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

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

[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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Rubi [A]  time = 0.105935, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a^4 A \log (x)+a^4 B x+2 a^3 A c x^2+\frac{4}{3} a^3 B c x^3+\frac{3}{2} a^2 A c^2 x^4+\frac{6}{5} a^2 B c^2 x^5+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]

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]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a^{4} \log{\left (x \right )} + 4 A a^{3} c \int x\, dx + \frac{3 A a^{2} c^{2} x^{4}}{2} + \frac{2 A a c^{3} x^{6}}{3} + \frac{A c^{4} x^{8}}{8} + \frac{4 B a^{3} c x^{3}}{3} + \frac{6 B a^{2} c^{2} x^{5}}{5} + \frac{4 B a c^{3} x^{7}}{7} + \frac{B c^{4} x^{9}}{9} + a^{4} \int B\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**4/x,x)

[Out]

A*a**4*log(x) + 4*A*a**3*c*Integral(x, x) + 3*A*a**2*c**2*x**4/2 + 2*A*a*c**3*x*
*6/3 + A*c**4*x**8/8 + 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 + a**4*Integral(B, x)

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Mathematica [A]  time = 0.0128156, size = 110, normalized size = 1. \[ a^4 A \log (x)+a^4 B x+2 a^3 A c x^2+\frac{4}{3} a^3 B c x^3+\frac{3}{2} a^2 A c^2 x^4+\frac{6}{5} a^2 B c^2 x^5+\frac{2}{3} a A c^3 x^6+\frac{4}{7} a B c^3 x^7+\frac{1}{8} A c^4 x^8+\frac{1}{9} B c^4 x^9 \]

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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Maple [A]  time = 0.004, size = 97, normalized size = 0.9 \[{a}^{4}Bx+2\,{a}^{3}Ac{x}^{2}+{\frac{4\,{a}^{3}Bc{x}^{3}}{3}}+{\frac{3\,{a}^{2}A{c}^{2}{x}^{4}}{2}}+{\frac{6\,{a}^{2}B{c}^{2}{x}^{5}}{5}}+{\frac{2\,aA{c}^{3}{x}^{6}}{3}}+{\frac{4\,aB{c}^{3}{x}^{7}}{7}}+{\frac{A{c}^{4}{x}^{8}}{8}}+{\frac{B{c}^{4}{x}^{9}}{9}}+{a}^{4}A\ln \left ( x \right ) \]

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 = 0.685199, size = 130, normalized size = 1.18 \[ \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^4*(B*x + A)/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*lo
g(x)

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Fricas [A]  time = 0.267559, size = 130, normalized size = 1.18 \[ \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^4*(B*x + A)/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*lo
g(x)

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Sympy [A]  time = 1.41325, size = 117, normalized size = 1.06 \[ A a^{4} \log{\left (x \right )} + 2 A a^{3} c x^{2} + \frac{3 A a^{2} c^{2} x^{4}}{2} + \frac{2 A a c^{3} x^{6}}{3} + \frac{A c^{4} x^{8}}{8} + B a^{4} x + \frac{4 B a^{3} c x^{3}}{3} + \frac{6 B a^{2} c^{2} x^{5}}{5} + \frac{4 B a c^{3} x^{7}}{7} + \frac{B c^{4} x^{9}}{9} \]

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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GIAC/XCAS [A]  time = 0.278248, size = 131, normalized size = 1.19 \[ \frac{1}{9} \, B c^{4} x^{9} + \frac{1}{8} \, A c^{4} x^{8} + \frac{4}{7} \, B a c^{3} x^{7} + \frac{2}{3} \, A a c^{3} x^{6} + \frac{6}{5} \, B a^{2} c^{2} x^{5} + \frac{3}{2} \, A a^{2} c^{2} x^{4} + \frac{4}{3} \, B a^{3} c x^{3} + 2 \, A a^{3} c x^{2} + B a^{4} x + A a^{4}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^4*(B*x + A)/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*ln
(abs(x))