3.339 \(\int x^m \left (a+b x^2\right )^5 \, dx\)

Optimal. Leaf size=97 \[ \frac{a^5 x^{m+1}}{m+1}+\frac{5 a^4 b x^{m+3}}{m+3}+\frac{10 a^3 b^2 x^{m+5}}{m+5}+\frac{10 a^2 b^3 x^{m+7}}{m+7}+\frac{5 a b^4 x^{m+9}}{m+9}+\frac{b^5 x^{m+11}}{m+11} \]

[Out]

(a^5*x^(1 + m))/(1 + m) + (5*a^4*b*x^(3 + m))/(3 + m) + (10*a^3*b^2*x^(5 + m))/(
5 + m) + (10*a^2*b^3*x^(7 + m))/(7 + m) + (5*a*b^4*x^(9 + m))/(9 + m) + (b^5*x^(
11 + m))/(11 + m)

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Rubi [A]  time = 0.11396, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^5 x^{m+1}}{m+1}+\frac{5 a^4 b x^{m+3}}{m+3}+\frac{10 a^3 b^2 x^{m+5}}{m+5}+\frac{10 a^2 b^3 x^{m+7}}{m+7}+\frac{5 a b^4 x^{m+9}}{m+9}+\frac{b^5 x^{m+11}}{m+11} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^2)^5,x]

[Out]

(a^5*x^(1 + m))/(1 + m) + (5*a^4*b*x^(3 + m))/(3 + m) + (10*a^3*b^2*x^(5 + m))/(
5 + m) + (10*a^2*b^3*x^(7 + m))/(7 + m) + (5*a*b^4*x^(9 + m))/(9 + m) + (b^5*x^(
11 + m))/(11 + m)

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Rubi in Sympy [A]  time = 17.008, size = 87, normalized size = 0.9 \[ \frac{a^{5} x^{m + 1}}{m + 1} + \frac{5 a^{4} b x^{m + 3}}{m + 3} + \frac{10 a^{3} b^{2} x^{m + 5}}{m + 5} + \frac{10 a^{2} b^{3} x^{m + 7}}{m + 7} + \frac{5 a b^{4} x^{m + 9}}{m + 9} + \frac{b^{5} x^{m + 11}}{m + 11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**2+a)**5,x)

[Out]

a**5*x**(m + 1)/(m + 1) + 5*a**4*b*x**(m + 3)/(m + 3) + 10*a**3*b**2*x**(m + 5)/
(m + 5) + 10*a**2*b**3*x**(m + 7)/(m + 7) + 5*a*b**4*x**(m + 9)/(m + 9) + b**5*x
**(m + 11)/(m + 11)

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Mathematica [A]  time = 0.0561545, size = 87, normalized size = 0.9 \[ x^m \left (\frac{a^5 x}{m+1}+\frac{5 a^4 b x^3}{m+3}+\frac{10 a^3 b^2 x^5}{m+5}+\frac{10 a^2 b^3 x^7}{m+7}+\frac{5 a b^4 x^9}{m+9}+\frac{b^5 x^{11}}{m+11}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^2)^5,x]

[Out]

x^m*((a^5*x)/(1 + m) + (5*a^4*b*x^3)/(3 + m) + (10*a^3*b^2*x^5)/(5 + m) + (10*a^
2*b^3*x^7)/(7 + m) + (5*a*b^4*x^9)/(9 + m) + (b^5*x^11)/(11 + m))

