3.15.78 \(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^9} \, dx\) [1478]

Optimal. Leaf size=117 \[ -\frac {(b d-a e)^4}{8 e^5 (d+e x)^8}+\frac {4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac {b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac {4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac {b^4}{4 e^5 (d+e x)^4} \]

[Out]

-1/8*(-a*e+b*d)^4/e^5/(e*x+d)^8+4/7*b*(-a*e+b*d)^3/e^5/(e*x+d)^7-b^2*(-a*e+b*d)^2/e^5/(e*x+d)^6+4/5*b^3*(-a*e+
b*d)/e^5/(e*x+d)^5-1/4*b^4/e^5/(e*x+d)^4

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Rubi [A]
time = 0.05, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \begin {gather*} \frac {4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac {b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac {4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac {(b d-a e)^4}{8 e^5 (d+e x)^8}-\frac {b^4}{4 e^5 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^9,x]

[Out]

-1/8*(b*d - a*e)^4/(e^5*(d + e*x)^8) + (4*b*(b*d - a*e)^3)/(7*e^5*(d + e*x)^7) - (b^2*(b*d - a*e)^2)/(e^5*(d +
e*x)^6) + (4*b^3*(b*d - a*e))/(5*e^5*(d + e*x)^5) - b^4/(4*e^5*(d + e*x)^4)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^9} \, dx\\ &=\int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^9}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^8}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^7}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^6}+\frac {b^4}{e^4 (d+e x)^5}\right ) \, dx\\ &=-\frac {(b d-a e)^4}{8 e^5 (d+e x)^8}+\frac {4 b (b d-a e)^3}{7 e^5 (d+e x)^7}-\frac {b^2 (b d-a e)^2}{e^5 (d+e x)^6}+\frac {4 b^3 (b d-a e)}{5 e^5 (d+e x)^5}-\frac {b^4}{4 e^5 (d+e x)^4}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 144, normalized size = 1.23 \begin {gather*} -\frac {35 a^4 e^4+20 a^3 b e^3 (d+8 e x)+10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a b^3 e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )}{280 e^5 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^9,x]

[Out]

-1/280*(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a*b^3*e*(d^3 +
8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^4*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4))/
(e^5*(d + e*x)^8)

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Maple [A]
time = 0.62, size = 186, normalized size = 1.59

method	result	size
risch	\(\frac {-\frac {b^{4} x^{4}}{4 e}-\frac {b^{3} \left (4 a e +b d \right ) x^{3}}{5 e^{2}}-\frac {b^{2} \left (10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) x^{2}}{10 e^{3}}-\frac {b \left (20 e^{3} a^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{35 e^{4}}-\frac {35 e^{4} a^{4}+20 a^{3} b d \,e^{3}+10 a^{2} b^{2} d^{2} e^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}}{280 e^{5}}}{\left (e x +d \right )^{8}}\)	\(171\)
gosper	\(-\frac {70 b^{4} x^{4} e^{4}+224 a \,b^{3} e^{4} x^{3}+56 b^{4} d \,e^{3} x^{3}+280 a^{2} b^{2} e^{4} x^{2}+112 a \,b^{3} d \,e^{3} x^{2}+28 b^{4} d^{2} e^{2} x^{2}+160 a^{3} b \,e^{4} x +80 a^{2} b^{2} d \,e^{3} x +32 a \,b^{3} d^{2} e^{2} x +8 b^{4} d^{3} e x +35 e^{4} a^{4}+20 a^{3} b d \,e^{3}+10 a^{2} b^{2} d^{2} e^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}}{280 e^{5} \left (e x +d \right )^{8}}\)	\(185\)
default	\(-\frac {4 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {4 b^{3} \left (a e -b d \right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{5} \left (e x +d \right )^{6}}-\frac {b^{4}}{4 e^{5} \left (e x +d \right )^{4}}-\frac {e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{8 e^{5} \left (e x +d \right )^{8}}\)	\(186\)
norman	\(\frac {-\frac {b^{4} x^{4}}{4 e}-\frac {\left (4 a \,b^{3} e^{4}+b^{4} d \,e^{3}\right ) x^{3}}{5 e^{5}}-\frac {\left (10 a^{2} b^{2} e^{5}+4 a \,b^{3} d \,e^{4}+b^{4} d^{2} e^{3}\right ) x^{2}}{10 e^{6}}-\frac {\left (20 a^{3} b \,e^{6}+10 a^{2} b^{2} d \,e^{5}+4 a \,b^{3} d^{2} e^{4}+b^{4} d^{3} e^{3}\right ) x}{35 e^{7}}-\frac {35 e^{7} a^{4}+20 a^{3} b d \,e^{6}+10 a^{2} b^{2} d^{2} e^{5}+4 a \,b^{3} d^{3} e^{4}+b^{4} d^{4} e^{3}}{280 e^{8}}}{\left (e x +d \right )^{8}}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^9,x,method=_RETURNVERBOSE)

