Integrand size = 20, antiderivative size = 99 \[ \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {x^3 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {x (3 A+4 B x)}{3 c^2 \sqrt {a+c x^2}}+\frac {8 B \sqrt {a+c x^2}}{3 c^3}+\frac {A \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \]
-1/3*x^3*(B*x+A)/c/(c*x^2+a)^(3/2)+A*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^ (5/2)-1/3*x*(4*B*x+3*A)/c^2/(c*x^2+a)^(1/2)+8/3*B*(c*x^2+a)^(1/2)/c^3
Time = 0.43 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {8 a^2 B-3 a c x (A-4 B x)+c^2 x^3 (-4 A+3 B x)}{3 c^3 \left (a+c x^2\right )^{3/2}}+\frac {2 A \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+c x^2}}\right )}{c^{5/2}} \]
(8*a^2*B - 3*a*c*x*(A - 4*B*x) + c^2*x^3*(-4*A + 3*B*x))/(3*c^3*(a + c*x^2 )^(3/2)) + (2*A*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + c*x^2])])/c^(5/2)
Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {530, 2345, 27, 455, 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.
| \(\displaystyle \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 530 |
| \(\displaystyle -\frac {\int \frac {-\frac {3 a B x^3}{c}-\frac {3 a A x^2}{c}+\frac {3 a^2 B x}{c^2}+\frac {a^2 A}{c^2}}{\left (c x^2+a\right )^{3/2}}dx}{3 a}-\frac {a (a B-A c x)}{3 c^3 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 a^2 (A+B x)}{c^2 \sqrt {c x^2+a}}dx}{a}-\frac {2 a (3 a B-2 A c x)}{c^3 \sqrt {a+c x^2}}}{3 a}-\frac {a (a B-A c x)}{3 c^3 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {3 a \int \frac {A+B x}{\sqrt {c x^2+a}}dx}{c^2}-\frac {2 a (3 a B-2 A c x)}{c^3 \sqrt {a+c x^2}}}{3 a}-\frac {a (a B-A c x)}{3 c^3 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {-\frac {3 a \left (A \int \frac {1}{\sqrt {c x^2+a}}dx+\frac {B \sqrt {a+c x^2}}{c}\right )}{c^2}-\frac {2 a (3 a B-2 A c x)}{c^3 \sqrt {a+c x^2}}}{3 a}-\frac {a (a B-A c x)}{3 c^3 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {-\frac {3 a \left (A \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}+\frac {B \sqrt {a+c x^2}}{c}\right )}{c^2}-\frac {2 a (3 a B-2 A c x)}{c^3 \sqrt {a+c x^2}}}{3 a}-\frac {a (a B-A c x)}{3 c^3 \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {3 a \left (\frac {A \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c}}+\frac {B \sqrt {a+c x^2}}{c}\right )}{c^2}-\frac {2 a (3 a B-2 A c x)}{c^3 \sqrt {a+c x^2}}}{3 a}-\frac {a (a B-A c x)}{3 c^3 \left (a+c x^2\right )^{3/2}}\) |
-1/3*(a*(a*B - A*c*x))/(c^3*(a + c*x^2)^(3/2)) - ((-2*a*(3*a*B - 2*A*c*x)) /(c^3*Sqrt[a + c*x^2]) - (3*a*((B*Sqrt[a + c*x^2])/c + (A*ArcTanh[(Sqrt[c] *x)/Sqrt[a + c*x^2]])/Sqrt[c]))/c^2)/(3*a)
3.4.74.3.1 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)^(-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_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po lynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x )*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 1] && IntegerQ[2*p]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.21
| method | result | size |
| default | \(B \left (\frac {x^{4}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{c}\right )+A \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}}{c}\right )\) | \(120\) |
| risch | \(\frac {B \sqrt {c \,x^{2}+a}}{c^{3}}+\frac {\frac {A \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}-\frac {a \left (A c -B \sqrt {-a c}\right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{3 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{3 a \left (x +\frac {\sqrt {-a c}}{c}\right )}\right )}{4 c^{2}}-\frac {a \left (A c +B \sqrt {-a c}\right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{3 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{3 a \left (x -\frac {\sqrt {-a c}}{c}\right )}\right )}{4 c^{2}}-\frac {\left (3 A c +4 B \sqrt {-a c}\right ) \sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{4 c^{2} \left (x -\frac {\sqrt {-a c}}{c}\right )}-\frac {\left (3 A c -4 B \sqrt {-a c}\right ) \sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{4 c^{2} \left (x +\frac {\sqrt {-a c}}{c}\right )}}{c^{2}}\) | \(472\) |
B*(x^4/c/(c*x^2+a)^(3/2)-4*a/c*(-x^2/c/(c*x^2+a)^(3/2)-2/3*a/c^2/(c*x^2+a) ^(3/2)))+A*(-1/3*x^3/c/(c*x^2+a)^(3/2)+1/c*(-x/c/(c*x^2+a)^(1/2)+1/c^(3/2) *ln(x*c^(1/2)+(c*x^2+a)^(1/2))))
Time = 0.31 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.61 \[ \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (A c^{2} x^{4} + 2 \, A a c x^{2} + A a^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (3 \, B c^{2} x^{4} - 4 \, A c^{2} x^{3} + 12 \, B a c x^{2} - 3 \, A a c x + 8 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{5} x^{4} + 2 \, a c^{4} x^{2} + a^{2} c^{3}\right )}}, -\frac {3 \, {\left (A c^{2} x^{4} + 2 \, A a c x^{2} + A a^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, B c^{2} x^{4} - 4 \, A c^{2} x^{3} + 12 \, B a c x^{2} - 3 \, A a c x + 8 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (c^{5} x^{4} + 2 \, a c^{4} x^{2} + a^{2} c^{3}\right )}}\right ] \]
[1/6*(3*(A*c^2*x^4 + 2*A*a*c*x^2 + A*a^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c* x^2 + a)*sqrt(c)*x - a) + 2*(3*B*c^2*x^4 - 4*A*c^2*x^3 + 12*B*a*c*x^2 - 3* A*a*c*x + 8*B*a^2)*sqrt(c*x^2 + a))/(c^5*x^4 + 2*a*c^4*x^2 + a^2*c^3), -1/ 3*(3*(A*c^2*x^4 + 2*A*a*c*x^2 + A*a^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x ^2 + a)) - (3*B*c^2*x^4 - 4*A*c^2*x^3 + 12*B*a*c*x^2 - 3*A*a*c*x + 8*B*a^2 )*sqrt(c*x^2 + a))/(c^5*x^4 + 2*a*c^4*x^2 + a^2*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (90) = 180\).
Time = 6.44 (sec) , antiderivative size = 445, normalized size of antiderivative = 4.49 \[ \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=A \left (\frac {3 a^{\frac {39}{2}} c^{11} \sqrt {1 + \frac {c x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} c^{\frac {27}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {37}{2}} c^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 a^{\frac {37}{2}} c^{12} x^{2} \sqrt {1 + \frac {c x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} c^{\frac {27}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {37}{2}} c^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {3 a^{19} c^{\frac {23}{2}} x}{3 a^{\frac {39}{2}} c^{\frac {27}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {37}{2}} c^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {4 a^{18} c^{\frac {25}{2}} x^{3}}{3 a^{\frac {39}{2}} c^{\frac {27}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {37}{2}} c^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {c x^{2}}{a}}}\right ) + B \left (\begin {cases} \frac {8 a^{2}}{3 a c^{3} \sqrt {a + c x^{2}} + 3 c^{4} x^{2} \sqrt {a + c x^{2}}} + \frac {12 a c x^{2}}{3 a c^{3} \sqrt {a + c x^{2}} + 3 c^{4} x^{2} \sqrt {a + c x^{2}}} + \frac {3 c^{2} x^{4}}{3 a c^{3} \sqrt {a + c x^{2}} + 3 c^{4} x^{2} \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{6}}{6 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \]
A*(3*a**(39/2)*c**11*sqrt(1 + c*x**2/a)*asinh(sqrt(c)*x/sqrt(a))/(3*a**(39 /2)*c**(27/2)*sqrt(1 + c*x**2/a) + 3*a**(37/2)*c**(29/2)*x**2*sqrt(1 + c*x **2/a)) + 3*a**(37/2)*c**12*x**2*sqrt(1 + c*x**2/a)*asinh(sqrt(c)*x/sqrt(a ))/(3*a**(39/2)*c**(27/2)*sqrt(1 + c*x**2/a) + 3*a**(37/2)*c**(29/2)*x**2* sqrt(1 + c*x**2/a)) - 3*a**19*c**(23/2)*x/(3*a**(39/2)*c**(27/2)*sqrt(1 + c*x**2/a) + 3*a**(37/2)*c**(29/2)*x**2*sqrt(1 + c*x**2/a)) - 4*a**18*c**(2 5/2)*x**3/(3*a**(39/2)*c**(27/2)*sqrt(1 + c*x**2/a) + 3*a**(37/2)*c**(29/2 )*x**2*sqrt(1 + c*x**2/a))) + B*Piecewise((8*a**2/(3*a*c**3*sqrt(a + c*x** 2) + 3*c**4*x**2*sqrt(a + c*x**2)) + 12*a*c*x**2/(3*a*c**3*sqrt(a + c*x**2 ) + 3*c**4*x**2*sqrt(a + c*x**2)) + 3*c**2*x**4/(3*a*c**3*sqrt(a + c*x**2) + 3*c**4*x**2*sqrt(a + c*x**2)), Ne(c, 0)), (x**6/(6*a**(5/2)), True))
Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.23 \[ \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, A x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, a}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}}\right )} + \frac {B x^{4}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {4 \, B a x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} - \frac {A x}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {A \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {5}{2}}} + \frac {8 \, B a^{2}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3}} \]
-1/3*A*x*(3*x^2/((c*x^2 + a)^(3/2)*c) + 2*a/((c*x^2 + a)^(3/2)*c^2)) + B*x ^4/((c*x^2 + a)^(3/2)*c) + 4*B*a*x^2/((c*x^2 + a)^(3/2)*c^2) - 1/3*A*x/(sq rt(c*x^2 + a)*c^2) + A*arcsinh(c*x/sqrt(a*c))/c^(5/2) + 8/3*B*a^2/((c*x^2 + a)^(3/2)*c^3)
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {{\left ({\left ({\left (\frac {3 \, B x}{c} - \frac {4 \, A}{c}\right )} x + \frac {12 \, B a}{c^{2}}\right )} x - \frac {3 \, A a}{c^{2}}\right )} x + \frac {8 \, B a^{2}}{c^{3}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} - \frac {A \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {5}{2}}} \]
1/3*((((3*B*x/c - 4*A/c)*x + 12*B*a/c^2)*x - 3*A*a/c^2)*x + 8*B*a^2/c^3)/( c*x^2 + a)^(3/2) - A*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)
Timed out. \[ \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]