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Maple [B]  time = 0.01, size = 432, normalized size = 4.5 \[{\frac{{x}^{1+m} \left ({b}^{5}{m}^{5}{x}^{10}+25\,{b}^{5}{m}^{4}{x}^{10}+5\,a{b}^{4}{m}^{5}{x}^{8}+230\,{b}^{5}{m}^{3}{x}^{10}+135\,a{b}^{4}{m}^{4}{x}^{8}+950\,{b}^{5}{m}^{2}{x}^{10}+10\,{a}^{2}{b}^{3}{m}^{5}{x}^{6}+1310\,a{b}^{4}{m}^{3}{x}^{8}+1689\,{b}^{5}m{x}^{10}+290\,{a}^{2}{b}^{3}{m}^{4}{x}^{6}+5610\,a{b}^{4}{m}^{2}{x}^{8}+945\,{b}^{5}{x}^{10}+10\,{a}^{3}{b}^{2}{m}^{5}{x}^{4}+3020\,{a}^{2}{b}^{3}{m}^{3}{x}^{6}+10205\,a{b}^{4}m{x}^{8}+310\,{a}^{3}{b}^{2}{m}^{4}{x}^{4}+13660\,{a}^{2}{b}^{3}{m}^{2}{x}^{6}+5775\,a{b}^{4}{x}^{8}+5\,{a}^{4}b{m}^{5}{x}^{2}+3500\,{a}^{3}{b}^{2}{m}^{3}{x}^{4}+25770\,{a}^{2}{b}^{3}m{x}^{6}+165\,{a}^{4}b{m}^{4}{x}^{2}+17300\,{a}^{3}{b}^{2}{m}^{2}{x}^{4}+14850\,{a}^{2}{b}^{3}{x}^{6}+{a}^{5}{m}^{5}+2030\,{a}^{4}b{m}^{3}{x}^{2}+34890\,{a}^{3}{b}^{2}m{x}^{4}+35\,{a}^{5}{m}^{4}+11310\,{a}^{4}b{m}^{2}{x}^{2}+20790\,{a}^{3}{b}^{2}{x}^{4}+470\,{a}^{5}{m}^{3}+26765\,{a}^{4}bm{x}^{2}+3010\,{a}^{5}{m}^{2}+17325\,{a}^{4}b{x}^{2}+9129\,{a}^{5}m+10395\,{a}^{5} \right ) }{ \left ( 11+m \right ) \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^2+a)^5,x)

[Out]

x^(1+m)*(b^5*m^5*x^10+25*b^5*m^4*x^10+5*a*b^4*m^5*x^8+230*b^5*m^3*x^10+135*a*b^4
*m^4*x^8+950*b^5*m^2*x^10+10*a^2*b^3*m^5*x^6+1310*a*b^4*m^3*x^8+1689*b^5*m*x^10+
290*a^2*b^3*m^4*x^6+5610*a*b^4*m^2*x^8+945*b^5*x^10+10*a^3*b^2*m^5*x^4+3020*a^2*
b^3*m^3*x^6+10205*a*b^4*m*x^8+310*a^3*b^2*m^4*x^4+13660*a^2*b^3*m^2*x^6+5775*a*b
^4*x^8+5*a^4*b*m^5*x^2+3500*a^3*b^2*m^3*x^4+25770*a^2*b^3*m*x^6+165*a^4*b*m^4*x^
2+17300*a^3*b^2*m^2*x^4+14850*a^2*b^3*x^6+a^5*m^5+2030*a^4*b*m^3*x^2+34890*a^3*b
^2*m*x^4+35*a^5*m^4+11310*a^4*b*m^2*x^2+20790*a^3*b^2*x^4+470*a^5*m^3+26765*a^4*
b*m*x^2+3010*a^5*m^2+17325*a^4*b*x^2+9129*a^5*m+10395*a^5)/(11+m)/(9+m)/(7+m)/(5
+m)/(3+m)/(1+m)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249803, size = 495, normalized size = 5.1 \[ \frac{{\left ({\left (b^{5} m^{5} + 25 \, b^{5} m^{4} + 230 \, b^{5} m^{3} + 950 \, b^{5} m^{2} + 1689 \, b^{5} m + 945 \, b^{5}\right )} x^{11} + 5 \,{\left (a b^{4} m^{5} + 27 \, a b^{4} m^{4} + 262 \, a b^{4} m^{3} + 1122 \, a b^{4} m^{2} + 2041 \, a b^{4} m + 1155 \, a b^{4}\right )} x^{9} + 10 \,{\left (a^{2} b^{3} m^{5} + 29 \, a^{2} b^{3} m^{4} + 302 \, a^{2} b^{3} m^{3} + 1366 \, a^{2} b^{3} m^{2} + 2577 \, a^{2} b^{3} m + 1485 \, a^{2} b^{3}\right )} x^{7} + 10 \,{\left (a^{3} b^{2} m^{5} + 31 \, a^{3} b^{2} m^{4} + 350 \, a^{3} b^{2} m^{3} + 1730 \, a^{3} b^{2} m^{2} + 3489 \, a^{3} b^{2} m + 2079 \, a^{3} b^{2}\right )} x^{5} + 5 \,{\left (a^{4} b m^{5} + 33 \, a^{4} b m^{4} + 406 \, a^{4} b m^{3} + 2262 \, a^{4} b m^{2} + 5353 \, a^{4} b m + 3465 \, a^{4} b\right )} x^{3} +{\left (a^{5} m^{5} + 35 \, a^{5} m^{4} + 470 \, a^{5} m^{3} + 3010 \, a^{5} m^{2} + 9129 \, a^{5} m + 10395 \, a^{5}\right )} x\right )} x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b^5*m^5 + 25*b^5*m^4 + 230*b^5*m^3 + 950*b^5*m^2 + 1689*b^5*m + 945*b^5)*x^11
+ 5*(a*b^4*m^5 + 27*a*b^4*m^4 + 262*a*b^4*m^3 + 1122*a*b^4*m^2 + 2041*a*b^4*m +
1155*a*b^4)*x^9 + 10*(a^2*b^3*m^5 + 29*a^2*b^3*m^4 + 302*a^2*b^3*m^3 + 1366*a^2*
b^3*m^2 + 2577*a^2*b^3*m + 1485*a^2*b^3)*x^7 + 10*(a^3*b^2*m^5 + 31*a^3*b^2*m^4
+ 350*a^3*b^2*m^3 + 1730*a^3*b^2*m^2 + 3489*a^3*b^2*m + 2079*a^3*b^2)*x^5 + 5*(a
^4*b*m^5 + 33*a^4*b*m^4 + 406*a^4*b*m^3 + 2262*a^4*b*m^2 + 5353*a^4*b*m + 3465*a
^4*b)*x^3 + (a^5*m^5 + 35*a^5*m^4 + 470*a^5*m^3 + 3010*a^5*m^2 + 9129*a^5*m + 10
395*a^5)*x)*x^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395
)

