3.87 \(\int (f x)^m \left (d+e x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx\)

Optimal. Leaf size=400 \[ \frac{10 a^2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (a e+b d)}{f^7 (m+7) \left (a+b x^2\right )}+\frac{b^5 e \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+13}}{f^{13} (m+13) \left (a+b x^2\right )}+\frac{b^4 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+11} (5 a e+b d)}{f^{11} (m+11) \left (a+b x^2\right )}+\frac{5 a b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+9} (2 a e+b d)}{f^9 (m+9) \left (a+b x^2\right )}+\frac{a^5 d \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left (a+b x^2\right )}+\frac{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+5 b d)}{f^3 (m+3) \left (a+b x^2\right )}+\frac{5 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+2 b d)}{f^5 (m+5) \left (a+b x^2\right )} \]

[Out]

(a^5*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^4*(5*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)) + (5*a^3*b*(2*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)) + (10*a^2*b^2*(b*d + 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)) + (5*a*b^3*(b*d + 2*a*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)) + (b^4*(b*d +
5*a*e)*(f*x)^(11 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f^11*(11 + m)*(a + b*x^
2)) + (b^5*e*(f*x)^(13 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f^13*(13 + m)*(a +
b*x^2))

_______________________________________________________________________________________

Rubi [A]  time = 0.587138, antiderivative size = 400, 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 \[ \frac{10 a^2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (a e+b d)}{f^7 (m+7) \left (a+b x^2\right )}+\frac{b^5 e \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+13}}{f^{13} (m+13) \left (a+b x^2\right )}+\frac{b^4 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+11} (5 a e+b d)}{f^{11} (m+11) \left (a+b x^2\right )}+\frac{5 a b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+9} (2 a e+b d)}{f^9 (m+9) \left (a+b x^2\right )}+\frac{a^5 d \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left (a+b x^2\right )}+\frac{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+5 b d)}{f^3 (m+3) \left (a+b x^2\right )}+\frac{5 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+2 b d)}{f^5 (m+5) \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(f*x)^m*(d + e*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(a^5*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^4*(5*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)) + (5*a^3*b*(2*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)) + (10*a^2*b^2*(b*d + 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)) + (5*a*b^3*(b*d + 2*a*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)) + (b^4*(b*d +
5*a*e)*(f*x)^(11 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f^11*(11 + m)*(a + b*x^
2)) + (b^5*e*(f*x)^(13 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f^13*(13 + m)*(a +
b*x^2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 85.5961, size = 374, normalized size = 0.94 \[ \frac{a^{5} d \left (f x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f \left (a + b x^{2}\right ) \left (m + 1\right )} + \frac{a^{4} \left (f x\right )^{m + 3} \left (a e + 5 b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{3} \left (a + b x^{2}\right ) \left (m + 3\right )} + \frac{5 a^{3} b \left (f x\right )^{m + 5} \left (a e + 2 b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{5} \left (a + b x^{2}\right ) \left (m + 5\right )} + \frac{10 a^{2} b^{2} \left (f x\right )^{m + 7} \left (a e + b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{7} \left (a + b x^{2}\right ) \left (m + 7\right )} + \frac{5 a b^{3} \left (f x\right )^{m + 9} \left (2 a e + b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{9} \left (a + b x^{2}\right ) \left (m + 9\right )} + \frac{b^{5} e \left (f x\right )^{m + 13} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{13} \left (a + b x^{2}\right ) \left (m + 13\right )} + \frac{b^{4} \left (f x\right )^{m + 11} \left (5 a e + b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{f^{11} \left (a + b x^{2}\right ) \left (m + 11\right )} \]

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)**(5/2),x)

[Out]

a**5*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**4*(f*x)**(m + 3)*(a*e + 5*b*d)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(f**3
*(a + b*x**2)*(m + 3)) + 5*a**3*b*(f*x)**(m + 5)*(a*e + 2*b*d)*sqrt(a**2 + 2*a*b
*x**2 + b**2*x**4)/(f**5*(a + b*x**2)*(m + 5)) + 10*a**2*b**2*(f*x)**(m + 7)*(a*
e + b*d)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(f**7*(a + b*x**2)*(m + 7)) + 5*a*b
**3*(f*x)**(m + 9)*(2*a*e + b*d)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(f**9*(a +
b*x**2)*(m + 9)) + b**5*e*(f*x)**(m + 13)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(f
**13*(a + b*x**2)*(m + 13)) + b**4*(f*x)**(m + 11)*(5*a*e + b*d)*sqrt(a**2 + 2*a
*b*x**2 + b**2*x**4)/(f**11*(a + b*x**2)*(m + 11))

