Integrand size = 20, antiderivative size = 175 \[ \int x^4 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {3 a^3 A x \sqrt {a+c x^2}}{128 c^2}+\frac {a^2 A x \left (a+c x^2\right )^{3/2}}{64 c^2}-\frac {4 a B x^2 \left (a+c x^2\right )^{5/2}}{63 c^2}+\frac {A x^3 \left (a+c x^2\right )^{5/2}}{8 c}+\frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}+\frac {a (128 a B-315 A c x) \left (a+c x^2\right )^{5/2}}{5040 c^3}+\frac {3 a^4 A \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}} \]
1/64*a^2*A*x*(c*x^2+a)^(3/2)/c^2-4/63*a*B*x^2*(c*x^2+a)^(5/2)/c^2+1/8*A*x^ 3*(c*x^2+a)^(5/2)/c+1/9*B*x^4*(c*x^2+a)^(5/2)/c+1/5040*a*(-315*A*c*x+128*B *a)*(c*x^2+a)^(5/2)/c^3+3/128*a^4*A*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^( 5/2)+3/128*a^3*A*x*(c*x^2+a)^(1/2)/c^2
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.72 \[ \int x^4 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {\sqrt {a+c x^2} \left (1024 a^4 B+560 c^4 x^7 (9 A+8 B x)+6 a^2 c^2 x^3 (105 A+64 B x)+40 a c^3 x^5 (189 A+160 B x)-a^3 c x (945 A+512 B x)\right )-945 a^4 A \sqrt {c} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{40320 c^3} \]
(Sqrt[a + c*x^2]*(1024*a^4*B + 560*c^4*x^7*(9*A + 8*B*x) + 6*a^2*c^2*x^3*( 105*A + 64*B*x) + 40*a*c^3*x^5*(189*A + 160*B*x) - a^3*c*x*(945*A + 512*B* x)) - 945*a^4*A*Sqrt[c]*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(40320*c^3)
Time = 0.32 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {533, 533, 25, 27, 533, 533, 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.
| \(\displaystyle \int x^4 \left (a+c x^2\right )^{3/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\int x^3 (4 a B-9 A c x) \left (c x^2+a\right )^{3/2}dx}{9 c}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {-\frac {\int -a c x^2 (27 A+32 B x) \left (c x^2+a\right )^{3/2}dx}{8 c}-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {\int a c x^2 (27 A+32 B x) \left (c x^2+a\right )^{3/2}dx}{8 c}-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {1}{8} a \int x^2 (27 A+32 B x) \left (c x^2+a\right )^{3/2}dx-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {1}{8} a \left (\frac {32 B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\int x (64 a B-189 A c x) \left (c x^2+a\right )^{3/2}dx}{7 c}\right )-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {1}{8} a \left (\frac {32 B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {-\frac {\int -3 a c (63 A+128 B x) \left (c x^2+a\right )^{3/2}dx}{6 c}-\frac {63}{2} A x \left (a+c x^2\right )^{5/2}}{7 c}\right )-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {1}{8} a \left (\frac {32 B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{2} a \int (63 A+128 B x) \left (c x^2+a\right )^{3/2}dx-\frac {63}{2} A x \left (a+c x^2\right )^{5/2}}{7 c}\right )-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {1}{8} a \left (\frac {32 B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{2} a \left (63 A \int \left (c x^2+a\right )^{3/2}dx+\frac {128 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {63}{2} A x \left (a+c x^2\right )^{5/2}}{7 c}\right )-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {1}{8} a \left (\frac {32 B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{2} a \left (63 A \left (\frac {3}{4} a \int \sqrt {c x^2+a}dx+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {128 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {63}{2} A x \left (a+c x^2\right )^{5/2}}{7 c}\right )-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {1}{8} a \left (\frac {32 B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{2} a \left (63 A \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {c x^2+a}}dx+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {128 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {63}{2} A x \left (a+c x^2\right )^{5/2}}{7 c}\right )-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {1}{8} a \left (\frac {32 B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{2} a \left (63 A \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {128 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {63}{2} A x \left (a+c x^2\right )^{5/2}}{7 c}\right )-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {B x^4 \left (a+c x^2\right )^{5/2}}{9 c}-\frac {\frac {1}{8} a \left (\frac {32 B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {\frac {1}{2} a \left (63 A \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {1}{2} x \sqrt {a+c x^2}\right )+\frac {1}{4} x \left (a+c x^2\right )^{3/2}\right )+\frac {128 B \left (a+c x^2\right )^{5/2}}{5 c}\right )-\frac {63}{2} A x \left (a+c x^2\right )^{5/2}}{7 c}\right )-\frac {9}{8} A x^3 \left (a+c x^2\right )^{5/2}}{9 c}\) |
(B*x^4*(a + c*x^2)^(5/2))/(9*c) - ((-9*A*x^3*(a + c*x^2)^(5/2))/8 + (a*((3 2*B*x^2*(a + c*x^2)^(5/2))/(7*c) - ((-63*A*x*(a + c*x^2)^(5/2))/2 + (a*((1 28*B*(a + c*x^2)^(5/2))/(5*c) + 63*A*((x*(a + c*x^2)^(3/2))/4 + (3*a*((x*S qrt[a + c*x^2])/2 + (a*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*Sqrt[c]))) /4)))/2)/(7*c)))/8)/(9*c)
3.4.26.3.1 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 = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(-\frac {\left (-4480 B \,c^{4} x^{8}-5040 A \,c^{4} x^{7}-6400 a B \,c^{3} x^{6}-7560 a A \,c^{3} x^{5}-384 a^{2} B \,c^{2} x^{4}-630 a^{2} A \,c^{2} x^{3}+512 a^{3} B c \,x^{2}+945 a^{3} A c x -1024 B \,a^{4}\right ) \sqrt {c \,x^{2}+a}}{40320 c^{3}}+\frac {3 a^{4} A \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {5}{2}}}\) | \(128\) |
| default | \(B \left (\frac {x^{4} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{9 c}-\frac {4 a \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{7 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{35 c^{2}}\right )}{9 c}\right )+A \left (\frac {x^{3} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{8 c}-\frac {3 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )}{8 c}\right )\) | \(160\) |
-1/40320*(-4480*B*c^4*x^8-5040*A*c^4*x^7-6400*B*a*c^3*x^6-7560*A*a*c^3*x^5 -384*B*a^2*c^2*x^4-630*A*a^2*c^2*x^3+512*B*a^3*c*x^2+945*A*a^3*c*x-1024*B* a^4)/c^3*(c*x^2+a)^(1/2)+3/128*a^4*A/c^(5/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))
Time = 0.39 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.55 \[ \int x^4 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\left [\frac {945 \, A a^{4} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (4480 \, B c^{4} x^{8} + 5040 \, A c^{4} x^{7} + 6400 \, B a c^{3} x^{6} + 7560 \, A a c^{3} x^{5} + 384 \, B a^{2} c^{2} x^{4} + 630 \, A a^{2} c^{2} x^{3} - 512 \, B a^{3} c x^{2} - 945 \, A a^{3} c x + 1024 \, B a^{4}\right )} \sqrt {c x^{2} + a}}{80640 \, c^{3}}, -\frac {945 \, A a^{4} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (4480 \, B c^{4} x^{8} + 5040 \, A c^{4} x^{7} + 6400 \, B a c^{3} x^{6} + 7560 \, A a c^{3} x^{5} + 384 \, B a^{2} c^{2} x^{4} + 630 \, A a^{2} c^{2} x^{3} - 512 \, B a^{3} c x^{2} - 945 \, A a^{3} c x + 1024 \, B a^{4}\right )} \sqrt {c x^{2} + a}}{40320 \, c^{3}}\right ] \]
