3.340 \(\int x^m (a+b x^2)^4 \, dx\)

Optimal. Leaf size=79 \[ \frac{6 a^2 b^2 x^{m+5}}{m+5}+\frac{4 a^3 b x^{m+3}}{m+3}+\frac{a^4 x^{m+1}}{m+1}+\frac{4 a b^3 x^{m+7}}{m+7}+\frac{b^4 x^{m+9}}{m+9} \]

[Out]

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

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Rubi [A]  time = 0.0315087, antiderivative size = 79, 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} \[ \frac{6 a^2 b^2 x^{m+5}}{m+5}+\frac{4 a^3 b x^{m+3}}{m+3}+\frac{a^4 x^{m+1}}{m+1}+\frac{4 a b^3 x^{m+7}}{m+7}+\frac{b^4 x^{m+9}}{m+9} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^2)^4,x]

[Out]

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

Mathematica [A]  time = 0.0323216, size = 72, normalized size = 0.91 \[ x^{m+1} \left (\frac{6 a^2 b^2 x^4}{m+5}+\frac{4 a^3 b x^2}{m+3}+\frac{a^4}{m+1}+\frac{4 a b^3 x^6}{m+7}+\frac{b^4 x^8}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^2)^4,x]

[Out]

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

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Maple [B]  time = 0.005, size = 291, normalized size = 3.7 \begin{align*}{\frac{{x}^{1+m} \left ({b}^{4}{m}^{4}{x}^{8}+16\,{b}^{4}{m}^{3}{x}^{8}+4\,a{b}^{3}{m}^{4}{x}^{6}+86\,{b}^{4}{m}^{2}{x}^{8}+72\,a{b}^{3}{m}^{3}{x}^{6}+176\,{b}^{4}m{x}^{8}+6\,{a}^{2}{b}^{2}{m}^{4}{x}^{4}+416\,a{b}^{3}{m}^{2}{x}^{6}+105\,{b}^{4}{x}^{8}+120\,{a}^{2}{b}^{2}{m}^{3}{x}^{4}+888\,a{b}^{3}m{x}^{6}+4\,{a}^{3}b{m}^{4}{x}^{2}+780\,{a}^{2}{b}^{2}{m}^{2}{x}^{4}+540\,a{b}^{3}{x}^{6}+88\,{a}^{3}b{m}^{3}{x}^{2}+1800\,{a}^{2}{b}^{2}m{x}^{4}+{a}^{4}{m}^{4}+656\,{a}^{3}b{m}^{2}{x}^{2}+1134\,{b}^{2}{a}^{2}{x}^{4}+24\,{a}^{4}{m}^{3}+1832\,{a}^{3}bm{x}^{2}+206\,{a}^{4}{m}^{2}+1260\,{a}^{3}b{x}^{2}+744\,{a}^{4}m+945\,{a}^{4} \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)^4,x)

[Out]

x^(1+m)*(b^4*m^4*x^8+16*b^4*m^3*x^8+4*a*b^3*m^4*x^6+86*b^4*m^2*x^8+72*a*b^3*m^3*x^6+176*b^4*m*x^8+6*a^2*b^2*m^
4*x^4+416*a*b^3*m^2*x^6+105*b^4*x^8+120*a^2*b^2*m^3*x^4+888*a*b^3*m*x^6+4*a^3*b*m^4*x^2+780*a^2*b^2*m^2*x^4+54
0*a*b^3*x^6+88*a^3*b*m^3*x^2+1800*a^2*b^2*m*x^4+a^4*m^4+656*a^3*b*m^2*x^2+1134*a^2*b^2*x^4+24*a^4*m^3+1832*a^3
*b*m*x^2+206*a^4*m^2+1260*a^3*b*x^2+744*a^4*m+945*a^4)/(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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.62544, size = 571, normalized size = 7.23 \begin{align*} \frac{{\left ({\left (b^{4} m^{4} + 16 \, b^{4} m^{3} + 86 \, b^{4} m^{2} + 176 \, b^{4} m + 105 \, b^{4}\right )} x^{9} + 4 \,{\left (a b^{3} m^{4} + 18 \, a b^{3} m^{3} + 104 \, a b^{3} m^{2} + 222 \, a b^{3} m + 135 \, a b^{3}\right )} x^{7} + 6 \,{\left (a^{2} b^{2} m^{4} + 20 \, a^{2} b^{2} m^{3} + 130 \, a^{2} b^{2} m^{2} + 300 \, a^{2} b^{2} m + 189 \, a^{2} b^{2}\right )} x^{5} + 4 \,{\left (a^{3} b m^{4} + 22 \, a^{3} b m^{3} + 164 \, a^{3} b m^{2} + 458 \, a^{3} b m + 315 \, a^{3} b\right )} x^{3} +{\left (a^{4} m^{4} + 24 \, a^{4} m^{3} + 206 \, a^{4} m^{2} + 744 \, a^{4} m + 945 \, a^{4}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^2+a)^4,x, algorithm="fricas")

[Out]

((b^4*m^4 + 16*b^4*m^3 + 86*b^4*m^2 + 176*b^4*m + 105*b^4)*x^9 + 4*(a*b^3*m^4 + 18*a*b^3*m^3 + 104*a*b^3*m^2 +
222*a*b^3*m + 135*a*b^3)*x^7 + 6*(a^2*b^2*m^4 + 20*a^2*b^2*m^3 + 130*a^2*b^2*m^2 + 300*a^2*b^2*m + 189*a^2*b^
2)*x^5 + 4*(a^3*b*m^4 + 22*a^3*b*m^3 + 164*a^3*b*m^2 + 458*a^3*b*m + 315*a^3*b)*x^3 + (a^4*m^4 + 24*a^4*m^3 +
206*a^4*m^2 + 744*a^4*m + 945*a^4)*x)*x^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)

