3.279 \(\int \frac {(a+b x^3)^5}{x^2} \, dx\)

Optimal. Leaf size=65 \[ -\frac {a^5}{x}+\frac {5}{2} a^4 b x^2+2 a^3 b^2 x^5+\frac {5}{4} a^2 b^3 x^8+\frac {5}{11} a b^4 x^{11}+\frac {b^5 x^{14}}{14} \]

[Out]

-a^5/x+5/2*a^4*b*x^2+2*a^3*b^2*x^5+5/4*a^2*b^3*x^8+5/11*a*b^4*x^11+1/14*b^5*x^14

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Rubi [A]  time = 0.02, antiderivative size = 65, 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 = {270} \[ \frac {5}{4} a^2 b^3 x^8+2 a^3 b^2 x^5+\frac {5}{2} a^4 b x^2-\frac {a^5}{x}+\frac {5}{11} a b^4 x^{11}+\frac {b^5 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^2,x]

[Out]

-(a^5/x) + (5*a^4*b*x^2)/2 + 2*a^3*b^2*x^5 + (5*a^2*b^3*x^8)/4 + (5*a*b^4*x^11)/11 + (b^5*x^14)/14

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 \frac {\left (a+b x^3\right )^5}{x^2} \, dx &=\int \left (\frac {a^5}{x^2}+5 a^4 b x+10 a^3 b^2 x^4+10 a^2 b^3 x^7+5 a b^4 x^{10}+b^5 x^{13}\right ) \, dx\\ &=-\frac {a^5}{x}+\frac {5}{2} a^4 b x^2+2 a^3 b^2 x^5+\frac {5}{4} a^2 b^3 x^8+\frac {5}{11} a b^4 x^{11}+\frac {b^5 x^{14}}{14}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 65, normalized size = 1.00 \[ -\frac {a^5}{x}+\frac {5}{2} a^4 b x^2+2 a^3 b^2 x^5+\frac {5}{4} a^2 b^3 x^8+\frac {5}{11} a b^4 x^{11}+\frac {b^5 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^2,x]

[Out]

-(a^5/x) + (5*a^4*b*x^2)/2 + 2*a^3*b^2*x^5 + (5*a^2*b^3*x^8)/4 + (5*a*b^4*x^11)/11 + (b^5*x^14)/14

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fricas [A]  time = 0.76, size = 59, normalized size = 0.91 \[ \frac {22 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 385 \, a^{2} b^{3} x^{9} + 616 \, a^{3} b^{2} x^{6} + 770 \, a^{4} b x^{3} - 308 \, a^{5}}{308 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/308*(22*b^5*x^15 + 140*a*b^4*x^12 + 385*a^2*b^3*x^9 + 616*a^3*b^2*x^6 + 770*a^4*b*x^3 - 308*a^5)/x

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giac [A]  time = 0.18, size = 57, normalized size = 0.88 \[ \frac {1}{14} \, b^{5} x^{14} + \frac {5}{11} \, a b^{4} x^{11} + \frac {5}{4} \, a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{5} + \frac {5}{2} \, a^{4} b x^{2} - \frac {a^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^5*x^14 + 5/11*a*b^4*x^11 + 5/4*a^2*b^3*x^8 + 2*a^3*b^2*x^5 + 5/2*a^4*b*x^2 - a^5/x

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maple [A]  time = 0.00, size = 58, normalized size = 0.89 \[ \frac {b^{5} x^{14}}{14}+\frac {5 a \,b^{4} x^{11}}{11}+\frac {5 a^{2} b^{3} x^{8}}{4}+2 a^{3} b^{2} x^{5}+\frac {5 a^{4} b \,x^{2}}{2}-\frac {a^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^2,x)

[Out]

-a^5/x+5/2*a^4*b*x^2+2*a^3*b^2*x^5+5/4*a^2*b^3*x^8+5/11*a*b^4*x^11+1/14*b^5*x^14

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maxima [A]  time = 1.29, size = 57, normalized size = 0.88 \[ \frac {1}{14} \, b^{5} x^{14} + \frac {5}{11} \, a b^{4} x^{11} + \frac {5}{4} \, a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{5} + \frac {5}{2} \, a^{4} b x^{2} - \frac {a^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^5*x^14 + 5/11*a*b^4*x^11 + 5/4*a^2*b^3*x^8 + 2*a^3*b^2*x^5 + 5/2*a^4*b*x^2 - a^5/x

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mupad [B]  time = 0.02, size = 57, normalized size = 0.88 \[ \frac {b^5\,x^{14}}{14}-\frac {a^5}{x}+\frac {5\,a^4\,b\,x^2}{2}+\frac {5\,a\,b^4\,x^{11}}{11}+2\,a^3\,b^2\,x^5+\frac {5\,a^2\,b^3\,x^8}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^5/x^2,x)

[Out]

(b^5*x^14)/14 - a^5/x + (5*a^4*b*x^2)/2 + (5*a*b^4*x^11)/11 + 2*a^3*b^2*x^5 + (5*a^2*b^3*x^8)/4

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sympy [A]  time = 0.15, size = 61, normalized size = 0.94 \[ - \frac {a^{5}}{x} + \frac {5 a^{4} b x^{2}}{2} + 2 a^{3} b^{2} x^{5} + \frac {5 a^{2} b^{3} x^{8}}{4} + \frac {5 a b^{4} x^{11}}{11} + \frac {b^{5} x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**5/x + 5*a**4*b*x**2/2 + 2*a**3*b**2*x**5 + 5*a**2*b**3*x**8/4 + 5*a*b**4*x**11/11 + b**5*x**14/14

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