[1/80640*(945*A*a^4*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a ) + 2*(4480*B*c^4*x^8 + 5040*A*c^4*x^7 + 6400*B*a*c^3*x^6 + 7560*A*a*c^3*x ^5 + 384*B*a^2*c^2*x^4 + 630*A*a^2*c^2*x^3 - 512*B*a^3*c*x^2 - 945*A*a^3*c *x + 1024*B*a^4)*sqrt(c*x^2 + a))/c^3, -1/40320*(945*A*a^4*sqrt(-c)*arctan (sqrt(-c)*x/sqrt(c*x^2 + a)) - (4480*B*c^4*x^8 + 5040*A*c^4*x^7 + 6400*B*a *c^3*x^6 + 7560*A*a*c^3*x^5 + 384*B*a^2*c^2*x^4 + 630*A*a^2*c^2*x^3 - 512* B*a^3*c*x^2 - 945*A*a^3*c*x + 1024*B*a^4)*sqrt(c*x^2 + a))/c^3]
Time = 0.62 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.05 \[ \int x^4 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 A a^{4} \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right )}{128 c^{2}} + \sqrt {a + c x^{2}} \left (- \frac {3 A a^{3} x}{128 c^{2}} + \frac {A a^{2} x^{3}}{64 c} + \frac {3 A a x^{5}}{16} + \frac {A c x^{7}}{8} + \frac {8 B a^{4}}{315 c^{3}} - \frac {4 B a^{3} x^{2}}{315 c^{2}} + \frac {B a^{2} x^{4}}{105 c} + \frac {10 B a x^{6}}{63} + \frac {B c x^{8}}{9}\right ) & \text {for}\: c \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{5}}{5} + \frac {B x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Piecewise((3*A*a**4*Piecewise((log(2*sqrt(c)*sqrt(a + c*x**2) + 2*c*x)/sqr t(c), Ne(a, 0)), (x*log(x)/sqrt(c*x**2), True))/(128*c**2) + sqrt(a + c*x* *2)*(-3*A*a**3*x/(128*c**2) + A*a**2*x**3/(64*c) + 3*A*a*x**5/16 + A*c*x** 7/8 + 8*B*a**4/(315*c**3) - 4*B*a**3*x**2/(315*c**2) + B*a**2*x**4/(105*c) + 10*B*a*x**6/63 + B*c*x**8/9), Ne(c, 0)), (a**(3/2)*(A*x**5/5 + B*x**6/6 ), True))
Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.84 \[ \int x^4 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B x^{4}}{9 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A x^{3}}{8 \, c} - \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B a x^{2}}{63 \, c^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A a x}{16 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A a^{2} x}{64 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} A a^{3} x}{128 \, c^{2}} + \frac {3 \, A a^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {5}{2}}} + \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B a^{2}}{315 \, c^{3}} \]
1/9*(c*x^2 + a)^(5/2)*B*x^4/c + 1/8*(c*x^2 + a)^(5/2)*A*x^3/c - 4/63*(c*x^ 2 + a)^(5/2)*B*a*x^2/c^2 - 1/16*(c*x^2 + a)^(5/2)*A*a*x/c^2 + 1/64*(c*x^2 + a)^(3/2)*A*a^2*x/c^2 + 3/128*sqrt(c*x^2 + a)*A*a^3*x/c^2 + 3/128*A*a^4*a rcsinh(c*x/sqrt(a*c))/c^(5/2) + 8/315*(c*x^2 + a)^(5/2)*B*a^2/c^3
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.74 \[ \int x^4 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=-\frac {3 \, A a^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {5}{2}}} + \frac {1}{40320} \, \sqrt {c x^{2} + a} {\left (\frac {1024 \, B a^{4}}{c^{3}} - {\left (\frac {945 \, A a^{3}}{c^{2}} + 2 \, {\left (\frac {256 \, B a^{3}}{c^{2}} - {\left (\frac {315 \, A a^{2}}{c} + 4 \, {\left (\frac {48 \, B a^{2}}{c} + 5 \, {\left (189 \, A a + 2 \, {\left (80 \, B a + 7 \, {\left (8 \, B c x + 9 \, A c\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
-3/128*A*a^4*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2) + 1/40320*sqrt (c*x^2 + a)*(1024*B*a^4/c^3 - (945*A*a^3/c^2 + 2*(256*B*a^3/c^2 - (315*A*a ^2/c + 4*(48*B*a^2/c + 5*(189*A*a + 2*(80*B*a + 7*(8*B*c*x + 9*A*c)*x)*x)* x)*x)*x)*x)*x)
Timed out. \[ \int x^4 (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\int x^4\,{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \]