Integrand size = 19, antiderivative size = 189 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\frac {5 a^2 \left (8 c d^2-a e^2\right ) x \sqrt {a+c x^2}}{128 c}+\frac {5 a \left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac {\left (8 c d^2-a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac {9 d e \left (a+c x^2\right )^{7/2}}{56 c}+\frac {e (d+e x) \left (a+c x^2\right )^{7/2}}{8 c}+\frac {5 a^3 \left (8 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{3/2}} \]
5/192*a*(-a*e^2+8*c*d^2)*x*(c*x^2+a)^(3/2)/c+1/48*(-a*e^2+8*c*d^2)*x*(c*x^ 2+a)^(5/2)/c+9/56*d*e*(c*x^2+a)^(7/2)/c+1/8*e*(e*x+d)*(c*x^2+a)^(7/2)/c+5/ 128*a^3*(-a*e^2+8*c*d^2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)+5/128* a^2*(-a*e^2+8*c*d^2)*x*(c*x^2+a)^(1/2)/c
Time = 0.61 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\frac {\sqrt {c} \sqrt {a+c x^2} \left (3 a^3 e (256 d+35 e x)+16 c^3 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+8 a c^2 x^3 \left (182 d^2+288 d e x+119 e^2 x^2\right )+2 a^2 c x \left (924 d^2+1152 d e x+413 e^2 x^2\right )\right )+105 a^3 \left (-8 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{2688 c^{3/2}} \]
(Sqrt[c]*Sqrt[a + c*x^2]*(3*a^3*e*(256*d + 35*e*x) + 16*c^3*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 8*a*c^2*x^3*(182*d^2 + 288*d*e*x + 119*e^2*x^2) + 2*a^2*c*x*(924*d^2 + 1152*d*e*x + 413*e^2*x^2)) + 105*a^3*(-8*c*d^2 + a*e ^2)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(2688*c^(3/2))
Time = 0.25 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.81, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {497, 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.
| \(\displaystyle \int \left (a+c x^2\right )^{5/2} (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 497 |
| \(\displaystyle \frac {\int \left (8 c d^2+9 c e x d-a e^2\right ) \left (c x^2+a\right )^{5/2}dx}{8 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {\left (8 c d^2-a e^2\right ) \int \left (c x^2+a\right )^{5/2}dx+\frac {9}{7} d e \left (a+c x^2\right )^{7/2}}{8 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\left (8 c d^2-a e^2\right ) \left (\frac {5}{6} a \int \left (c x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {9}{7} d e \left (a+c x^2\right )^{7/2}}{8 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\left (8 c d^2-a e^2\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {c x^2+a}dx+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {9}{7} d e \left (a+c x^2\right )^{7/2}}{8 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\left (8 c d^2-a e^2\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {c x^2+a}}dx+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {9}{7} d e \left (a+c x^2\right )^{7/2}}{8 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\left (8 c d^2-a e^2\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right )+\frac {9}{7} d e \left (a+c x^2\right )^{7/2}}{8 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+c x^2\right )^{5/2}\right ) \left (8 c d^2-a e^2\right )+\frac {9}{7} d e \left (a+c x^2\right )^{7/2}}{8 c}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)}{8 c}\) |
