Integrand size = 35, antiderivative size = 276 \[ \int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {a^3 d (f x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f (1+m) \left (a+b x^2\right )}+\frac {a^2 (3 b d+a e) (f x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^3 (3+m) \left (a+b x^2\right )}+\frac {3 a b (b d+a e) (f x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^5 (5+m) \left (a+b x^2\right )}+\frac {b^2 (b d+3 a e) (f x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^7 (7+m) \left (a+b x^2\right )}+\frac {b^3 e (f x)^{9+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^9 (9+m) \left (a+b x^2\right )} \]
a^3*d*(f*x)^(1+m)*((b*x^2+a)^2)^(1/2)/f/(1+m)/(b*x^2+a)+a^2*(a*e+3*b*d)*(f *x)^(3+m)*((b*x^2+a)^2)^(1/2)/f^3/(3+m)/(b*x^2+a)+3*a*b*(a*e+b*d)*(f*x)^(5 +m)*((b*x^2+a)^2)^(1/2)/f^5/(5+m)/(b*x^2+a)+b^2*(3*a*e+b*d)*(f*x)^(7+m)*(( b*x^2+a)^2)^(1/2)/f^7/(7+m)/(b*x^2+a)+b^3*e*(f*x)^(9+m)*((b*x^2+a)^2)^(1/2 )/f^9/(9+m)/(b*x^2+a)
Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.41 \[ \int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {x (f x)^m \left (\left (a+b x^2\right )^2\right )^{3/2} \left (\frac {a^3 d}{1+m}+\frac {a^2 (3 b d+a e) x^2}{3+m}+\frac {3 a b (b d+a e) x^4}{5+m}+\frac {b^2 (b d+3 a e) x^6}{7+m}+\frac {b^3 e x^8}{9+m}\right )}{\left (a+b x^2\right )^3} \]
(x*(f*x)^m*((a + b*x^2)^2)^(3/2)*((a^3*d)/(1 + m) + (a^2*(3*b*d + a*e)*x^2 )/(3 + m) + (3*a*b*(b*d + a*e)*x^4)/(5 + m) + (b^2*(b*d + 3*a*e)*x^6)/(7 + m) + (b^3*e*x^8)/(9 + m)))/(a + b*x^2)^3
Time = 0.33 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1384, 27, 355, 2009}
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.
| \(\displaystyle \int \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \left (d+e x^2\right ) (f x)^m \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int b^3 (f x)^m \left (b x^2+a\right )^3 \left (e x^2+d\right )dx}{b^3 \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (f x)^m \left (b x^2+a\right )^3 \left (e x^2+d\right )dx}{a+b x^2}\) |
| \(\Big \downarrow \) 355 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^3 d (f x)^m+\frac {a^2 (3 b d+a e) (f x)^{m+2}}{f^2}+\frac {3 a b (b d+a e) (f x)^{m+4}}{f^4}+\frac {b^2 (b d+3 a e) (f x)^{m+6}}{f^6}+\frac {b^3 e (f x)^{m+8}}{f^8}\right )dx}{a+b x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (\frac {a^3 d (f x)^{m+1}}{f (m+1)}+\frac {a^2 (f x)^{m+3} (a e+3 b d)}{f^3 (m+3)}+\frac {b^2 (f x)^{m+7} (3 a e+b d)}{f^7 (m+7)}+\frac {3 a b (f x)^{m+5} (a e+b d)}{f^5 (m+5)}+\frac {b^3 e (f x)^{m+9}}{f^9 (m+9)}\right )}{a+b x^2}\) |
(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*((a^3*d*(f*x)^(1 + m))/(f*(1 + m)) + (a^2 *(3*b*d + a*e)*(f*x)^(3 + m))/(f^3*(3 + m)) + (3*a*b*(b*d + a*e)*(f*x)^(5 + m))/(f^5*(5 + m)) + (b^2*(b*d + 3*a*e)*(f*x)^(7 + m))/(f^7*(7 + m)) + (b ^3*e*(f*x)^(9 + m))/(f^9*(9 + m))))/(a + b*x^2)
3.1.88.3.1 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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & & IGtQ[q, 0]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Leaf count of result is larger than twice the leaf count of optimal. \(494\) vs. \(2(221)=442\).
