3.1.22 \(\int \frac {(A+B x) (b x+c x^2)^2}{x^4} \, dx\) [22]

Optimal. Leaf size=44 \[ -\frac {A b^2}{x}+c (2 b B+A c) x+\frac {1}{2} B c^2 x^2+b (b B+2 A c) \log (x) \]

[Out]

-A*b^2/x+c*(A*c+2*B*b)*x+1/2*B*c^2*x^2+b*(2*A*c+B*b)*ln(x)

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Rubi [A]
time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {779} \begin {gather*} -\frac {A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac {1}{2} B c^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x^4,x]

[Out]

-((A*b^2)/x) + c*(2*b*B + A*c)*x + (B*c^2*x^2)/2 + b*(b*B + 2*A*c)*Log[x]

Rule 779

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{x^4} \, dx &=\int \left (c (2 b B+A c)+\frac {A b^2}{x^2}+\frac {b (b B+2 A c)}{x}+B c^2 x\right ) \, dx\\ &=-\frac {A b^2}{x}+c (2 b B+A c) x+\frac {1}{2} B c^2 x^2+b (b B+2 A c) \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 43, normalized size = 0.98 \begin {gather*} \frac {1}{2} B c x (4 b+c x)+A \left (-\frac {b^2}{x}+c^2 x\right )+b (b B+2 A c) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*c*x*(4*b + c*x))/2 + A*(-(b^2/x) + c^2*x) + b*(b*B + 2*A*c)*Log[x]

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Maple [A]
time = 0.50, size = 44, normalized size = 1.00

method	result	size
default	\(\frac {B \,c^{2} x^{2}}{2}+A \,c^{2} x +2 b B c x +b \left (2 A c +B b \right ) \ln \left (x \right )-\frac {A \,b^{2}}{x}\)	\(44\)
risch	\(\frac {B \,c^{2} x^{2}}{2}+A \,c^{2} x +2 b B c x -\frac {A \,b^{2}}{x}+2 A \ln \left (x \right ) b c +b^{2} B \ln \left (x \right )\)	\(46\)
norman	\(\frac {\left (A \,c^{2}+2 b B c \right ) x^{4}+\frac {B \,c^{2} x^{5}}{2}-b^{2} A \,x^{2}}{x^{3}}+\left (2 A b c +b^{2} B \right ) \ln \left (x \right )\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

1/2*B*c^2*x^2+A*c^2*x+2*b*B*c*x+b*(2*A*c+B*b)*ln(x)-A*b^2/x

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Maxima [A]
time = 0.29, size = 46, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, B c^{2} x^{2} - \frac {A b^{2}}{x} + {\left (2 \, B b c + A c^{2}\right )} x + {\left (B b^{2} + 2 \, A b c\right )} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*c^2*x^2 - A*b^2/x + (2*B*b*c + A*c^2)*x + (B*b^2 + 2*A*b*c)*log(x)

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Fricas [A]
time = 3.10, size = 52, normalized size = 1.18 \begin {gather*} \frac {B c^{2} x^{3} - 2 \, A b^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} x \log \left (x\right )}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(B*c^2*x^3 - 2*A*b^2 + 2*(2*B*b*c + A*c^2)*x^2 + 2*(B*b^2 + 2*A*b*c)*x*log(x))/x

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Sympy [A]
time = 0.08, size = 42, normalized size = 0.95 \begin {gather*} - \frac {A b^{2}}{x} + \frac {B c^{2} x^{2}}{2} + b \left (2 A c + B b\right ) \log {\left (x \right )} + x \left (A c^{2} + 2 B b c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*b**2/x + B*c**2*x**2/2 + b*(2*A*c + B*b)*log(x) + x*(A*c**2 + 2*B*b*c)

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Giac [A]
time = 1.22, size = 46, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, B c^{2} x^{2} + 2 \, B b c x + A c^{2} x - \frac {A b^{2}}{x} + {\left (B b^{2} + 2 \, A b c\right )} \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+b*x)^2/x^4,x, algorithm="giac")

[Out]

1/2*B*c^2*x^2 + 2*B*b*c*x + A*c^2*x - A*b^2/x + (B*b^2 + 2*A*b*c)*log(abs(x))

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Mupad [B]
time = 0.05, size = 46, normalized size = 1.05 \begin {gather*} x\,\left (A\,c^2+2\,B\,b\,c\right )+\ln \left (x\right )\,\left (B\,b^2+2\,A\,c\,b\right )-\frac {A\,b^2}{x}+\frac {B\,c^2\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(A*c^2 + 2*B*b*c) + log(x)*(B*b^2 + 2*A*b*c) - (A*b^2)/x + (B*c^2*x^2)/2

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