3.3.76 \(\int \frac {(A+B x) (a+c x^2)^4}{x} \, dx\)

Optimal. Leaf size=110 \[ a^4 A \log (x)+a^4 B x+2 a^3 A c x^2+\frac {4}{3} a^3 B c x^3+\frac {3}{2} a^2 A c^2 x^4+\frac {6}{5} a^2 B c^2 x^5+\frac {2}{3} a A c^3 x^6+\frac {4}{7} a B c^3 x^7+\frac {1}{8} A c^4 x^8+\frac {1}{9} B c^4 x^9 \]

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Rubi [A]  time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {766} \begin {gather*} \frac {3}{2} a^2 A c^2 x^4+2 a^3 A c x^2+a^4 A \log (x)+\frac {6}{5} a^2 B c^2 x^5+\frac {4}{3} a^3 B c x^3+a^4 B x+\frac {2}{3} a A c^3 x^6+\frac {4}{7} a B c^3 x^7+\frac {1}{8} A c^4 x^8+\frac {1}{9} B c^4 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^4)/x,x]

[Out]

a^4*B*x + 2*a^3*A*c*x^2 + (4*a^3*B*c*x^3)/3 + (3*a^2*A*c^2*x^4)/2 + (6*a^2*B*c^2*x^5)/5 + (2*a*A*c^3*x^6)/3 +
(4*a*B*c^3*x^7)/7 + (A*c^4*x^8)/8 + (B*c^4*x^9)/9 + a^4*A*Log[x]

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^4}{x} \, dx &=\int \left (a^4 B+\frac {a^4 A}{x}+4 a^3 A c x+4 a^3 B c x^2+6 a^2 A c^2 x^3+6 a^2 B c^2 x^4+4 a A c^3 x^5+4 a B c^3 x^6+A c^4 x^7+B c^4 x^8\right ) \, dx\\ &=a^4 B x+2 a^3 A c x^2+\frac {4}{3} a^3 B c x^3+\frac {3}{2} a^2 A c^2 x^4+\frac {6}{5} a^2 B c^2 x^5+\frac {2}{3} a A c^3 x^6+\frac {4}{7} a B c^3 x^7+\frac {1}{8} A c^4 x^8+\frac {1}{9} B c^4 x^9+a^4 A \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 110, normalized size = 1.00 \begin {gather*} a^4 A \log (x)+a^4 B x+2 a^3 A c x^2+\frac {4}{3} a^3 B c x^3+\frac {3}{2} a^2 A c^2 x^4+\frac {6}{5} a^2 B c^2 x^5+\frac {2}{3} a A c^3 x^6+\frac {4}{7} a B c^3 x^7+\frac {1}{8} A c^4 x^8+\frac {1}{9} B c^4 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^4)/x,x]

[Out]

a^4*B*x + 2*a^3*A*c*x^2 + (4*a^3*B*c*x^3)/3 + (3*a^2*A*c^2*x^4)/2 + (6*a^2*B*c^2*x^5)/5 + (2*a*A*c^3*x^6)/3 +
(4*a*B*c^3*x^7)/7 + (A*c^4*x^8)/8 + (B*c^4*x^9)/9 + a^4*A*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )^4}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^4)/x,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^4)/x, x]

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fricas [A]  time = 0.40, size = 96, normalized size = 0.87 \begin {gather*} \frac {1}{9} \, B c^{4} x^{9} + \frac {1}{8} \, A c^{4} x^{8} + \frac {4}{7} \, B a c^{3} x^{7} + \frac {2}{3} \, A a c^{3} x^{6} + \frac {6}{5} \, B a^{2} c^{2} x^{5} + \frac {3}{2} \, A a^{2} c^{2} x^{4} + \frac {4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^4/x,x, algorithm="fricas")

[Out]

1/9*B*c^4*x^9 + 1/8*A*c^4*x^8 + 4/7*B*a*c^3*x^7 + 2/3*A*a*c^3*x^6 + 6/5*B*a^2*c^2*x^5 + 3/2*A*a^2*c^2*x^4 + 4/
3*B*a^3*c*x^3 + 2*A*a^3*c*x^2 + B*a^4*x + A*a^4*log(x)

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giac [A]  time = 0.15, size = 97, normalized size = 0.88 \begin {gather*} \frac {1}{9} \, B c^{4} x^{9} + \frac {1}{8} \, A c^{4} x^{8} + \frac {4}{7} \, B a c^{3} x^{7} + \frac {2}{3} \, A a c^{3} x^{6} + \frac {6}{5} \, B a^{2} c^{2} x^{5} + \frac {3}{2} \, A a^{2} c^{2} x^{4} + \frac {4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^4/x,x, algorithm="giac")

