3.16.1 \(\int \frac {(b+2 c x) (a+b x+c x^2)}{(d+e x)^3} \, dx\) [1501]

Optimal. Leaf size=111 \[ \frac {2 c^2 x}{e^3}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{2 e^4 (d+e x)^2}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)}-\frac {3 c (2 c d-b e) \log (d+e x)}{e^4} \]

[Out]

2*c^2*x/e^3+1/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)/e^4/(e*x+d)^2+(-6*c^2*d^2-b^2*e^2+2*c*e*(-a*e+3*b*d))/e^4/(e*
x+d)-3*c*(-b*e+2*c*d)*ln(e*x+d)/e^4

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Rubi [A]
time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \begin {gather*} -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^4 (d+e x)}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^4 (d+e x)^2}-\frac {3 c (2 c d-b e) \log (d+e x)}{e^4}+\frac {2 c^2 x}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^3,x]

[Out]

(2*c^2*x)/e^3 + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(2*e^4*(d + e*x)^2) - (6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*
b*d - a*e))/(e^4*(d + e*x)) - (3*c*(2*c*d - b*e)*Log[d + e*x])/e^4

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^3} \, dx &=\int \left (\frac {2 c^2}{e^3}+\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^3}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 (d+e x)^2}-\frac {3 c (2 c d-b e)}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {2 c^2 x}{e^3}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{2 e^4 (d+e x)^2}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)}-\frac {3 c (2 c d-b e) \log (d+e x)}{e^4}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 118, normalized size = 1.06 \begin {gather*} \frac {c^2 \left (-10 d^3-8 d^2 e x+8 d e^2 x^2+4 e^3 x^3\right )-b e^2 (a e+b (d+2 e x))+c e (-2 a e (d+2 e x)+3 b d (3 d+4 e x))-6 c (2 c d-b e) (d+e x)^2 \log (d+e x)}{2 e^4 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^3,x]

[Out]

(c^2*(-10*d^3 - 8*d^2*e*x + 8*d*e^2*x^2 + 4*e^3*x^3) - b*e^2*(a*e + b*(d + 2*e*x)) + c*e*(-2*a*e*(d + 2*e*x) +
3*b*d*(3*d + 4*e*x)) - 6*c*(2*c*d - b*e)*(d + e*x)^2*Log[d + e*x])/(2*e^4*(d + e*x)^2)

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Maple [A]
time = 0.57, size = 124, normalized size = 1.12

method	result	size
norman	\(\frac {-\frac {a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}-9 d^{2} e b c +18 c^{2} d^{3}}{2 e^{4}}+\frac {2 c^{2} x^{3}}{e}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +12 c^{2} d^{2}\right ) x}{e^{3}}}{\left (e x +d \right )^{2}}+\frac {3 c \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{4}}\)	\(121\)
default	\(\frac {2 c^{2} x}{e^{3}}-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{e^{4} \left (e x +d \right )}+\frac {3 c \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{4}}-\frac {a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}}{2 e^{4} \left (e x +d \right )^{2}}\)	\(124\)
risch	\(\frac {2 c^{2} x}{e^{3}}+\frac {\left (-2 a c \,e^{2}-b^{2} e^{2}+6 b c d e -6 c^{2} d^{2}\right ) x -\frac {a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}-9 d^{2} e b c +10 c^{2} d^{3}}{2 e}}{e^{3} \left (e x +d \right )^{2}}+\frac {3 c \ln \left (e x +d \right ) b}{e^{3}}-\frac {6 c^{2} \ln \left (e x +d \right ) d}{e^{4}}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

2*c^2*x/e^3-1/e^4*(2*a*c*e^2+b^2*e^2-6*b*c*d*e+6*c^2*d^2)/(e*x+d)+3*c*(b*e-2*c*d)/e^4*ln(e*x+d)-1/2*(a*b*e^3-2
*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^4/(e*x+d)^2

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Maxima [A]
time = 0.29, size = 127, normalized size = 1.14 \begin {gather*} 2 \, c^{2} x e^{\left (-3\right )} - 3 \, {\left (2 \, c^{2} d - b c e\right )} e^{\left (-4\right )} \log \left (x e + d\right ) - \frac {10 \, c^{2} d^{3} - 9 \, b c d^{2} e + a b e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d + 2 \, {\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} + b^{2} e^{3} + 2 \, a c e^{3}\right )} x}{2 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c^2*x*e^(-3) - 3*(2*c^2*d - b*c*e)*e^(-4)*log(x*e + d) - 1/2*(10*c^2*d^3 - 9*b*c*d^2*e + a*b*e^3 + (b^2*e^2
+ 2*a*c*e^2)*d + 2*(6*c^2*d^2*e - 6*b*c*d*e^2 + b^2*e^3 + 2*a*c*e^3)*x)/(x^2*e^6 + 2*d*x*e^5 + d^2*e^4)