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Sympy [A]  time = 2.5335, size = 1221, normalized size = 15.46 \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)**4,x)

[Out]

Piecewise((-a**4/(8*x**8) - 2*a**3*b/(3*x**6) - 3*a**2*b**2/(2*x**4) - 2*a*b**3/x**2 + b**4*log(x), Eq(m, -9))
, (-a**4/(6*x**6) - a**3*b/x**4 - 3*a**2*b**2/x**2 + 4*a*b**3*log(x) + b**4*x**2/2, Eq(m, -7)), (-a**4/(4*x**4
) - 2*a**3*b/x**2 + 6*a**2*b**2*log(x) + 2*a*b**3*x**2 + b**4*x**4/4, Eq(m, -5)), (-a**4/(2*x**2) + 4*a**3*b*l
og(x) + 3*a**2*b**2*x**2 + a*b**3*x**4 + b**4*x**6/6, Eq(m, -3)), (a**4*log(x) + 2*a**3*b*x**2 + 3*a**2*b**2*x
**4/2 + 2*a*b**3*x**6/3 + b**4*x**8/8, Eq(m, -1)), (a**4*m**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1
689*m + 945) + 24*a**4*m**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*a**4*m**2*x*x**
m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 744*a**4*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 945*a**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 4*a**3*b*m**4*x*
*3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 88*a**3*b*m**3*x**3*x**m/(m**5 + 25*m**4 + 230
*m**3 + 950*m**2 + 1689*m + 945) + 656*a**3*b*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m +
945) + 1832*a**3*b*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1260*a**3*b*x**3*x**m/(
m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 6*a**2*b**2*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 120*a**2*b**2*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 780*a**2*b**2*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1800*a**2*b**2*m*x**5*
x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1134*a**2*b**2*x**5*x**m/(m**5 + 25*m**4 + 230*m*
*3 + 950*m**2 + 1689*m + 945) + 4*a*b**3*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 72*a*b**3*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 416*a*b**3*m**2*x**7*x**m/(
m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 888*a*b**3*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950
*m**2 + 1689*m + 945) + 540*a*b**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + b**4*m**4
*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 16*b**4*m**3*x**9*x**m/(m**5 + 25*m**4 + 23
0*m**3 + 950*m**2 + 1689*m + 945) + 86*b**4*m**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 176*b**4*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 105*b**4*x**9*x**m/(m**5 + 2
5*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945), True))

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Giac [B]  time = 2.94744, size = 493, normalized size = 6.24 \begin{align*} \frac{b^{4} m^{4} x^{9} x^{m} + 16 \, b^{4} m^{3} x^{9} x^{m} + 4 \, a b^{3} m^{4} x^{7} x^{m} + 86 \, b^{4} m^{2} x^{9} x^{m} + 72 \, a b^{3} m^{3} x^{7} x^{m} + 176 \, b^{4} m x^{9} x^{m} + 6 \, a^{2} b^{2} m^{4} x^{5} x^{m} + 416 \, a b^{3} m^{2} x^{7} x^{m} + 105 \, b^{4} x^{9} x^{m} + 120 \, a^{2} b^{2} m^{3} x^{5} x^{m} + 888 \, a b^{3} m x^{7} x^{m} + 4 \, a^{3} b m^{4} x^{3} x^{m} + 780 \, a^{2} b^{2} m^{2} x^{5} x^{m} + 540 \, a b^{3} x^{7} x^{m} + 88 \, a^{3} b m^{3} x^{3} x^{m} + 1800 \, a^{2} b^{2} m x^{5} x^{m} + a^{4} m^{4} x x^{m} + 656 \, a^{3} b m^{2} x^{3} x^{m} + 1134 \, a^{2} b^{2} x^{5} x^{m} + 24 \, a^{4} m^{3} x x^{m} + 1832 \, a^{3} b m x^{3} x^{m} + 206 \, a^{4} m^{2} x x^{m} + 1260 \, a^{3} b x^{3} x^{m} + 744 \, a^{4} m x x^{m} + 945 \, a^{4} x x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^2+a)^4,x, algorithm="giac")

[Out]

(b^4*m^4*x^9*x^m + 16*b^4*m^3*x^9*x^m + 4*a*b^3*m^4*x^7*x^m + 86*b^4*m^2*x^9*x^m + 72*a*b^3*m^3*x^7*x^m + 176*
b^4*m*x^9*x^m + 6*a^2*b^2*m^4*x^5*x^m + 416*a*b^3*m^2*x^7*x^m + 105*b^4*x^9*x^m + 120*a^2*b^2*m^3*x^5*x^m + 88
8*a*b^3*m*x^7*x^m + 4*a^3*b*m^4*x^3*x^m + 780*a^2*b^2*m^2*x^5*x^m + 540*a*b^3*x^7*x^m + 88*a^3*b*m^3*x^3*x^m +
1800*a^2*b^2*m*x^5*x^m + a^4*m^4*x*x^m + 656*a^3*b*m^2*x^3*x^m + 1134*a^2*b^2*x^5*x^m + 24*a^4*m^3*x*x^m + 18
32*a^3*b*m*x^3*x^m + 206*a^4*m^2*x*x^m + 1260*a^3*b*x^3*x^m + 744*a^4*m*x*x^m + 945*a^4*x*x^m)/(m^5 + 25*m^4 +
230*m^3 + 950*m^2 + 1689*m + 945)