[Out]

-4/7*b/e^5*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/(e*x+d)^7-4/5*b^3/e^5*(a*e-b*d)/(e*x+d)^5-b^2/e^5*(a^
2*e^2-2*a*b*d*e+b^2*d^2)/(e*x+d)^6-1/4*b^4/e^5/(e*x+d)^4-1/8*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*
d^3*e+b^4*d^4)/e^5/(e*x+d)^8

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (113) = 226\).
time = 0.29, size = 239, normalized size = 2.04 \begin {gather*} -\frac {70 \, b^{4} x^{4} e^{4} + b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 10 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 56 \, {\left (b^{4} d e^{3} + 4 \, a b^{3} e^{4}\right )} x^{3} + 28 \, {\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 10 \, a^{2} b^{2} e^{4}\right )} x^{2} + 8 \, {\left (b^{4} d^{3} e + 4 \, a b^{3} d^{2} e^{2} + 10 \, a^{2} b^{2} d e^{3} + 20 \, a^{3} b e^{4}\right )} x}{280 \, {\left (x^{8} e^{13} + 8 \, d x^{7} e^{12} + 28 \, d^{2} x^{6} e^{11} + 56 \, d^{3} x^{5} e^{10} + 70 \, d^{4} x^{4} e^{9} + 56 \, d^{5} x^{3} e^{8} + 28 \, d^{6} x^{2} e^{7} + 8 \, d^{7} x e^{6} + d^{8} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^9,x, algorithm="maxima")

[Out]

-1/280*(70*b^4*x^4*e^4 + b^4*d^4 + 4*a*b^3*d^3*e + 10*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + 35*a^4*e^4 + 56*(b^4*
d*e^3 + 4*a*b^3*e^4)*x^3 + 28*(b^4*d^2*e^2 + 4*a*b^3*d*e^3 + 10*a^2*b^2*e^4)*x^2 + 8*(b^4*d^3*e + 4*a*b^3*d^2*
e^2 + 10*a^2*b^2*d*e^3 + 20*a^3*b*e^4)*x)/(x^8*e^13 + 8*d*x^7*e^12 + 28*d^2*x^6*e^11 + 56*d^3*x^5*e^10 + 70*d^
4*x^4*e^9 + 56*d^5*x^3*e^8 + 28*d^6*x^2*e^7 + 8*d^7*x*e^6 + d^8*e^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (113) = 226\).
time = 3.33, size = 235, normalized size = 2.01 \begin {gather*} -\frac {b^{4} d^{4} + {\left (70 \, b^{4} x^{4} + 224 \, a b^{3} x^{3} + 280 \, a^{2} b^{2} x^{2} + 160 \, a^{3} b x + 35 \, a^{4}\right )} e^{4} + 4 \, {\left (14 \, b^{4} d x^{3} + 28 \, a b^{3} d x^{2} + 20 \, a^{2} b^{2} d x + 5 \, a^{3} b d\right )} e^{3} + 2 \, {\left (14 \, b^{4} d^{2} x^{2} + 16 \, a b^{3} d^{2} x + 5 \, a^{2} b^{2} d^{2}\right )} e^{2} + 4 \, {\left (2 \, b^{4} d^{3} x + a b^{3} d^{3}\right )} e}{280 \, {\left (x^{8} e^{13} + 8 \, d x^{7} e^{12} + 28 \, d^{2} x^{6} e^{11} + 56 \, d^{3} x^{5} e^{10} + 70 \, d^{4} x^{4} e^{9} + 56 \, d^{5} x^{3} e^{8} + 28 \, d^{6} x^{2} e^{7} + 8 \, d^{7} x e^{6} + d^{8} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/280*(b^4*d^4 + (70*b^4*x^4 + 224*a*b^3*x^3 + 280*a^2*b^2*x^2 + 160*a^3*b*x + 35*a^4)*e^4 + 4*(14*b^4*d*x^3
+ 28*a*b^3*d*x^2 + 20*a^2*b^2*d*x + 5*a^3*b*d)*e^3 + 2*(14*b^4*d^2*x^2 + 16*a*b^3*d^2*x + 5*a^2*b^2*d^2)*e^2 +
4*(2*b^4*d^3*x + a*b^3*d^3)*e)/(x^8*e^13 + 8*d*x^7*e^12 + 28*d^2*x^6*e^11 + 56*d^3*x^5*e^10 + 70*d^4*x^4*e^9
+ 56*d^5*x^3*e^8 + 28*d^6*x^2*e^7 + 8*d^7*x*e^6 + d^8*e^5)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**9,x)