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Fricas [A]
time = 2.55, size = 177, normalized size = 1.59 \begin {gather*} -\frac {10 \, c^{2} d^{3} - {\left (4 \, c^{2} x^{3} - a b - 2 \, {\left (b^{2} + 2 \, a c\right )} x\right )} e^{3} - {\left (8 \, c^{2} d x^{2} + 12 \, b c d x - {\left (b^{2} + 2 \, a c\right )} d\right )} e^{2} + {\left (8 \, c^{2} d^{2} x - 9 \, b c d^{2}\right )} e + 6 \, {\left (2 \, c^{2} d^{3} - b c x^{2} e^{3} + 2 \, {\left (c^{2} d x^{2} - b c d x\right )} e^{2} + {\left (4 \, c^{2} d^{2} x - b c d^{2}\right )} e\right )} \log \left (x e + d\right )}{2 \, {\left (x^{2} e^{6} + 2 \, d x e^{5} + d^{2} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(10*c^2*d^3 - (4*c^2*x^3 - a*b - 2*(b^2 + 2*a*c)*x)*e^3 - (8*c^2*d*x^2 + 12*b*c*d*x - (b^2 + 2*a*c)*d)*e^
2 + (8*c^2*d^2*x - 9*b*c*d^2)*e + 6*(2*c^2*d^3 - b*c*x^2*e^3 + 2*(c^2*d*x^2 - b*c*d*x)*e^2 + (4*c^2*d^2*x - b*
c*d^2)*e)*log(x*e + d))/(x^2*e^6 + 2*d*x*e^5 + d^2*e^4)

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Sympy [A]
time = 0.84, size = 139, normalized size = 1.25 \begin {gather*} \frac {2 c^{2} x}{e^{3}} + \frac {3 c \left (b e - 2 c d\right ) \log {\left (d + e x \right )}}{e^{4}} + \frac {- a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} + 9 b c d^{2} e - 10 c^{2} d^{3} + x \left (- 4 a c e^{3} - 2 b^{2} e^{3} + 12 b c d e^{2} - 12 c^{2} d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)/(e*x+d)**3,x)

[Out]

2*c**2*x/e**3 + 3*c*(b*e - 2*c*d)*log(d + e*x)/e**4 + (-a*b*e**3 - 2*a*c*d*e**2 - b**2*d*e**2 + 9*b*c*d**2*e -
10*c**2*d**3 + x*(-4*a*c*e**3 - 2*b**2*e**3 + 12*b*c*d*e**2 - 12*c**2*d**2*e))/(2*d**2*e**4 + 4*d*e**5*x + 2*
e**6*x**2)

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Giac [A]
time = 1.70, size = 116, normalized size = 1.05 \begin {gather*} 2 \, c^{2} x e^{\left (-3\right )} - 3 \, {\left (2 \, c^{2} d - b c e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (10 \, c^{2} d^{3} - 9 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} + a b e^{3} + 2 \, {\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} + b^{2} e^{3} + 2 \, a c e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c^2*x*e^(-3) - 3*(2*c^2*d - b*c*e)*e^(-4)*log(abs(x*e + d)) - 1/2*(10*c^2*d^3 - 9*b*c*d^2*e + b^2*d*e^2 + 2*
a*c*d*e^2 + a*b*e^3 + 2*(6*c^2*d^2*e - 6*b*c*d*e^2 + b^2*e^3 + 2*a*c*e^3)*x)*e^(-4)/(x*e + d)^2

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Mupad [B]
time = 0.11, size = 135, normalized size = 1.22 \begin {gather*} \frac {2\,c^2\,x}{e^3}-\frac {\frac {b^2\,d\,e^2-9\,b\,c\,d^2\,e+a\,b\,e^3+10\,c^2\,d^3+2\,a\,c\,d\,e^2}{2\,e}+x\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2}-\frac {\ln \left (d+e\,x\right )\,\left (6\,c^2\,d-3\,b\,c\,e\right )}{e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^3,x)

[Out]

(2*c^2*x)/e^3 - ((10*c^2*d^3 + b^2*d*e^2 + a*b*e^3 + 2*a*c*d*e^2 - 9*b*c*d^2*e)/(2*e) + x*(b^2*e^2 + 6*c^2*d^2
+ 2*a*c*e^2 - 6*b*c*d*e))/(d^2*e^3 + e^5*x^2 + 2*d*e^4*x) - (log(d + e*x)*(6*c^2*d - 3*b*c*e))/e^4

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