3.4.41 \(\int x^m (a+b x^2)^3 \, dx\) [341]

Optimal. Leaf size=61 \[ \frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{3+m}}{3+m}+\frac {3 a b^2 x^{5+m}}{5+m}+\frac {b^3 x^{7+m}}{7+m} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 = {276} \begin {gather*} \frac {a^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+3}}{m+3}+\frac {3 a b^2 x^{m+5}}{m+5}+\frac {b^3 x^{m+7}}{m+7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^2)^3,x]

[Out]

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

Rule 276

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

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Mathematica [A]
time = 0.04, size = 56, normalized size = 0.92 \begin {gather*} x^{1+m} \left (\frac {a^3}{1+m}+\frac {3 a^2 b x^2}{3+m}+\frac {3 a b^2 x^4}{5+m}+\frac {b^3 x^6}{7+m}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^2)^3,x]

[Out]

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

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Maple [A]
time = 0.03, size = 72, normalized size = 1.18

method	result	size
norman	\(\frac {a^{3} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{3} x^{7} {\mathrm e}^{m \ln \left (x \right )}}{7+m}+\frac {3 a \,b^{2} x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {3 a^{2} b \,x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}\)	\(72\)
risch	\(\frac {x \left (b^{3} m^{3} x^{6}+9 b^{3} m^{2} x^{6}+3 a \,b^{2} m^{3} x^{4}+23 m \,x^{6} b^{3}+33 a \,b^{2} m^{2} x^{4}+15 b^{3} x^{6}+3 a^{2} b \,m^{3} x^{2}+93 m \,x^{4} a \,b^{2}+39 a^{2} b \,m^{2} x^{2}+63 a \,b^{2} x^{4}+a^{3} m^{3}+141 m \,x^{2} a^{2} b +15 a^{3} m^{2}+105 a^{2} b \,x^{2}+71 m \,a^{3}+105 a^{3}\right ) x^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(177\)
gosper	\(\frac {x^{1+m} \left (b^{3} m^{3} x^{6}+9 b^{3} m^{2} x^{6}+3 a \,b^{2} m^{3} x^{4}+23 m \,x^{6} b^{3}+33 a \,b^{2} m^{2} x^{4}+15 b^{3} x^{6}+3 a^{2} b \,m^{3} x^{2}+93 m \,x^{4} a \,b^{2}+39 a^{2} b \,m^{2} x^{2}+63 a \,b^{2} x^{4}+a^{3} m^{3}+141 m \,x^{2} a^{2} b +15 a^{3} m^{2}+105 a^{2} b \,x^{2}+71 m \,a^{3}+105 a^{3}\right )}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(178\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

a^3/(1+m)*x*exp(m*ln(x))+b^3/(7+m)*x^7*exp(m*ln(x))+3*a*b^2/(5+m)*x^5*exp(m*ln(x))+3*a^2*b/(3+m)*x^3*exp(m*ln(
x))

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Maxima [A]
time = 0.27, size = 61, normalized size = 1.00 \begin {gather*} \frac {b^{3} x^{m + 7}}{m + 7} + \frac {3 \, a b^{2} x^{m + 5}}{m + 5} + \frac {3 \, a^{2} b x^{m + 3}}{m + 3} + \frac {a^{3} x^{m + 1}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (61) = 122\).
time = 1.20, size = 157, normalized size = 2.57 \begin {gather*} \frac {{\left ({\left (b^{3} m^{3} + 9 \, b^{3} m^{2} + 23 \, b^{3} m + 15 \, b^{3}\right )} x^{7} + 3 \, {\left (a b^{2} m^{3} + 11 \, a b^{2} m^{2} + 31 \, a b^{2} m + 21 \, a b^{2}\right )} x^{5} + 3 \, {\left (a^{2} b m^{3} + 13 \, a^{2} b m^{2} + 47 \, a^{2} b m + 35 \, a^{2} b\right )} x^{3} + {\left (a^{3} m^{3} + 15 \, a^{3} m^{2} + 71 \, a^{3} m + 105 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^2+a)^3,x, algorithm="fricas")

[Out]

((b^3*m^3 + 9*b^3*m^2 + 23*b^3*m + 15*b^3)*x^7 + 3*(a*b^2*m^3 + 11*a*b^2*m^2 + 31*a*b^2*m + 21*a*b^2)*x^5 + 3*
(a^2*b*m^3 + 13*a^2*b*m^2 + 47*a^2*b*m + 35*a^2*b)*x^3 + (a^3*m^3 + 15*a^3*m^2 + 71*a^3*m + 105*a^3)*x)*x^m/(m
^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (53) = 106\).
time = 0.35, size = 683, normalized size = 11.20 \begin {gather*} \begin {cases} - \frac {a^{3}}{6 x^{6}} - \frac {3 a^{2} b}{4 x^{4}} - \frac {3 a b^{2}}{2 x^{2}} + b^{3} \log {\left (x \right )} & \text {for}\: m = -7 \\- \frac {a^{3}}{4 x^{4}} - \frac {3 a^{2} b}{2 x^{2}} + 3 a b^{2} \log {\left (x \right )} + \frac {b^{3} x^{2}}{2} & \text {for}\: m = -5 \\- \frac {a^{3}}{2 x^{2}} + 3 a^{2} b \log {\left (x \right )} + \frac {3 a b^{2} x^{2}}{2} + \frac {b^{3} x^{4}}{4} & \text {for}\: m = -3 \\a^{3} \log {\left (x \right )} + \frac {3 a^{2} b x^{2}}{2} + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{6}}{6} & \text {for}\: m = -1 \\\frac {a^{3} m^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 a^{3} m^{2} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {71 a^{3} m x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {105 a^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {3 a^{2} b m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {39 a^{2} b m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {141 a^{2} b m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {105 a^{2} b x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {3 a b^{2} m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {33 a b^{2} m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {93 a b^{2} m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {63 a b^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {b^{3} m^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {9 b^{3} m^{2} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {23 b^{3} m x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 b^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**2+a)**3,x)

