3.88 \(\int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=276 \[ \frac{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (3 a e+b d)}{f^7 (m+7) \left (a+b x^2\right )}+\frac{3 a b \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+b d)}{f^5 (m+5) \left (a+b x^2\right )}+\frac{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+3 b d)}{f^3 (m+3) \left (a+b x^2\right )}+\frac{b^3 e \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+9}}{f^9 (m+9) \left (a+b x^2\right )}+\frac{a^3 d \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left (a+b x^2\right )} \]

[Out]

(a^3*d*(f*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f*(1 + m)*(a + b*x^2)) +
(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)*(
a + b*x^2)) + (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)*(a + b*x^2)) + (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)*(a + b*x^2)) + (b^3*e*(f*x)^(9 + m)*Sqrt[a^2 + 2*a*b*
x^2 + b^2*x^4])/(f^9*(9 + m)*(a + b*x^2))

_______________________________________________________________________________________

Rubi [A]  time = 0.398662, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (3 a e+b d)}{f^7 (m+7) \left (a+b x^2\right )}+\frac{3 a b \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+b d)}{f^5 (m+5) \left (a+b x^2\right )}+\frac{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+3 b d)}{f^3 (m+3) \left (a+b x^2\right )}+\frac{b^3 e \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+9}}{f^9 (m+9) \left (a+b x^2\right )}+\frac{a^3 d \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(f*x)^m*(d + e*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(a^3*d*(f*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f*(1 + m)*(a + b*x^2)) +
(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)*(
a + b*x^2)) + (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)*(a + b*x^2)) + (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)*(a + b*x^2)) + (b^3*e*(f*x)^(9 + m)*Sqrt[a^2 + 2*a*b*
x^2 + b^2*x^4])/(f^9*(9 + m)*(a + b*x^2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 61.7697, size = 255, normalized size = 0.92 \[ \frac{a^{3} d \left (f x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f \left (a + b x^{2}\right ) \left (m + 1\right )} + \frac{a^{2} \left (f x\right )^{m + 3} \left (a e + 3 b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{3} \left (a + b x^{2}\right ) \left (m + 3\right )} + \frac{3 a b \left (f x\right )^{m + 5} \left (a e + b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{5} \left (a + b x^{2}\right ) \left (m + 5\right )} + \frac{b^{3} e \left (f x\right )^{m + 9} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{9} \left (a + b x^{2}\right ) \left (m + 9\right )} + \frac{b^{2} \left (f x\right )^{m + 7} \left (3 a e + b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{7} \left (a + b x^{2}\right ) \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x)**m*(e*x**2+d)*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

a**3*d*(f*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(f*(a + b*x**2)*(m + 1
)) + a**2*(f*x)**(m + 3)*(a*e + 3*b*d)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(f**3
*(a + b*x**2)*(m + 3)) + 3*a*b*(f*x)**(m + 5)*(a*e + b*d)*sqrt(a**2 + 2*a*b*x**2
+ b**2*x**4)/(f**5*(a + b*x**2)*(m + 5)) + b**3*e*(f*x)**(m + 9)*sqrt(a**2 + 2*
a*b*x**2 + b**2*x**4)/(f**9*(a + b*x**2)*(m + 9)) + b**2*(f*x)**(m + 7)*(3*a*e +
b*d)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(f**7*(a + b*x**2)*(m + 7))

_______________________________________________________________________________________

Mathematica [A]  time = 0.179688, size = 112, normalized size = 0.41 \[ \frac{\left (\left (a+b x^2\right )^2\right )^{3/2} (f x)^m \left (\frac{a^3 d x}{m+1}+\frac{a^2 x^3 (a e+3 b d)}{m+3}+\frac{b^2 x^7 (3 a e+b d)}{m+7}+\frac{3 a b x^5 (a e+b d)}{m+5}+\frac{b^3 e x^9}{m+9}\right )}{\left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(f*x)^m*(d + e*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

((f*x)^m*((a + b*x^2)^2)^(3/2)*((a^3*d*x)/(1 + m) + (a^2*(3*b*d + a*e)*x^3)/(3 +
m) + (3*a*b*(b*d + a*e)*x^5)/(5 + m) + (b^2*(b*d + 3*a*e)*x^7)/(7 + m) + (b^3*e
*x^9)/(9 + m)))/(a + b*x^2)^3