[Out]

1/9*B*c^4*x^9 + 1/8*A*c^4*x^8 + 4/7*B*a*c^3*x^7 + 2/3*A*a*c^3*x^6 + 6/5*B*a^2*c^2*x^5 + 3/2*A*a^2*c^2*x^4 + 4/
3*B*a^3*c*x^3 + 2*A*a^3*c*x^2 + B*a^4*x + A*a^4*log(abs(x))

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maple [A]  time = 0.04, size = 97, normalized size = 0.88 \begin {gather*} \frac {B \,c^{4} x^{9}}{9}+\frac {A \,c^{4} x^{8}}{8}+\frac {4 B a \,c^{3} x^{7}}{7}+\frac {2 A a \,c^{3} x^{6}}{3}+\frac {6 B \,a^{2} c^{2} x^{5}}{5}+\frac {3 A \,a^{2} c^{2} x^{4}}{2}+\frac {4 B \,a^{3} c \,x^{3}}{3}+2 A \,a^{3} c \,x^{2}+A \,a^{4} \ln \relax (x )+B \,a^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^4/x,x)

[Out]

a^4*B*x+2*a^3*A*c*x^2+4/3*a^3*B*c*x^3+3/2*a^2*A*c^2*x^4+6/5*a^2*B*c^2*x^5+2/3*a*A*c^3*x^6+4/7*a*B*c^3*x^7+1/8*
A*c^4*x^8+1/9*B*c^4*x^9+a^4*A*ln(x)

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maxima [A]  time = 0.48, size = 96, normalized size = 0.87 \begin {gather*} \frac {1}{9} \, B c^{4} x^{9} + \frac {1}{8} \, A c^{4} x^{8} + \frac {4}{7} \, B a c^{3} x^{7} + \frac {2}{3} \, A a c^{3} x^{6} + \frac {6}{5} \, B a^{2} c^{2} x^{5} + \frac {3}{2} \, A a^{2} c^{2} x^{4} + \frac {4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^4/x,x, algorithm="maxima")

[Out]

1/9*B*c^4*x^9 + 1/8*A*c^4*x^8 + 4/7*B*a*c^3*x^7 + 2/3*A*a*c^3*x^6 + 6/5*B*a^2*c^2*x^5 + 3/2*A*a^2*c^2*x^4 + 4/
3*B*a^3*c*x^3 + 2*A*a^3*c*x^2 + B*a^4*x + A*a^4*log(x)

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mupad [B]  time = 0.05, size = 96, normalized size = 0.87 \begin {gather*} \frac {A\,c^4\,x^8}{8}+\frac {B\,c^4\,x^9}{9}+A\,a^4\,\ln \relax (x)+B\,a^4\,x+2\,A\,a^3\,c\,x^2+\frac {2\,A\,a\,c^3\,x^6}{3}+\frac {4\,B\,a^3\,c\,x^3}{3}+\frac {4\,B\,a\,c^3\,x^7}{7}+\frac {3\,A\,a^2\,c^2\,x^4}{2}+\frac {6\,B\,a^2\,c^2\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^4*(A + B*x))/x,x)

[Out]

(A*c^4*x^8)/8 + (B*c^4*x^9)/9 + A*a^4*log(x) + B*a^4*x + 2*A*a^3*c*x^2 + (2*A*a*c^3*x^6)/3 + (4*B*a^3*c*x^3)/3
+ (4*B*a*c^3*x^7)/7 + (3*A*a^2*c^2*x^4)/2 + (6*B*a^2*c^2*x^5)/5

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sympy [A]  time = 0.22, size = 117, normalized size = 1.06 \begin {gather*} A a^{4} \log {\relax (x )} + 2 A a^{3} c x^{2} + \frac {3 A a^{2} c^{2} x^{4}}{2} + \frac {2 A a c^{3} x^{6}}{3} + \frac {A c^{4} x^{8}}{8} + B a^{4} x + \frac {4 B a^{3} c x^{3}}{3} + \frac {6 B a^{2} c^{2} x^{5}}{5} + \frac {4 B a c^{3} x^{7}}{7} + \frac {B c^{4} x^{9}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**4/x,x)

[Out]

A*a**4*log(x) + 2*A*a**3*c*x**2 + 3*A*a**2*c**2*x**4/2 + 2*A*a*c**3*x**6/3 + A*c**4*x**8/8 + B*a**4*x + 4*B*a*
*3*c*x**3/3 + 6*B*a**2*c**2*x**5/5 + 4*B*a*c**3*x**7/7 + B*c**4*x**9/9

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