[Out]

Timed out

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Giac [A]
time = 1.29, size = 174, normalized size = 1.49 \begin {gather*} -\frac {{\left (70 \, b^{4} x^{4} e^{4} + 56 \, b^{4} d x^{3} e^{3} + 28 \, b^{4} d^{2} x^{2} e^{2} + 8 \, b^{4} d^{3} x e + b^{4} d^{4} + 224 \, a b^{3} x^{3} e^{4} + 112 \, a b^{3} d x^{2} e^{3} + 32 \, a b^{3} d^{2} x e^{2} + 4 \, a b^{3} d^{3} e + 280 \, a^{2} b^{2} x^{2} e^{4} + 80 \, a^{2} b^{2} d x e^{3} + 10 \, a^{2} b^{2} d^{2} e^{2} + 160 \, a^{3} b x e^{4} + 20 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{280 \, {\left (x e + d\right )}^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/280*(70*b^4*x^4*e^4 + 56*b^4*d*x^3*e^3 + 28*b^4*d^2*x^2*e^2 + 8*b^4*d^3*x*e + b^4*d^4 + 224*a*b^3*x^3*e^4 +
112*a*b^3*d*x^2*e^3 + 32*a*b^3*d^2*x*e^2 + 4*a*b^3*d^3*e + 280*a^2*b^2*x^2*e^4 + 80*a^2*b^2*d*x*e^3 + 10*a^2*
b^2*d^2*e^2 + 160*a^3*b*x*e^4 + 20*a^3*b*d*e^3 + 35*a^4*e^4)*e^(-5)/(x*e + d)^8

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Mupad [B]
time = 0.62, size = 248, normalized size = 2.12 \begin {gather*} -\frac {\frac {35\,a^4\,e^4+20\,a^3\,b\,d\,e^3+10\,a^2\,b^2\,d^2\,e^2+4\,a\,b^3\,d^3\,e+b^4\,d^4}{280\,e^5}+\frac {b^4\,x^4}{4\,e}+\frac {b^3\,x^3\,\left (4\,a\,e+b\,d\right )}{5\,e^2}+\frac {b\,x\,\left (20\,a^3\,e^3+10\,a^2\,b\,d\,e^2+4\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{35\,e^4}+\frac {b^2\,x^2\,\left (10\,a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )}{10\,e^3}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^2/(d + e*x)^9,x)

[Out]

-((35*a^4*e^4 + b^4*d^4 + 10*a^2*b^2*d^2*e^2 + 4*a*b^3*d^3*e + 20*a^3*b*d*e^3)/(280*e^5) + (b^4*x^4)/(4*e) + (
b^3*x^3*(4*a*e + b*d))/(5*e^2) + (b*x*(20*a^3*e^3 + b^3*d^3 + 4*a*b^2*d^2*e + 10*a^2*b*d*e^2))/(35*e^4) + (b^2
*x^2*(10*a^2*e^2 + b^2*d^2 + 4*a*b*d*e))/(10*e^3))/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5*e^3*
x^3 + 70*d^4*e^4*x^4 + 56*d^3*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x)

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