3.1.23 \(\int \frac {(A+B x) (b x+c x^2)^2}{x^5} \, dx\) [23]

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

[Out]

-1/2*A*b^2/x^2-b*(2*A*c+B*b)/x+B*c^2*x+c*(A*c+2*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}{2 x^2}-\frac {b (2 A c+b B)}{x}+c \log (x) (A c+2 b B)+B c^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/x^5,x]

[Out]

-1/2*(A*b^2)/x^2 - (b*(b*B + 2*A*c))/x + B*c^2*x + c*(2*b*B + 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^5} \, dx &=\int \left (B c^2+\frac {A b^2}{x^3}+\frac {b (b B+2 A c)}{x^2}+\frac {c (2 b B+A c)}{x}\right ) \, dx\\ &=-\frac {A b^2}{2 x^2}-\frac {b (b B+2 A c)}{x}+B c^2 x+c (2 b B+A c) \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 44, normalized size = 1.00 \begin {gather*} -\frac {A b^2}{2 x^2}-\frac {b (b B+2 A c)}{x}+B c^2 x+c (2 b B+A c) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^5,x]

[Out]

-1/2*(A*b^2)/x^2 - (b*(b*B + 2*A*c))/x + B*c^2*x + c*(2*b*B + A*c)*Log[x]

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Maple [A]
time = 0.51, size = 43, normalized size = 0.98

method	result	size
default	\(-\frac {A \,b^{2}}{2 x^{2}}-\frac {b \left (2 A c +B b \right )}{x}+B \,c^{2} x +c \left (A c +2 B b \right ) \ln \left (x \right )\)	\(43\)
risch	\(B \,c^{2} x +\frac {\left (-2 A b c -b^{2} B \right ) x -\frac {b^{2} A}{2}}{x^{2}}+A \ln \left (x \right ) c^{2}+2 B \ln \left (x \right ) b c\)	\(47\)
norman	\(\frac {\left (-2 A b c -b^{2} B \right ) x^{3}+B \,c^{2} x^{5}-\frac {b^{2} A \,x^{2}}{2}}{x^{4}}+\left (A \,c^{2}+2 b B c \right ) \ln \left (x \right )\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/x^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 46, normalized size = 1.05 \begin {gather*} B c^{2} x + {\left (2 \, B b c + A c^{2}\right )} \log \left (x\right ) - \frac {A b^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} x}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.48, size = 53, normalized size = 1.20 \begin {gather*} \frac {2 \, B c^{2} x^{3} + 2 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} \log \left (x\right ) - A b^{2} - 2 \, {\left (B b^{2} + 2 \, A b c\right )} x}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.17, size = 46, normalized size = 1.05 \begin {gather*} B c^{2} x + c \left (A c + 2 B b\right ) \log {\left (x \right )} + \frac {- A b^{2} + x \left (- 4 A b c - 2 B b^{2}\right )}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/x**5,x)

[Out]

B*c**2*x + c*(A*c + 2*B*b)*log(x) + (-A*b**2 + x*(-4*A*b*c - 2*B*b**2))/(2*x**2)

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Giac [A]
time = 0.99, size = 47, normalized size = 1.07 \begin {gather*} B c^{2} x + {\left (2 \, B b c + A c^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A b^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} x}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^5,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.04, size = 46, normalized size = 1.05 \begin {gather*} \ln \left (x\right )\,\left (A\,c^2+2\,B\,b\,c\right )-\frac {\frac {A\,b^2}{2}+x\,\left (B\,b^2+2\,A\,c\,b\right )}{x^2}+B\,c^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^2*(A + B*x))/x^5,x)

[Out]

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

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