_______________________________________________________________________________________

Mathematica [A]  time = 0.302714, size = 160, normalized size = 0.4 \[ \frac{\sqrt{\left (a+b x^2\right )^2} (f x)^m \left (\frac{a^5 d x}{m+1}+\frac{a^4 x^3 (a e+5 b d)}{m+3}+\frac{5 a^3 b x^5 (a e+2 b d)}{m+5}+\frac{10 a^2 b^2 x^7 (a e+b d)}{m+7}+\frac{b^4 x^{11} (5 a e+b d)}{m+11}+\frac{5 a b^3 x^9 (2 a e+b d)}{m+9}+\frac{b^5 e x^{13}}{m+13}\right )}{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(f*x)^m*(d + e*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

((f*x)^m*Sqrt[(a + b*x^2)^2]*((a^5*d*x)/(1 + m) + (a^4*(5*b*d + a*e)*x^3)/(3 + m
) + (5*a^3*b*(2*b*d + a*e)*x^5)/(5 + m) + (10*a^2*b^2*(b*d + a*e)*x^7)/(7 + m) +
(5*a*b^3*(b*d + 2*a*e)*x^9)/(9 + m) + (b^4*(b*d + 5*a*e)*x^11)/(11 + m) + (b^5*
e*x^13)/(13 + m)))/(a + b*x^2)

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 1099, normalized size = 2.8 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

x*(b^5*e*m^6*x^12+36*b^5*e*m^5*x^12+5*a*b^4*e*m^6*x^10+b^5*d*m^6*x^10+505*b^5*e*
m^4*x^12+190*a*b^4*e*m^5*x^10+38*b^5*d*m^5*x^10+3480*b^5*e*m^3*x^12+10*a^2*b^3*e
*m^6*x^8+5*a*b^4*d*m^6*x^8+2775*a*b^4*e*m^4*x^10+555*b^5*d*m^4*x^10+12139*b^5*e*
m^2*x^12+400*a^2*b^3*e*m^5*x^8+200*a*b^4*d*m^5*x^8+19700*a*b^4*e*m^3*x^10+3940*b
^5*d*m^3*x^10+19524*b^5*e*m*x^12+10*a^3*b^2*e*m^6*x^6+10*a^2*b^3*d*m^6*x^6+6130*
a^2*b^3*e*m^4*x^8+3065*a*b^4*d*m^4*x^8+70195*a*b^4*e*m^2*x^10+14039*b^5*d*m^2*x^
10+10395*b^5*e*x^12+420*a^3*b^2*e*m^5*x^6+420*a^2*b^3*d*m^5*x^6+45280*a^2*b^3*e*
m^3*x^8+22640*a*b^4*d*m^3*x^8+114510*a*b^4*e*m*x^10+22902*b^5*d*m*x^10+5*a^4*b*e
*m^6*x^4+10*a^3*b^2*d*m^6*x^4+6790*a^3*b^2*e*m^4*x^6+6790*a^2*b^3*d*m^4*x^6+1662
70*a^2*b^3*e*m^2*x^8+83135*a*b^4*d*m^2*x^8+61425*a*b^4*e*x^10+12285*b^5*d*x^10+2
20*a^4*b*e*m^5*x^4+440*a^3*b^2*d*m^5*x^4+52920*a^3*b^2*e*m^3*x^6+52920*a^2*b^3*d
*m^3*x^6+276880*a^2*b^3*e*m*x^8+138440*a*b^4*d*m*x^8+a^5*e*m^6*x^2+5*a^4*b*d*m^6
*x^2+3765*a^4*b*e*m^4*x^4+7530*a^3*b^2*d*m^4*x^4+203350*a^3*b^2*e*m^2*x^6+203350
*a^2*b^3*d*m^2*x^6+150150*a^2*b^3*e*x^8+75075*a*b^4*d*x^8+46*a^5*e*m^5*x^2+230*a
^4*b*d*m^5*x^2+31400*a^4*b*e*m^3*x^4+62800*a^3*b^2*d*m^3*x^4+349860*a^3*b^2*e*m*
x^6+349860*a^2*b^3*d*m*x^6+a^5*d*m^6+835*a^5*e*m^4*x^2+4175*a^4*b*d*m^4*x^2+1298
95*a^4*b*e*m^2*x^4+259790*a^3*b^2*d*m^2*x^4+193050*a^3*b^2*e*x^6+193050*a^2*b^3*
d*x^6+48*a^5*d*m^5+7540*a^5*e*m^3*x^2+37700*a^4*b*d*m^3*x^2+237180*a^4*b*e*m*x^4
+474360*a^3*b^2*d*m*x^4+925*a^5*d*m^4+34759*a^5*e*m^2*x^2+173795*a^4*b*d*m^2*x^2
+135135*a^4*b*e*x^4+270270*a^3*b^2*d*x^4+9120*a^5*d*m^3+73054*a^5*e*m*x^2+365270
*a^4*b*d*m*x^2+48259*a^5*d*m^2+45045*a^5*e*x^2+225225*a^4*b*d*x^2+129072*a^5*d*m
+135135*a^5*d)*(f*x)^m*((b*x^2+a)^2)^(5/2)/(13+m)/(11+m)/(9+m)/(7+m)/(5+m)/(3+m)
/(1+m)/(b*x^2+a)^5