Time = 0.03 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.79
| method | result | size |
| gosper | \(\frac {x \left (b^{3} e \,m^{4} x^{8}+16 b^{3} e \,m^{3} x^{8}+3 a \,b^{2} e \,m^{4} x^{6}+b^{3} d \,m^{4} x^{6}+86 b^{3} e \,m^{2} x^{8}+54 a \,b^{2} e \,m^{3} x^{6}+18 b^{3} d \,m^{3} x^{6}+176 m \,x^{8} b^{3} e +3 a^{2} b e \,m^{4} x^{4}+3 a \,b^{2} d \,m^{4} x^{4}+312 a \,b^{2} e \,m^{2} x^{6}+104 b^{3} d \,m^{2} x^{6}+105 b^{3} e \,x^{8}+60 a^{2} b e \,m^{3} x^{4}+60 a \,b^{2} d \,m^{3} x^{4}+666 a \,b^{2} e \,x^{6} m +222 b^{3} d \,x^{6} m +a^{3} e \,m^{4} x^{2}+3 a^{2} b d \,m^{4} x^{2}+390 a^{2} b e \,m^{2} x^{4}+390 a \,b^{2} d \,m^{2} x^{4}+405 a \,b^{2} e \,x^{6}+135 b^{3} d \,x^{6}+22 a^{3} e \,m^{3} x^{2}+66 a^{2} b d \,m^{3} x^{2}+900 a^{2} b e \,x^{4} m +900 a \,b^{2} d \,x^{4} m +a^{3} d \,m^{4}+164 a^{3} e \,m^{2} x^{2}+492 a^{2} b d \,m^{2} x^{2}+567 a^{2} b e \,x^{4}+567 a \,b^{2} d \,x^{4}+24 a^{3} d \,m^{3}+458 a^{3} e \,x^{2} m +1374 a^{2} b d \,x^{2} m +206 a^{3} d \,m^{2}+315 a^{3} e \,x^{2}+945 a^{2} b d \,x^{2}+744 a^{3} d m +945 a^{3} d \right ) \left (f x \right )^{m} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right ) \left (b \,x^{2}+a \right )^{3}}\) | \(495\) |
| risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (b^{3} e \,m^{4} x^{8}+16 b^{3} e \,m^{3} x^{8}+3 a \,b^{2} e \,m^{4} x^{6}+b^{3} d \,m^{4} x^{6}+86 b^{3} e \,m^{2} x^{8}+54 a \,b^{2} e \,m^{3} x^{6}+18 b^{3} d \,m^{3} x^{6}+176 m \,x^{8} b^{3} e +3 a^{2} b e \,m^{4} x^{4}+3 a \,b^{2} d \,m^{4} x^{4}+312 a \,b^{2} e \,m^{2} x^{6}+104 b^{3} d \,m^{2} x^{6}+105 b^{3} e \,x^{8}+60 a^{2} b e \,m^{3} x^{4}+60 a \,b^{2} d \,m^{3} x^{4}+666 a \,b^{2} e \,x^{6} m +222 b^{3} d \,x^{6} m +a^{3} e \,m^{4} x^{2}+3 a^{2} b d \,m^{4} x^{2}+390 a^{2} b e \,m^{2} x^{4}+390 a \,b^{2} d \,m^{2} x^{4}+405 a \,b^{2} e \,x^{6}+135 b^{3} d \,x^{6}+22 a^{3} e \,m^{3} x^{2}+66 a^{2} b d \,m^{3} x^{2}+900 a^{2} b e \,x^{4} m +900 a \,b^{2} d \,x^{4} m +a^{3} d \,m^{4}+164 a^{3} e \,m^{2} x^{2}+492 a^{2} b d \,m^{2} x^{2}+567 a^{2} b e \,x^{4}+567 a \,b^{2} d \,x^{4}+24 a^{3} d \,m^{3}+458 a^{3} e \,x^{2} m +1374 a^{2} b d \,x^{2} m +206 a^{3} d \,m^{2}+315 a^{3} e \,x^{2}+945 a^{2} b d \,x^{2}+744 a^{3} d m +945 a^{3} d \right ) x \left (f x \right )^{m}}{\left (b \,x^{2}+a \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(495\) |
x*(b^3*e*m^4*x^8+16*b^3*e*m^3*x^8+3*a*b^2*e*m^4*x^6+b^3*d*m^4*x^6+86*b^3*e *m^2*x^8+54*a*b^2*e*m^3*x^6+18*b^3*d*m^3*x^6+176*b^3*e*m*x^8+3*a^2*b*e*m^4 *x^4+3*a*b^2*d*m^4*x^4+312*a*b^2*e*m^2*x^6+104*b^3*d*m^2*x^6+105*b^3*e*x^8 +60*a^2*b*e*m^3*x^4+60*a*b^2*d*m^3*x^4+666*a*b^2*e*m*x^6+222*b^3*d*m*x^6+a ^3*e*m^4*x^2+3*a^2*b*d*m^4*x^2+390*a^2*b*e*m^2*x^4+390*a*b^2*d*m^2*x^4+405 *a*b^2*e*x^6+135*b^3*d*x^6+22*a^3*e*m^3*x^2+66*a^2*b*d*m^3*x^2+900*a^2*b*e *m*x^4+900*a*b^2*d*m*x^4+a^3*d*m^4+164*a^3*e*m^2*x^2+492*a^2*b*d*m^2*x^2+5 67*a^2*b*e*x^4+567*a*b^2*d*x^4+24*a^3*d*m^3+458*a^3*e*m*x^2+1374*a^2*b*d*m *x^2+206*a^3*d*m^2+315*a^3*e*x^2+945*a^2*b*d*x^2+744*a^3*d*m+945*a^3*d)*(f *x)^m*((b*x^2+a)^2)^(3/2)/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)/(b*x^2+a)^3