[Out]

Piecewise((-a**3/(6*x**6) - 3*a**2*b/(4*x**4) - 3*a*b**2/(2*x**2) + b**3*log(x), Eq(m, -7)), (-a**3/(4*x**4) -
3*a**2*b/(2*x**2) + 3*a*b**2*log(x) + b**3*x**2/2, Eq(m, -5)), (-a**3/(2*x**2) + 3*a**2*b*log(x) + 3*a*b**2*x
**2/2 + b**3*x**4/4, Eq(m, -3)), (a**3*log(x) + 3*a**2*b*x**2/2 + 3*a*b**2*x**4/4 + b**3*x**6/6, Eq(m, -1)), (
a**3*m**3*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*a**3*m**2*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 17
6*m + 105) + 71*a**3*m*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 105*a**3*x*x**m/(m**4 + 16*m**3 + 86*
m**2 + 176*m + 105) + 3*a**2*b*m**3*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 39*a**2*b*m**2*x**3*x
**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 141*a**2*b*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105)
+ 105*a**2*b*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 3*a*b**2*m**3*x**5*x**m/(m**4 + 16*m**3 + 8
6*m**2 + 176*m + 105) + 33*a*b**2*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 93*a*b**2*m*x**5*x
**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 63*a*b**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) +
b**3*m**3*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 9*b**3*m**2*x**7*x**m/(m**4 + 16*m**3 + 86*m**2
+ 176*m + 105) + 23*b**3*m*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*b**3*x**7*x**m/(m**4 + 16*
m**3 + 86*m**2 + 176*m + 105), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (61) = 122\).
time = 1.65, size = 224, normalized size = 3.67 \begin {gather*} \frac {b^{3} m^{3} x^{7} x^{m} + 9 \, b^{3} m^{2} x^{7} x^{m} + 3 \, a b^{2} m^{3} x^{5} x^{m} + 23 \, b^{3} m x^{7} x^{m} + 33 \, a b^{2} m^{2} x^{5} x^{m} + 15 \, b^{3} x^{7} x^{m} + 3 \, a^{2} b m^{3} x^{3} x^{m} + 93 \, a b^{2} m x^{5} x^{m} + 39 \, a^{2} b m^{2} x^{3} x^{m} + 63 \, a b^{2} x^{5} x^{m} + a^{3} m^{3} x x^{m} + 141 \, a^{2} b m x^{3} x^{m} + 15 \, a^{3} m^{2} x x^{m} + 105 \, a^{2} b x^{3} x^{m} + 71 \, a^{3} m x x^{m} + 105 \, a^{3} x x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^2+a)^3,x, algorithm="giac")

[Out]

(b^3*m^3*x^7*x^m + 9*b^3*m^2*x^7*x^m + 3*a*b^2*m^3*x^5*x^m + 23*b^3*m*x^7*x^m + 33*a*b^2*m^2*x^5*x^m + 15*b^3*
x^7*x^m + 3*a^2*b*m^3*x^3*x^m + 93*a*b^2*m*x^5*x^m + 39*a^2*b*m^2*x^3*x^m + 63*a*b^2*x^5*x^m + a^3*m^3*x*x^m +
141*a^2*b*m*x^3*x^m + 15*a^3*m^2*x*x^m + 105*a^2*b*x^3*x^m + 71*a^3*m*x*x^m + 105*a^3*x*x^m)/(m^4 + 16*m^3 +
86*m^2 + 176*m + 105)

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Mupad [B]
time = 4.79, size = 167, normalized size = 2.74 \begin {gather*} x^m\,\left (\frac {a^3\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {b^3\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {3\,a\,b^2\,x^5\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {3\,a^2\,b\,x^3\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^2)^3,x)

[Out]

x^m*((a^3*x*(71*m + 15*m^2 + m^3 + 105))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (b^3*x^7*(23*m + 9*m^2 + m^3
+ 15))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (3*a*b^2*x^5*(31*m + 11*m^2 + m^3 + 21))/(176*m + 86*m^2 + 16*m
^3 + m^4 + 105) + (3*a^2*b*x^3*(47*m + 13*m^2 + m^3 + 35))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105))

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