_______________________________________________________________________________________

Maxima [A]  time = 0.736163, size = 663, normalized size = 1.66 \[ \frac{{\left ({\left (m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945\right )} b^{5} f^{m} x^{11} + 5 \,{\left (m^{5} + 27 \, m^{4} + 262 \, m^{3} + 1122 \, m^{2} + 2041 \, m + 1155\right )} a b^{4} f^{m} x^{9} + 10 \,{\left (m^{5} + 29 \, m^{4} + 302 \, m^{3} + 1366 \, m^{2} + 2577 \, m + 1485\right )} a^{2} b^{3} f^{m} x^{7} + 10 \,{\left (m^{5} + 31 \, m^{4} + 350 \, m^{3} + 1730 \, m^{2} + 3489 \, m + 2079\right )} a^{3} b^{2} f^{m} x^{5} + 5 \,{\left (m^{5} + 33 \, m^{4} + 406 \, m^{3} + 2262 \, m^{2} + 5353 \, m + 3465\right )} a^{4} b f^{m} x^{3} +{\left (m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395\right )} a^{5} f^{m} x\right )} d x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} + \frac{{\left ({\left (m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395\right )} b^{5} f^{m} x^{13} + 5 \,{\left (m^{5} + 37 \, m^{4} + 518 \, m^{3} + 3422 \, m^{2} + 10617 \, m + 12285\right )} a b^{4} f^{m} x^{11} + 10 \,{\left (m^{5} + 39 \, m^{4} + 574 \, m^{3} + 3954 \, m^{2} + 12673 \, m + 15015\right )} a^{2} b^{3} f^{m} x^{9} + 10 \,{\left (m^{5} + 41 \, m^{4} + 638 \, m^{3} + 4654 \, m^{2} + 15681 \, m + 19305\right )} a^{3} b^{2} f^{m} x^{7} + 5 \,{\left (m^{5} + 43 \, m^{4} + 710 \, m^{3} + 5570 \, m^{2} + 20409 \, m + 27027\right )} a^{4} b f^{m} x^{5} +{\left (m^{5} + 45 \, m^{4} + 790 \, m^{3} + 6750 \, m^{2} + 28009 \, m + 45045\right )} a^{5} f^{m} x^{3}\right )} e x^{m}}{m^{6} + 48 \, m^{5} + 925 \, m^{4} + 9120 \, m^{3} + 48259 \, m^{2} + 129072 \, m + 135135} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(e*x^2 + d)*(f*x)^m,x, algorithm="maxima")

[Out]

((m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)*b^5*f^m*x^11 + 5*(m^5 + 27*m^
4 + 262*m^3 + 1122*m^2 + 2041*m + 1155)*a*b^4*f^m*x^9 + 10*(m^5 + 29*m^4 + 302*m
^3 + 1366*m^2 + 2577*m + 1485)*a^2*b^3*f^m*x^7 + 10*(m^5 + 31*m^4 + 350*m^3 + 17
30*m^2 + 3489*m + 2079)*a^3*b^2*f^m*x^5 + 5*(m^5 + 33*m^4 + 406*m^3 + 2262*m^2 +
5353*m + 3465)*a^4*b*f^m*x^3 + (m^5 + 35*m^4 + 470*m^3 + 3010*m^2 + 9129*m + 10
395)*a^5*f^m*x)*d*x^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m +
10395) + ((m^5 + 35*m^4 + 470*m^3 + 3010*m^2 + 9129*m + 10395)*b^5*f^m*x^13 + 5
*(m^5 + 37*m^4 + 518*m^3 + 3422*m^2 + 10617*m + 12285)*a*b^4*f^m*x^11 + 10*(m^5
+ 39*m^4 + 574*m^3 + 3954*m^2 + 12673*m + 15015)*a^2*b^3*f^m*x^9 + 10*(m^5 + 41*
m^4 + 638*m^3 + 4654*m^2 + 15681*m + 19305)*a^3*b^2*f^m*x^7 + 5*(m^5 + 43*m^4 +
710*m^3 + 5570*m^2 + 20409*m + 27027)*a^4*b*f^m*x^5 + (m^5 + 45*m^4 + 790*m^3 +
6750*m^2 + 28009*m + 45045)*a^5*f^m*x^3)*e*x^m/(m^6 + 48*m^5 + 925*m^4 + 9120*m^
3 + 48259*m^2 + 129072*m + 135135)