Time = 0.26 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.38 \[ \int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {{\left ({\left (b^{3} e m^{4} + 16 \, b^{3} e m^{3} + 86 \, b^{3} e m^{2} + 176 \, b^{3} e m + 105 \, b^{3} e\right )} x^{9} + {\left ({\left (b^{3} d + 3 \, a b^{2} e\right )} m^{4} + 135 \, b^{3} d + 405 \, a b^{2} e + 18 \, {\left (b^{3} d + 3 \, a b^{2} e\right )} m^{3} + 104 \, {\left (b^{3} d + 3 \, a b^{2} e\right )} m^{2} + 222 \, {\left (b^{3} d + 3 \, a b^{2} e\right )} m\right )} x^{7} + 3 \, {\left ({\left (a b^{2} d + a^{2} b e\right )} m^{4} + 189 \, a b^{2} d + 189 \, a^{2} b e + 20 \, {\left (a b^{2} d + a^{2} b e\right )} m^{3} + 130 \, {\left (a b^{2} d + a^{2} b e\right )} m^{2} + 300 \, {\left (a b^{2} d + a^{2} b e\right )} m\right )} x^{5} + {\left ({\left (3 \, a^{2} b d + a^{3} e\right )} m^{4} + 945 \, a^{2} b d + 315 \, a^{3} e + 22 \, {\left (3 \, a^{2} b d + a^{3} e\right )} m^{3} + 164 \, {\left (3 \, a^{2} b d + a^{3} e\right )} m^{2} + 458 \, {\left (3 \, a^{2} b d + a^{3} e\right )} m\right )} x^{3} + {\left (a^{3} d m^{4} + 24 \, a^{3} d m^{3} + 206 \, a^{3} d m^{2} + 744 \, a^{3} d m + 945 \, a^{3} d\right )} x\right )} \left (f x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
((b^3*e*m^4 + 16*b^3*e*m^3 + 86*b^3*e*m^2 + 176*b^3*e*m + 105*b^3*e)*x^9 + ((b^3*d + 3*a*b^2*e)*m^4 + 135*b^3*d + 405*a*b^2*e + 18*(b^3*d + 3*a*b^2* e)*m^3 + 104*(b^3*d + 3*a*b^2*e)*m^2 + 222*(b^3*d + 3*a*b^2*e)*m)*x^7 + 3* ((a*b^2*d + a^2*b*e)*m^4 + 189*a*b^2*d + 189*a^2*b*e + 20*(a*b^2*d + a^2*b *e)*m^3 + 130*(a*b^2*d + a^2*b*e)*m^2 + 300*(a*b^2*d + a^2*b*e)*m)*x^5 + ( (3*a^2*b*d + a^3*e)*m^4 + 945*a^2*b*d + 315*a^3*e + 22*(3*a^2*b*d + a^3*e) *m^3 + 164*(3*a^2*b*d + a^3*e)*m^2 + 458*(3*a^2*b*d + a^3*e)*m)*x^3 + (a^3 *d*m^4 + 24*a^3*d*m^3 + 206*a^3*d*m^2 + 744*a^3*d*m + 945*a^3*d)*x)*(f*x)^ m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)
\[ \int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\int \left (f x\right )^{m} \left (d + e x^{2}\right ) \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.88 \[ \int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b^{3} f^{m} x^{7} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} a b^{2} f^{m} x^{5} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} a^{2} b f^{m} x^{3} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} a^{3} f^{m} x\right )} d x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} + \frac {{\left ({\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b^{3} f^{m} x^{9} + 3 \, {\left (m^{3} + 17 \, m^{2} + 87 \, m + 135\right )} a b^{2} f^{m} x^{7} + 3 \, {\left (m^{3} + 19 \, m^{2} + 111 \, m + 189\right )} a^{2} b f^{m} x^{5} + {\left (m^{3} + 21 \, m^{2} + 143 \, m + 315\right )} a^{3} f^{m} x^{3}\right )} e x^{m}}{m^{4} + 24 \, m^{3} + 206 \, m^{2} + 744 \, m + 945} \]
((m^3 + 9*m^2 + 23*m + 15)*b^3*f^m*x^7 + 3*(m^3 + 11*m^2 + 31*m + 21)*a*b^ 2*f^m*x^5 + 3*(m^3 + 13*m^2 + 47*m + 35)*a^2*b*f^m*x^3 + (m^3 + 15*m^2 + 7 1*m + 105)*a^3*f^m*x)*d*x^m/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105) + ((m^3 + 15*m^2 + 71*m + 105)*b^3*f^m*x^9 + 3*(m^3 + 17*m^2 + 87*m + 135)*a*b^2*f ^m*x^7 + 3*(m^3 + 19*m^2 + 111*m + 189)*a^2*b*f^m*x^5 + (m^3 + 21*m^2 + 14 3*m + 315)*a^3*f^m*x^3)*e*x^m/(m^4 + 24*m^3 + 206*m^2 + 744*m + 945)
Leaf count of result is larger than twice the leaf count of optimal. 993 vs. \(2 (221) = 442\).