_______________________________________________________________________________________

Maple [B]  time = 0.012, size = 495, normalized size = 1.8 \[{\frac{ \left ({b}^{3}e{m}^{4}{x}^{8}+16\,{b}^{3}e{m}^{3}{x}^{8}+3\,a{b}^{2}e{m}^{4}{x}^{6}+{b}^{3}d{m}^{4}{x}^{6}+86\,{b}^{3}e{m}^{2}{x}^{8}+54\,a{b}^{2}e{m}^{3}{x}^{6}+18\,{b}^{3}d{m}^{3}{x}^{6}+176\,{b}^{3}em{x}^{8}+3\,{a}^{2}be{m}^{4}{x}^{4}+3\,a{b}^{2}d{m}^{4}{x}^{4}+312\,a{b}^{2}e{m}^{2}{x}^{6}+104\,{b}^{3}d{m}^{2}{x}^{6}+105\,{b}^{3}e{x}^{8}+60\,{a}^{2}be{m}^{3}{x}^{4}+60\,a{b}^{2}d{m}^{3}{x}^{4}+666\,a{b}^{2}em{x}^{6}+222\,{b}^{3}dm{x}^{6}+{a}^{3}e{m}^{4}{x}^{2}+3\,{a}^{2}bd{m}^{4}{x}^{2}+390\,{a}^{2}be{m}^{2}{x}^{4}+390\,a{b}^{2}d{m}^{2}{x}^{4}+405\,a{b}^{2}e{x}^{6}+135\,{b}^{3}d{x}^{6}+22\,{a}^{3}e{m}^{3}{x}^{2}+66\,{a}^{2}bd{m}^{3}{x}^{2}+900\,{a}^{2}bem{x}^{4}+900\,a{b}^{2}dm{x}^{4}+{a}^{3}d{m}^{4}+164\,{a}^{3}e{m}^{2}{x}^{2}+492\,{a}^{2}bd{m}^{2}{x}^{2}+567\,{x}^{4}{a}^{2}be+567\,{x}^{4}a{b}^{2}d+24\,{a}^{3}d{m}^{3}+458\,{a}^{3}em{x}^{2}+1374\,{a}^{2}bdm{x}^{2}+206\,{a}^{3}d{m}^{2}+315\,{x}^{2}{a}^{3}e+945\,{x}^{2}{a}^{2}bd+744\,{a}^{3}dm+945\,{a}^{3}d \right ) x \left ( fx \right ) ^{m}}{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x)^m*(e*x^2+d)*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

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+1
64*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*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

_______________________________________________________________________________________

Maxima [A]  time = 0.713054, size = 328, normalized size = 1.19 \[ \frac{{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b^{3} f^{m} x^{7} + 3 \,{\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} a b^{2} f^{m} x^{5} + 3 \,{\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} a^{2} b f^{m} x^{3} +{\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} a^{3} f^{m} x\right )} d x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} + \frac{{\left ({\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} b^{3} f^{m} x^{9} + 3 \,{\left (m^{3} + 17 \, m^{2} + 87 \, m + 135\right )} a b^{2} f^{m} x^{7} + 3 \,{\left (m^{3} + 19 \, m^{2} + 111 \, m + 189\right )} a^{2} b f^{m} x^{5} +{\left (m^{3} + 21 \, m^{2} + 143 \, m + 315\right )} a^{3} f^{m} x^{3}\right )} e x^{m}}{m^{4} + 24 \, m^{3} + 206 \, m^{2} + 744 \, m + 945} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*(e*x^2 + d)*(f*x)^m,x, algorithm="maxima")

[Out]

((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 + 71*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 + 143*m + 315)*a^3*f^m*x^3)*e*x^m/
(m^4 + 24*m^3 + 206*m^2 + 744*m + 945)

_______________________________________________________________________________________

Fricas [A]  time = 0.284129, size = 514, normalized size = 1.86 \[ \frac{{\left ({\left (b^{3} e m^{4} + 16 \, b^{3} e m^{3} + 86 \, b^{3} e m^{2} + 176 \, b^{3} e m + 105 \, b^{3} e\right )} x^{9} +{\left ({\left (b^{3} d + 3 \, a b^{2} e\right )} m^{4} + 135 \, b^{3} d + 405 \, a b^{2} e + 18 \,{\left (b^{3} d + 3 \, a b^{2} e\right )} m^{3} + 104 \,{\left (b^{3} d + 3 \, a b^{2} e\right )} m^{2} + 222 \,{\left (b^{3} d + 3 \, a b^{2} e\right )} m\right )} x^{7} + 3 \,{\left ({\left (a b^{2} d + a^{2} b e\right )} m^{4} + 189 \, a b^{2} d + 189 \, a^{2} b e + 20 \,{\left (a b^{2} d + a^{2} b e\right )} m^{3} + 130 \,{\left (a b^{2} d + a^{2} b e\right )} m^{2} + 300 \,{\left (a b^{2} d + a^{2} b e\right )} m\right )} x^{5} +{\left ({\left (3 \, a^{2} b d + a^{3} e\right )} m^{4} + 945 \, a^{2} b d + 315 \, a^{3} e + 22 \,{\left (3 \, a^{2} b d + a^{3} e\right )} m^{3} + 164 \,{\left (3 \, a^{2} b d + a^{3} e\right )} m^{2} + 458 \,{\left (3 \, a^{2} b d + a^{3} e\right )} m\right )} x^{3} +{\left (a^{3} d m^{4} + 24 \, a^{3} d m^{3} + 206 \, a^{3} d m^{2} + 744 \, a^{3} d m + 945 \, a^{3} d\right )} x\right )} \left (f x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*(e*x^2 + d)*(f*x)^m,x, algorithm="fricas")

[Out]

((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 + 74
4*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 + 9
45)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x)**m*(e*x**2+d)*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.284723, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*(e*x^2 + d)*(f*x)^m,x, algorithm="giac")

[Out]

Done