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Sympy [A]  time = 13.4555, size = 1999, normalized size = 20.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**2+a)**5,x)

[Out]

Piecewise((-a**5/(10*x**10) - 5*a**4*b/(8*x**8) - 5*a**3*b**2/(3*x**6) - 5*a**2*
b**3/(2*x**4) - 5*a*b**4/(2*x**2) + b**5*log(x), Eq(m, -11)), (-a**5/(8*x**8) -
5*a**4*b/(6*x**6) - 5*a**3*b**2/(2*x**4) - 5*a**2*b**3/x**2 + 5*a*b**4*log(x) +
b**5*x**2/2, Eq(m, -9)), (-a**5/(6*x**6) - 5*a**4*b/(4*x**4) - 5*a**3*b**2/x**2
+ 10*a**2*b**3*log(x) + 5*a*b**4*x**2/2 + b**5*x**4/4, Eq(m, -7)), (-a**5/(4*x**
4) - 5*a**4*b/(2*x**2) + 10*a**3*b**2*log(x) + 5*a**2*b**3*x**2 + 5*a*b**4*x**4/
4 + b**5*x**6/6, Eq(m, -5)), (-a**5/(2*x**2) + 5*a**4*b*log(x) + 5*a**3*b**2*x**
2 + 5*a**2*b**3*x**4/2 + 5*a*b**4*x**6/6 + b**5*x**8/8, Eq(m, -3)), (a**5*log(x)
+ 5*a**4*b*x**2/2 + 5*a**3*b**2*x**4/2 + 5*a**2*b**3*x**6/3 + 5*a*b**4*x**8/8 +
b**5*x**10/10, Eq(m, -1)), (a**5*m**5*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*
m**3 + 12139*m**2 + 19524*m + 10395) + 35*a**5*m**4*x*x**m/(m**6 + 36*m**5 + 505
*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 470*a**5*m**3*x*x**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3010*a**5*m**2
*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +
9129*a**5*m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*
m + 10395) + 10395*a**5*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m*
*2 + 19524*m + 10395) + 5*a**4*b*m**5*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 348
0*m**3 + 12139*m**2 + 19524*m + 10395) + 165*a**4*b*m**4*x**3*x**m/(m**6 + 36*m*
*5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2030*a**4*b*m**3*x**
3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +
11310*a**4*b*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2
+ 19524*m + 10395) + 26765*a**4*b*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*
m**3 + 12139*m**2 + 19524*m + 10395) + 17325*a**4*b*x**3*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10*a**3*b**2*m**5*x**5*x*
*m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 310*
a**3*b**2*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1
9524*m + 10395) + 3500*a**3*b**2*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 348
0*m**3 + 12139*m**2 + 19524*m + 10395) + 17300*a**3*b**2*m**2*x**5*x**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 34890*a**3*b**2
*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103
95) + 20790*a**3*b**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m
**2 + 19524*m + 10395) + 10*a**2*b**3*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4
+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 290*a**2*b**3*m**4*x**7*x**m/(m**6
+ 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3020*a**2*b*
*3*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m
+ 10395) + 13660*a**2*b**3*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 25770*a**2*b**3*m*x**7*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 14850*a**2*b**3*x**7*x**
m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5*a*b
**4*m**5*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m
+ 10395) + 135*a*b**4*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1
2139*m**2 + 19524*m + 10395) + 1310*a*b**4*m**3*x**9*x**m/(m**6 + 36*m**5 + 505*
m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5610*a*b**4*m**2*x**9*x**m/(m
**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10205*a*b
**4*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +
10395) + 5775*a*b**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m*
*2 + 19524*m + 10395) + b**5*m**5*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m
**3 + 12139*m**2 + 19524*m + 10395) + 25*b**5*m**4*x**11*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 230*b**5*m**3*x**11*x**m/
(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 950*b**
5*m**2*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m
+ 10395) + 1689*b**5*m*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139
*m**2 + 19524*m + 10395) + 945*b**5*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480
*m**3 + 12139*m**2 + 19524*m + 10395), True))