Time = 0.30 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.60 \[ \int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\text {Too large to display} \]
((f*x)^m*b^3*e*m^4*x^9*sgn(b*x^2 + a) + 16*(f*x)^m*b^3*e*m^3*x^9*sgn(b*x^2 + a) + (f*x)^m*b^3*d*m^4*x^7*sgn(b*x^2 + a) + 3*(f*x)^m*a*b^2*e*m^4*x^7*s gn(b*x^2 + a) + 86*(f*x)^m*b^3*e*m^2*x^9*sgn(b*x^2 + a) + 18*(f*x)^m*b^3*d *m^3*x^7*sgn(b*x^2 + a) + 54*(f*x)^m*a*b^2*e*m^3*x^7*sgn(b*x^2 + a) + 176* (f*x)^m*b^3*e*m*x^9*sgn(b*x^2 + a) + 3*(f*x)^m*a*b^2*d*m^4*x^5*sgn(b*x^2 + a) + 3*(f*x)^m*a^2*b*e*m^4*x^5*sgn(b*x^2 + a) + 104*(f*x)^m*b^3*d*m^2*x^7 *sgn(b*x^2 + a) + 312*(f*x)^m*a*b^2*e*m^2*x^7*sgn(b*x^2 + a) + 105*(f*x)^m *b^3*e*x^9*sgn(b*x^2 + a) + 60*(f*x)^m*a*b^2*d*m^3*x^5*sgn(b*x^2 + a) + 60 *(f*x)^m*a^2*b*e*m^3*x^5*sgn(b*x^2 + a) + 222*(f*x)^m*b^3*d*m*x^7*sgn(b*x^ 2 + a) + 666*(f*x)^m*a*b^2*e*m*x^7*sgn(b*x^2 + a) + 3*(f*x)^m*a^2*b*d*m^4* x^3*sgn(b*x^2 + a) + (f*x)^m*a^3*e*m^4*x^3*sgn(b*x^2 + a) + 390*(f*x)^m*a* b^2*d*m^2*x^5*sgn(b*x^2 + a) + 390*(f*x)^m*a^2*b*e*m^2*x^5*sgn(b*x^2 + a) + 135*(f*x)^m*b^3*d*x^7*sgn(b*x^2 + a) + 405*(f*x)^m*a*b^2*e*x^7*sgn(b*x^2 + a) + 66*(f*x)^m*a^2*b*d*m^3*x^3*sgn(b*x^2 + a) + 22*(f*x)^m*a^3*e*m^3*x ^3*sgn(b*x^2 + a) + 900*(f*x)^m*a*b^2*d*m*x^5*sgn(b*x^2 + a) + 900*(f*x)^m *a^2*b*e*m*x^5*sgn(b*x^2 + a) + (f*x)^m*a^3*d*m^4*x*sgn(b*x^2 + a) + 492*( f*x)^m*a^2*b*d*m^2*x^3*sgn(b*x^2 + a) + 164*(f*x)^m*a^3*e*m^2*x^3*sgn(b*x^ 2 + a) + 567*(f*x)^m*a*b^2*d*x^5*sgn(b*x^2 + a) + 567*(f*x)^m*a^2*b*e*x^5* sgn(b*x^2 + a) + 24*(f*x)^m*a^3*d*m^3*x*sgn(b*x^2 + a) + 1374*(f*x)^m*a^2* b*d*m*x^3*sgn(b*x^2 + a) + 458*(f*x)^m*a^3*e*m*x^3*sgn(b*x^2 + a) + 206...
Timed out. \[ \int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx=\int {\left (f\,x\right )}^m\,\left (e\,x^2+d\right )\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \]