3.1.57 \(\int x^3 (a+b x^2)^5 \, dx\) [57]

Optimal. Leaf size=34 \[ -\frac {a \left (a+b x^2\right )^6}{12 b^2}+\frac {\left (a+b x^2\right )^7}{14 b^2} \]

[Out]

-1/12*a*(b*x^2+a)^6/b^2+1/14*(b*x^2+a)^7/b^2

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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} \frac {\left (a+b x^2\right )^7}{14 b^2}-\frac {a \left (a+b x^2\right )^6}{12 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)^5,x]

[Out]

-1/12*(a*(a + b*x^2)^6)/b^2 + (a + b*x^2)^7/(14*b^2)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^3 \left (a+b x^2\right )^5 \, dx &=\frac {1}{2} \text {Subst}\left (\int x (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a (a+b x)^5}{b}+\frac {(a+b x)^6}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac {a \left (a+b x^2\right )^6}{12 b^2}+\frac {\left (a+b x^2\right )^7}{14 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 66, normalized size = 1.94 \begin {gather*} \frac {a^5 x^4}{4}+\frac {5}{6} a^4 b x^6+\frac {5}{4} a^3 b^2 x^8+a^2 b^3 x^{10}+\frac {5}{12} a b^4 x^{12}+\frac {b^5 x^{14}}{14} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^2)^5,x]

[Out]

(a^5*x^4)/4 + (5*a^4*b*x^6)/6 + (5*a^3*b^2*x^8)/4 + a^2*b^3*x^10 + (5*a*b^4*x^12)/12 + (b^5*x^14)/14

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Maple [A]
time = 0.03, size = 57, normalized size = 1.68

method	result	size
gosper	\(\frac {1}{4} a^{5} x^{4}+\frac {5}{6} a^{4} b \,x^{6}+\frac {5}{4} a^{3} b^{2} x^{8}+a^{2} b^{3} x^{10}+\frac {5}{12} a \,b^{4} x^{12}+\frac {1}{14} b^{5} x^{14}\)	\(57\)
default	\(\frac {1}{4} a^{5} x^{4}+\frac {5}{6} a^{4} b \,x^{6}+\frac {5}{4} a^{3} b^{2} x^{8}+a^{2} b^{3} x^{10}+\frac {5}{12} a \,b^{4} x^{12}+\frac {1}{14} b^{5} x^{14}\)	\(57\)
norman	\(\frac {1}{4} a^{5} x^{4}+\frac {5}{6} a^{4} b \,x^{6}+\frac {5}{4} a^{3} b^{2} x^{8}+a^{2} b^{3} x^{10}+\frac {5}{12} a \,b^{4} x^{12}+\frac {1}{14} b^{5} x^{14}\)	\(57\)
risch	\(\frac {1}{4} a^{5} x^{4}+\frac {5}{6} a^{4} b \,x^{6}+\frac {5}{4} a^{3} b^{2} x^{8}+a^{2} b^{3} x^{10}+\frac {5}{12} a \,b^{4} x^{12}+\frac {1}{14} b^{5} x^{14}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/4*a^5*x^4+5/6*a^4*b*x^6+5/4*a^3*b^2*x^8+a^2*b^3*x^10+5/12*a*b^4*x^12+1/14*b^5*x^14

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Maxima [A]
time = 0.28, size = 56, normalized size = 1.65 \begin {gather*} \frac {1}{14} \, b^{5} x^{14} + \frac {5}{12} \, a b^{4} x^{12} + a^{2} b^{3} x^{10} + \frac {5}{4} \, a^{3} b^{2} x^{8} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{4} \, a^{5} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^5,x, algorithm="maxima")

[Out]

1/14*b^5*x^14 + 5/12*a*b^4*x^12 + a^2*b^3*x^10 + 5/4*a^3*b^2*x^8 + 5/6*a^4*b*x^6 + 1/4*a^5*x^4

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Fricas [A]
time = 0.94, size = 56, normalized size = 1.65 \begin {gather*} \frac {1}{14} \, b^{5} x^{14} + \frac {5}{12} \, a b^{4} x^{12} + a^{2} b^{3} x^{10} + \frac {5}{4} \, a^{3} b^{2} x^{8} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{4} \, a^{5} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^5,x, algorithm="fricas")

[Out]

1/14*b^5*x^14 + 5/12*a*b^4*x^12 + a^2*b^3*x^10 + 5/4*a^3*b^2*x^8 + 5/6*a^4*b*x^6 + 1/4*a^5*x^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
time = 0.01, size = 63, normalized size = 1.85 \begin {gather*} \frac {a^{5} x^{4}}{4} + \frac {5 a^{4} b x^{6}}{6} + \frac {5 a^{3} b^{2} x^{8}}{4} + a^{2} b^{3} x^{10} + \frac {5 a b^{4} x^{12}}{12} + \frac {b^{5} x^{14}}{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**2+a)**5,x)

[Out]

a**5*x**4/4 + 5*a**4*b*x**6/6 + 5*a**3*b**2*x**8/4 + a**2*b**3*x**10 + 5*a*b**4*x**12/12 + b**5*x**14/14

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Giac [A]
time = 0.53, size = 56, normalized size = 1.65 \begin {gather*} \frac {1}{14} \, b^{5} x^{14} + \frac {5}{12} \, a b^{4} x^{12} + a^{2} b^{3} x^{10} + \frac {5}{4} \, a^{3} b^{2} x^{8} + \frac {5}{6} \, a^{4} b x^{6} + \frac {1}{4} \, a^{5} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^5,x, algorithm="giac")

[Out]

1/14*b^5*x^14 + 5/12*a*b^4*x^12 + a^2*b^3*x^10 + 5/4*a^3*b^2*x^8 + 5/6*a^4*b*x^6 + 1/4*a^5*x^4

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Mupad [B]
time = 0.02, size = 56, normalized size = 1.65 \begin {gather*} \frac {a^5\,x^4}{4}+\frac {5\,a^4\,b\,x^6}{6}+\frac {5\,a^3\,b^2\,x^8}{4}+a^2\,b^3\,x^{10}+\frac {5\,a\,b^4\,x^{12}}{12}+\frac {b^5\,x^{14}}{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^2)^5,x)

[Out]

(a^5*x^4)/4 + (b^5*x^14)/14 + (5*a^4*b*x^6)/6 + (5*a*b^4*x^12)/12 + (5*a^3*b^2*x^8)/4 + a^2*b^3*x^10

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