(e*(d + e*x)*(a + c*x^2)^(7/2))/(8*c) + ((9*d*e*(a + c*x^2)^(7/2))/7 + (8* c*d^2 - a*e^2)*((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)
3.6.47.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Time = 1.96 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.89
| method | result | size |
| risch | \(\frac {\left (336 e^{2} c^{3} x^{7}+768 d e \,c^{3} x^{6}+952 e^{2} c^{2} a \,x^{5}+448 c^{3} d^{2} x^{5}+2304 a \,c^{2} d e \,x^{4}+826 a^{2} c \,e^{2} x^{3}+1456 a \,c^{2} d^{2} x^{3}+2304 x^{2} a^{2} c d e +105 a^{3} e^{2} x +1848 c \,a^{2} d^{2} x +768 d e \,a^{3}\right ) \sqrt {c \,x^{2}+a}}{2688 c}-\frac {5 a^{3} \left (e^{2} a -8 c \,d^{2}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {3}{2}}}\) | \(169\) |
| default | \(d^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{8 c}\right )+\frac {2 d e \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 c}\) | \(182\) |
1/2688/c*(336*c^3*e^2*x^7+768*c^3*d*e*x^6+952*a*c^2*e^2*x^5+448*c^3*d^2*x^ 5+2304*a*c^2*d*e*x^4+826*a^2*c*e^2*x^3+1456*a*c^2*d^2*x^3+2304*a^2*c*d*e*x ^2+105*a^3*e^2*x+1848*a^2*c*d^2*x+768*a^3*d*e)*(c*x^2+a)^(1/2)-5/128*a^3*( a*e^2-8*c*d^2)/c^(3/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))
Time = 0.31 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.01 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\left [\frac {105 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (336 \, c^{4} e^{2} x^{7} + 768 \, c^{4} d e x^{6} + 2304 \, a c^{3} d e x^{4} + 2304 \, a^{2} c^{2} d e x^{2} + 768 \, a^{3} c d e + 56 \, {\left (8 \, c^{4} d^{2} + 17 \, a c^{3} e^{2}\right )} x^{5} + 14 \, {\left (104 \, a c^{3} d^{2} + 59 \, a^{2} c^{2} e^{2}\right )} x^{3} + 21 \, {\left (88 \, a^{2} c^{2} d^{2} + 5 \, a^{3} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{5376 \, c^{2}}, -\frac {105 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (336 \, c^{4} e^{2} x^{7} + 768 \, c^{4} d e x^{6} + 2304 \, a c^{3} d e x^{4} + 2304 \, a^{2} c^{2} d e x^{2} + 768 \, a^{3} c d e + 56 \, {\left (8 \, c^{4} d^{2} + 17 \, a c^{3} e^{2}\right )} x^{5} + 14 \, {\left (104 \, a c^{3} d^{2} + 59 \, a^{2} c^{2} e^{2}\right )} x^{3} + 21 \, {\left (88 \, a^{2} c^{2} d^{2} + 5 \, a^{3} c e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2688 \, c^{2}}\right ] \]
[1/5376*(105*(8*a^3*c*d^2 - a^4*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(336*c^4*e^2*x^7 + 768*c^4*d*e*x^6 + 2304*a*c^3*d*e *x^4 + 2304*a^2*c^2*d*e*x^2 + 768*a^3*c*d*e + 56*(8*c^4*d^2 + 17*a*c^3*e^2 )*x^5 + 14*(104*a*c^3*d^2 + 59*a^2*c^2*e^2)*x^3 + 21*(88*a^2*c^2*d^2 + 5*a ^3*c*e^2)*x)*sqrt(c*x^2 + a))/c^2, -1/2688*(105*(8*a^3*c*d^2 - a^4*e^2)*sq rt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (336*c^4*e^2*x^7 + 768*c^4*d*e *x^6 + 2304*a*c^3*d*e*x^4 + 2304*a^2*c^2*d*e*x^2 + 768*a^3*c*d*e + 56*(8*c ^4*d^2 + 17*a*c^3*e^2)*x^5 + 14*(104*a*c^3*d^2 + 59*a^2*c^2*e^2)*x^3 + 21* (88*a^2*c^2*d^2 + 5*a^3*c*e^2)*x)*sqrt(c*x^2 + a))/c^2]
Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (170) = 340\).
Time = 0.61 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.96 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\begin {cases} \sqrt {a + c x^{2}} \cdot \left (\frac {2 a^{3} d e}{7 c} + \frac {6 a^{2} d e x^{2}}{7} + \frac {6 a c d e x^{4}}{7} + \frac {2 c^{2} d e x^{6}}{7} + \frac {c^{2} e^{2} x^{7}}{8} + \frac {x^{5} \cdot \left (\frac {17 a c^{2} e^{2}}{8} + c^{3} d^{2}\right )}{6 c} + \frac {x^{3} \cdot \left (3 a^{2} c e^{2} + 3 a c^{2} d^{2} - \frac {5 a \left (\frac {17 a c^{2} e^{2}}{8} + c^{3} d^{2}\right )}{6 c}\right )}{4 c} + \frac {x \left (a^{3} e^{2} + 3 a^{2} c d^{2} - \frac {3 a \left (3 a^{2} c e^{2} + 3 a c^{2} d^{2} - \frac {5 a \left (\frac {17 a c^{2} e^{2}}{8} + c^{3} d^{2}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) + \left (a^{3} d^{2} - \frac {a \left (a^{3} e^{2} + 3 a^{2} c d^{2} - \frac {3 a \left (3 a^{2} c e^{2} + 3 a c^{2} d^{2} - \frac {5 a \left (\frac {17 a c^{2} e^{2}}{8} + c^{3} d^{2}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\a^{\frac {5}{2}} \left (\begin {cases} d^{2} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{3}}{3 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((sqrt(a + c*x**2)*(2*a**3*d*e/(7*c) + 6*a**2*d*e*x**2/7 + 6*a*c* d*e*x**4/7 + 2*c**2*d*e*x**6/7 + c**2*e**2*x**7/8 + x**5*(17*a*c**2*e**2/8 + c**3*d**2)/(6*c) + x**3*(3*a**2*c*e**2 + 3*a*c**2*d**2 - 5*a*(17*a*c**2 *e**2/8 + c**3*d**2)/(6*c))/(4*c) + x*(a**3*e**2 + 3*a**2*c*d**2 - 3*a*(3* a**2*c*e**2 + 3*a*c**2*d**2 - 5*a*(17*a*c**2*e**2/8 + c**3*d**2)/(6*c))/(4 *c))/(2*c)) + (a**3*d**2 - a*(a**3*e**2 + 3*a**2*c*d**2 - 3*a*(3*a**2*c*e* *2 + 3*a*c**2*d**2 - 5*a*(17*a*c**2*e**2/8 + c**3*d**2)/(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)*Piecewise((d**2*x, Eq (e, 0)), ((d + e*x)**3/(3*e), True)), True))
Time = 0.20 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{2} x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d^{2} x + \frac {5}{16} \, \sqrt {c x^{2} + a} a^{2} d^{2} x + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} e^{2} x}{8 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a e^{2} x}{48 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} e^{2} x}{192 \, c} - \frac {5 \, \sqrt {c x^{2} + a} a^{3} e^{2} x}{128 \, c} + \frac {5 \, a^{3} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} - \frac {5 \, a^{4} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {3}{2}}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d e}{7 \, c} \]
1/6*(c*x^2 + a)^(5/2)*d^2*x + 5/24*(c*x^2 + a)^(3/2)*a*d^2*x + 5/16*sqrt(c *x^2 + a)*a^2*d^2*x + 1/8*(c*x^2 + a)^(7/2)*e^2*x/c - 1/48*(c*x^2 + a)^(5/ 2)*a*e^2*x/c - 5/192*(c*x^2 + a)^(3/2)*a^2*e^2*x/c - 5/128*sqrt(c*x^2 + a) *a^3*e^2*x/c + 5/16*a^3*d^2*arcsinh(c*x/sqrt(a*c))/sqrt(c) - 5/128*a^4*e^2 *arcsinh(c*x/sqrt(a*c))/c^(3/2) + 2/7*(c*x^2 + a)^(7/2)*d*e/c
Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01 \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\frac {1}{2688} \, {\left (\frac {768 \, a^{3} d e}{c} + {\left (2 \, {\left (1152 \, a^{2} d e + {\left (4 \, {\left (288 \, a c d e + {\left (6 \, {\left (7 \, c^{2} e^{2} x + 16 \, c^{2} d e\right )} x + \frac {7 \, {\left (8 \, c^{8} d^{2} + 17 \, a c^{7} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac {7 \, {\left (104 \, a c^{7} d^{2} + 59 \, a^{2} c^{6} e^{2}\right )}}{c^{6}}\right )} x\right )} x + \frac {21 \, {\left (88 \, a^{2} c^{6} d^{2} + 5 \, a^{3} c^{5} e^{2}\right )}}{c^{6}}\right )} x\right )} \sqrt {c x^{2} + a} - \frac {5 \, {\left (8 \, a^{3} c d^{2} - a^{4} e^{2}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {3}{2}}} \]
1/2688*(768*a^3*d*e/c + (2*(1152*a^2*d*e + (4*(288*a*c*d*e + (6*(7*c^2*e^2 *x + 16*c^2*d*e)*x + 7*(8*c^8*d^2 + 17*a*c^7*e^2)/c^6)*x)*x + 7*(104*a*c^7 *d^2 + 59*a^2*c^6*e^2)/c^6)*x)*x + 21*(88*a^2*c^6*d^2 + 5*a^3*c^5*e^2)/c^6 )*x)*sqrt(c*x^2 + a) - 5/128*(8*a^3*c*d^2 - a^4*e^2)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)
Timed out. \[ \int (d+e x)^2 \left (a+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^2 \,d x \]