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GIAC/XCAS [A]  time = 0.217624, size = 826, normalized size = 8.52 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^m,x, algorithm="giac")

[Out]

(b^5*m^5*x^11*e^(m*ln(x)) + 25*b^5*m^4*x^11*e^(m*ln(x)) + 5*a*b^4*m^5*x^9*e^(m*l
n(x)) + 230*b^5*m^3*x^11*e^(m*ln(x)) + 135*a*b^4*m^4*x^9*e^(m*ln(x)) + 950*b^5*m
^2*x^11*e^(m*ln(x)) + 10*a^2*b^3*m^5*x^7*e^(m*ln(x)) + 1310*a*b^4*m^3*x^9*e^(m*l
n(x)) + 1689*b^5*m*x^11*e^(m*ln(x)) + 290*a^2*b^3*m^4*x^7*e^(m*ln(x)) + 5610*a*b
^4*m^2*x^9*e^(m*ln(x)) + 945*b^5*x^11*e^(m*ln(x)) + 10*a^3*b^2*m^5*x^5*e^(m*ln(x
)) + 3020*a^2*b^3*m^3*x^7*e^(m*ln(x)) + 10205*a*b^4*m*x^9*e^(m*ln(x)) + 310*a^3*
b^2*m^4*x^5*e^(m*ln(x)) + 13660*a^2*b^3*m^2*x^7*e^(m*ln(x)) + 5775*a*b^4*x^9*e^(
m*ln(x)) + 5*a^4*b*m^5*x^3*e^(m*ln(x)) + 3500*a^3*b^2*m^3*x^5*e^(m*ln(x)) + 2577
0*a^2*b^3*m*x^7*e^(m*ln(x)) + 165*a^4*b*m^4*x^3*e^(m*ln(x)) + 17300*a^3*b^2*m^2*
x^5*e^(m*ln(x)) + 14850*a^2*b^3*x^7*e^(m*ln(x)) + a^5*m^5*x*e^(m*ln(x)) + 2030*a
^4*b*m^3*x^3*e^(m*ln(x)) + 34890*a^3*b^2*m*x^5*e^(m*ln(x)) + 35*a^5*m^4*x*e^(m*l
n(x)) + 11310*a^4*b*m^2*x^3*e^(m*ln(x)) + 20790*a^3*b^2*x^5*e^(m*ln(x)) + 470*a^
5*m^3*x*e^(m*ln(x)) + 26765*a^4*b*m*x^3*e^(m*ln(x)) + 3010*a^5*m^2*x*e^(m*ln(x))
+ 17325*a^4*b*x^3*e^(m*ln(x)) + 9129*a^5*m*x*e^(m*ln(x)) + 10395*a^5*x*e^(m*ln(
x)))/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)