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

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

[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 = 0.0430813, 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, Rules used = {270} \[ \frac{10 a^3 b^2 x^{m+5}}{m+5}+\frac{10 a^2 b^3 x^{m+7}}{m+7}+\frac{5 a^4 b x^{m+3}}{m+3}+\frac{a^5 x^{m+1}}{m+1}+\frac{5 a b^4 x^{m+9}}{m+9}+\frac{b^5 x^{m+11}}{m+11} \]

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)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^m \left (a+b x^2\right )^5 \, dx &=\int \left (a^5 x^m+5 a^4 b x^{2+m}+10 a^3 b^2 x^{4+m}+10 a^2 b^3 x^{6+m}+5 a b^4 x^{8+m}+b^5 x^{10+m}\right ) \, dx\\ &=\frac{a^5 x^{1+m}}{1+m}+\frac{5 a^4 b x^{3+m}}{3+m}+\frac{10 a^3 b^2 x^{5+m}}{5+m}+\frac{10 a^2 b^3 x^{7+m}}{7+m}+\frac{5 a b^4 x^{9+m}}{9+m}+\frac{b^5 x^{11+m}}{11+m}\\ \end{align*}

Mathematica [A]  time = 0.0504206, size = 88, normalized size = 0.91 \[ x^{m+1} \left (\frac{10 a^2 b^3 x^6}{m+7}+\frac{10 a^3 b^2 x^4}{m+5}+\frac{5 a^4 b x^2}{m+3}+\frac{a^5}{m+1}+\frac{5 a b^4 x^8}{m+9}+\frac{b^5 x^{10}}{m+11}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^2)^5,x]

[Out]

x^(1 + m)*(a^5/(1 + m) + (5*a^4*b*x^2)/(3 + m) + (10*a^3*b^2*x^4)/(5 + m) + (10*a^2*b^3*x^6)/(7 + m) + (5*a*b^
4*x^8)/(9 + m) + (b^5*x^10)/(11 + m))

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Maple [B]  time = 0.004, size = 432, normalized size = 4.5 \begin{align*}{\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 ) }} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.59317, size = 868, normalized size = 8.95 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^2+a)^5,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 + 10395*a^5)*x)*x^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

________________________________________________________________________________________

Sympy [A]  time = 4.06092, size = 1999, normalized size = 20.61 \begin{align*} \text{result too large to display} \end{align*}

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*lo
g(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 + 3480*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 + 195
24*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 + 19524*m + 1
0395) + 3500*a**3*b**2*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*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 + 10395) + 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 + 12139*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 + 10
395) + 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 [B]  time = 2.89287, size = 729, normalized size = 7.52 \begin{align*} \frac{b^{5} m^{5} x^{11} x^{m} + 25 \, b^{5} m^{4} x^{11} x^{m} + 5 \, a b^{4} m^{5} x^{9} x^{m} + 230 \, b^{5} m^{3} x^{11} x^{m} + 135 \, a b^{4} m^{4} x^{9} x^{m} + 950 \, b^{5} m^{2} x^{11} x^{m} + 10 \, a^{2} b^{3} m^{5} x^{7} x^{m} + 1310 \, a b^{4} m^{3} x^{9} x^{m} + 1689 \, b^{5} m x^{11} x^{m} + 290 \, a^{2} b^{3} m^{4} x^{7} x^{m} + 5610 \, a b^{4} m^{2} x^{9} x^{m} + 945 \, b^{5} x^{11} x^{m} + 10 \, a^{3} b^{2} m^{5} x^{5} x^{m} + 3020 \, a^{2} b^{3} m^{3} x^{7} x^{m} + 10205 \, a b^{4} m x^{9} x^{m} + 310 \, a^{3} b^{2} m^{4} x^{5} x^{m} + 13660 \, a^{2} b^{3} m^{2} x^{7} x^{m} + 5775 \, a b^{4} x^{9} x^{m} + 5 \, a^{4} b m^{5} x^{3} x^{m} + 3500 \, a^{3} b^{2} m^{3} x^{5} x^{m} + 25770 \, a^{2} b^{3} m x^{7} x^{m} + 165 \, a^{4} b m^{4} x^{3} x^{m} + 17300 \, a^{3} b^{2} m^{2} x^{5} x^{m} + 14850 \, a^{2} b^{3} x^{7} x^{m} + a^{5} m^{5} x x^{m} + 2030 \, a^{4} b m^{3} x^{3} x^{m} + 34890 \, a^{3} b^{2} m x^{5} x^{m} + 35 \, a^{5} m^{4} x x^{m} + 11310 \, a^{4} b m^{2} x^{3} x^{m} + 20790 \, a^{3} b^{2} x^{5} x^{m} + 470 \, a^{5} m^{3} x x^{m} + 26765 \, a^{4} b m x^{3} x^{m} + 3010 \, a^{5} m^{2} x x^{m} + 17325 \, a^{4} b x^{3} x^{m} + 9129 \, a^{5} m x x^{m} + 10395 \, a^{5} x x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^5*m^5*x^11*x^m + 25*b^5*m^4*x^11*x^m + 5*a*b^4*m^5*x^9*x^m + 230*b^5*m^3*x^11*x^m + 135*a*b^4*m^4*x^9*x^m +
950*b^5*m^2*x^11*x^m + 10*a^2*b^3*m^5*x^7*x^m + 1310*a*b^4*m^3*x^9*x^m + 1689*b^5*m*x^11*x^m + 290*a^2*b^3*m^
4*x^7*x^m + 5610*a*b^4*m^2*x^9*x^m + 945*b^5*x^11*x^m + 10*a^3*b^2*m^5*x^5*x^m + 3020*a^2*b^3*m^3*x^7*x^m + 10
205*a*b^4*m*x^9*x^m + 310*a^3*b^2*m^4*x^5*x^m + 13660*a^2*b^3*m^2*x^7*x^m + 5775*a*b^4*x^9*x^m + 5*a^4*b*m^5*x
^3*x^m + 3500*a^3*b^2*m^3*x^5*x^m + 25770*a^2*b^3*m*x^7*x^m + 165*a^4*b*m^4*x^3*x^m + 17300*a^3*b^2*m^2*x^5*x^
m + 14850*a^2*b^3*x^7*x^m + a^5*m^5*x*x^m + 2030*a^4*b*m^3*x^3*x^m + 34890*a^3*b^2*m*x^5*x^m + 35*a^5*m^4*x*x^
m + 11310*a^4*b*m^2*x^3*x^m + 20790*a^3*b^2*x^5*x^m + 470*a^5*m^3*x*x^m + 26765*a^4*b*m*x^3*x^m + 3010*a^5*m^2
*x*x^m + 17325*a^4*b*x^3*x^m + 9129*a^5*m*x*x^m + 10395*a^5*x*x^m)/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*
m^2 + 19524*m + 10395)