{5} \left (A c^{3} + \frac {17 C a c^{2}}{8}\right )}{6 c} + \frac {x^{3} \cdot \left (3 A a c^{2} + 3 C a^{2} c - \frac {5 a \left (A c^{3} + \frac {17 C a c^{2}}{8}\right )}{6 c}\right )}{4 c} + \frac {x \left (3 A a^{2} c + C a^{3} - \frac {3 a \left (3 A a c^{2} + 3 C a^{2} c - \frac {5 a \left (A c^{3} + \frac {17 C a c^{2}}{8}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) + \left (A a^{3} - \frac {a \left (3 A a^{2} c + C a^{3} - \frac {3 a \left (3 A a c^{2} + 3 C a^{2} c - \frac {5 a \left (A c^{3} + \frac {17 C a c^{2}}{8}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\a^{\frac {5}{2}} \left (A x + \frac {B x^{2}}{2} + \frac {C x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((sqrt(a + c*x**2)*(B*a**3/(7*c) + 3*B*a**2*x**2/7 + 3*B*a*c*x**4 /7 + B*c**2*x**6/7 + C*c**2*x**7/8 + x**5*(A*c**3 + 17*C*a*c**2/8)/(6*c) + x**3*(3*A*a*c**2 + 3*C*a**2*c - 5*a*(A*c**3 + 17*C*a*c**2/8)/(6*c))/(4*c) + x*(3*A*a**2*c + C*a**3 - 3*a*(3*A*a*c**2 + 3*C*a**2*c - 5*a*(A*c**3 + 1 7*C*a*c**2/8)/(6*c))/(4*c))/(2*c)) + (A*a**3 - a*(3*A*a**2*c + C*a**3 - 3* a*(3*A*a*c**2 + 3*C*a**2*c - 5*a*(A*c**3 + 17*C*a*c**2/8)/(6*c))/(4*c))/(2 *c))*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)), Ne(c, 0)), (a**(5/2)*(A*x + B*x**2/2 + C *x**3/3), True))
Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99 \[ \int \left (a+c x^2\right )^{5/2} \left (A+B x+C x^2\right ) \, dx=\frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} A x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A a x + \frac {5}{16} \, \sqrt {c x^{2} + a} A a^{2} x + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} C x}{8 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} C a x}{48 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} C a^{2} x}{192 \, c} - \frac {5 \, \sqrt {c x^{2} + a} C a^{3} x}{128 \, c} - \frac {5 \, C a^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {3}{2}}} + \frac {5 \, A a^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{7 \, c} \]
1/6*(c*x^2 + a)^(5/2)*A*x + 5/24*(c*x^2 + a)^(3/2)*A*a*x + 5/16*sqrt(c*x^2 + a)*A*a^2*x + 1/8*(c*x^2 + a)^(7/2)*C*x/c - 1/48*(c*x^2 + a)^(5/2)*C*a*x /c - 5/192*(c*x^2 + a)^(3/2)*C*a^2*x/c - 5/128*sqrt(c*x^2 + a)*C*a^3*x/c - 5/128*C*a^4*arcsinh(c*x/sqrt(a*c))/c^(3/2) + 5/16*A*a^3*arcsinh(c*x/sqrt( a*c))/sqrt(c) + 1/7*(c*x^2 + a)^(7/2)*B/c
Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00 \[ \int \left (a+c x^2\right )^{5/2} \left (A+B x+C x^2\right ) \, dx=\frac {1}{2688} \, {\left (\frac {384 \, B a^{3}}{c} + {\left (2 \, {\left (576 \, B a^{2} + {\left (4 \, {\left (144 \, B a c + {\left (6 \, {\left (7 \, C c^{2} x + 8 \, B c^{2}\right )} x + \frac {7 \, {\left (17 \, C a c^{7} + 8 \, A c^{8}\right )}}{c^{6}}\right )} x\right )} x + \frac {7 \, {\left (59 \, C a^{2} c^{6} + 104 \, A a c^{7}\right )}}{c^{6}}\right )} x\right )} x + \frac {21 \, {\left (5 \, C a^{3} c^{5} + 88 \, A a^{2} c^{6}\right )}}{c^{6}}\right )} x\right )} \sqrt {c x^{2} + a} + \frac {5 \, {\left (C a^{4} - 8 \, A a^{3} c\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {3}{2}}} \]
1/2688*(384*B*a^3/c + (2*(576*B*a^2 + (4*(144*B*a*c + (6*(7*C*c^2*x + 8*B* c^2)*x + 7*(17*C*a*c^7 + 8*A*c^8)/c^6)*x)*x + 7*(59*C*a^2*c^6 + 104*A*a*c^ 7)/c^6)*x)*x + 21*(5*C*a^3*c^5 + 88*A*a^2*c^6)/c^6)*x)*sqrt(c*x^2 + a) + 5 /128*(C*a^4 - 8*A*a^3*c)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)
Timed out. \[ \int \left (a+c x^2\right )^{5/2} \left (A+B x+C x^2\right ) \, dx=\int {\left (c\,x^2+a\right )}^{5/2}\,\left (C\,x^2+B\,x+A\right ) \,d x \]