_______________________________________________________________________________________

Fricas [A]  time = 0.287381, size = 1152, normalized size = 2.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(e*x^2 + d)*(f*x)^m,x, algorithm="fricas")

[Out]

((b^5*e*m^6 + 36*b^5*e*m^5 + 505*b^5*e*m^4 + 3480*b^5*e*m^3 + 12139*b^5*e*m^2 +
19524*b^5*e*m + 10395*b^5*e)*x^13 + ((b^5*d + 5*a*b^4*e)*m^6 + 12285*b^5*d + 614
25*a*b^4*e + 38*(b^5*d + 5*a*b^4*e)*m^5 + 555*(b^5*d + 5*a*b^4*e)*m^4 + 3940*(b^
5*d + 5*a*b^4*e)*m^3 + 14039*(b^5*d + 5*a*b^4*e)*m^2 + 22902*(b^5*d + 5*a*b^4*e)
*m)*x^11 + 5*((a*b^4*d + 2*a^2*b^3*e)*m^6 + 15015*a*b^4*d + 30030*a^2*b^3*e + 40
*(a*b^4*d + 2*a^2*b^3*e)*m^5 + 613*(a*b^4*d + 2*a^2*b^3*e)*m^4 + 4528*(a*b^4*d +
2*a^2*b^3*e)*m^3 + 16627*(a*b^4*d + 2*a^2*b^3*e)*m^2 + 27688*(a*b^4*d + 2*a^2*b
^3*e)*m)*x^9 + 10*((a^2*b^3*d + a^3*b^2*e)*m^6 + 19305*a^2*b^3*d + 19305*a^3*b^2
*e + 42*(a^2*b^3*d + a^3*b^2*e)*m^5 + 679*(a^2*b^3*d + a^3*b^2*e)*m^4 + 5292*(a^
2*b^3*d + a^3*b^2*e)*m^3 + 20335*(a^2*b^3*d + a^3*b^2*e)*m^2 + 34986*(a^2*b^3*d
+ a^3*b^2*e)*m)*x^7 + 5*((2*a^3*b^2*d + a^4*b*e)*m^6 + 54054*a^3*b^2*d + 27027*a
^4*b*e + 44*(2*a^3*b^2*d + a^4*b*e)*m^5 + 753*(2*a^3*b^2*d + a^4*b*e)*m^4 + 6280
*(2*a^3*b^2*d + a^4*b*e)*m^3 + 25979*(2*a^3*b^2*d + a^4*b*e)*m^2 + 47436*(2*a^3*
b^2*d + a^4*b*e)*m)*x^5 + ((5*a^4*b*d + a^5*e)*m^6 + 225225*a^4*b*d + 45045*a^5*
e + 46*(5*a^4*b*d + a^5*e)*m^5 + 835*(5*a^4*b*d + a^5*e)*m^4 + 7540*(5*a^4*b*d +
a^5*e)*m^3 + 34759*(5*a^4*b*d + a^5*e)*m^2 + 73054*(5*a^4*b*d + a^5*e)*m)*x^3 +
(a^5*d*m^6 + 48*a^5*d*m^5 + 925*a^5*d*m^4 + 9120*a^5*d*m^3 + 48259*a^5*d*m^2 +
129072*a^5*d*m + 135135*a^5*d)*x)*(f*x)^m/(m^7 + 49*m^6 + 973*m^5 + 10045*m^4 +
57379*m^3 + 177331*m^2 + 264207*m + 135135)

_______________________________________________________________________________________

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)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.308387, 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)^(5/2)*(e*x^2 + d)*(f*x)^m,x, algorithm="giac")

[Out]

Done