3.4.89 \(\int \frac {(a+c x^2)^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=110 \[ -\frac {2 c \left (a e^2+3 c d^2\right )}{3 e^5 (d+e x)^3}+\frac {c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^4}-\frac {\left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^5}-\frac {c^2}{e^5 (d+e x)}+\frac {2 c^2 d}{e^5 (d+e x)^2} \]

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Rubi [A]  time = 0.07, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \begin {gather*} -\frac {2 c \left (a e^2+3 c d^2\right )}{3 e^5 (d+e x)^3}+\frac {c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^4}-\frac {\left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^5}-\frac {c^2}{e^5 (d+e x)}+\frac {2 c^2 d}{e^5 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x)^6,x]

[Out]

-(c*d^2 + a*e^2)^2/(5*e^5*(d + e*x)^5) + (c*d*(c*d^2 + a*e^2))/(e^5*(d + e*x)^4) - (2*c*(3*c*d^2 + a*e^2))/(3*
e^5*(d + e*x)^3) + (2*c^2*d)/(e^5*(d + e*x)^2) - c^2/(e^5*(d + e*x))

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 90, normalized size = 0.82 \begin {gather*} -\frac {3 a^2 e^4+a c e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{15 e^5 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^2/(d + e*x)^6,x]

[Out]

-1/15*(3*a^2*e^4 + a*c*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x
^3 + 5*e^4*x^4))/(e^5*(d + e*x)^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+c x^2\right )^2}{(d+e x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + c*x^2)^2/(d + e*x)^6,x]

[Out]

IntegrateAlgebraic[(a + c*x^2)^2/(d + e*x)^6, x]

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fricas [A]  time = 0.39, size = 151, normalized size = 1.37 \begin {gather*} -\frac {15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/15*(15*c^2*e^4*x^4 + 30*c^2*d*e^3*x^3 + 3*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4 + 10*(3*c^2*d^2*e^2 + a*c*e^4)*
x^2 + 5*(3*c^2*d^3*e + a*c*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6*x +
d^5*e^5)

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giac [A]  time = 0.16, size = 98, normalized size = 0.89 \begin {gather*} -\frac {{\left (15 \, c^{2} x^{4} e^{4} + 30 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 15 \, c^{2} d^{3} x e + 3 \, c^{2} d^{4} + 10 \, a c x^{2} e^{4} + 5 \, a c d x e^{3} + a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \, {\left (x e + d\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/15*(15*c^2*x^4*e^4 + 30*c^2*d*x^3*e^3 + 30*c^2*d^2*x^2*e^2 + 15*c^2*d^3*x*e + 3*c^2*d^4 + 10*a*c*x^2*e^4 +
5*a*c*d*x*e^3 + a*c*d^2*e^2 + 3*a^2*e^4)*e^(-5)/(x*e + d)^5

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maple [A]  time = 0.05, size = 119, normalized size = 1.08 \begin {gather*} \frac {2 c^{2} d}{\left (e x +d \right )^{2} e^{5}}-\frac {c^{2}}{\left (e x +d \right ) e^{5}}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) c d}{\left (e x +d \right )^{4} e^{5}}-\frac {2 \left (a \,e^{2}+3 c \,d^{2}\right ) c}{3 \left (e x +d \right )^{3} e^{5}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{5 \left (e x +d \right )^{5} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^2/(e*x+d)^6,x)

[Out]

2*c^2*d/e^5/(e*x+d)^2+c*d*(a*e^2+c*d^2)/e^5/(e*x+d)^4-1/(e*x+d)*c^2/e^5-2/3*c*(a*e^2+3*c*d^2)/e^5/(e*x+d)^3-1/
5*(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/e^5/(e*x+d)^5

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maxima [A]  time = 1.42, size = 151, normalized size = 1.37 \begin {gather*} -\frac {15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/15*(15*c^2*e^4*x^4 + 30*c^2*d*e^3*x^3 + 3*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4 + 10*(3*c^2*d^2*e^2 + a*c*e^4)*
x^2 + 5*(3*c^2*d^3*e + a*c*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6*x +
d^5*e^5)

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mupad [B]  time = 0.07, size = 148, normalized size = 1.35 \begin {gather*} -\frac {\frac {3\,a^2\,e^4+a\,c\,d^2\,e^2+3\,c^2\,d^4}{15\,e^5}+\frac {c^2\,x^4}{e}+\frac {2\,c^2\,d\,x^3}{e^2}+\frac {2\,c\,x^2\,\left (3\,c\,d^2+a\,e^2\right )}{3\,e^3}+\frac {c\,d\,x\,\left (3\,c\,d^2+a\,e^2\right )}{3\,e^4}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2/(d + e*x)^6,x)

[Out]

-((3*a^2*e^4 + 3*c^2*d^4 + a*c*d^2*e^2)/(15*e^5) + (c^2*x^4)/e + (2*c^2*d*x^3)/e^2 + (2*c*x^2*(a*e^2 + 3*c*d^2
))/(3*e^3) + (c*d*x*(a*e^2 + 3*c*d^2))/(3*e^4))/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d^2*e^3*x^3
+ 5*d^4*e*x)

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sympy [A]  time = 1.87, size = 162, normalized size = 1.47 \begin {gather*} \frac {- 3 a^{2} e^{4} - a c d^{2} e^{2} - 3 c^{2} d^{4} - 30 c^{2} d e^{3} x^{3} - 15 c^{2} e^{4} x^{4} + x^{2} \left (- 10 a c e^{4} - 30 c^{2} d^{2} e^{2}\right ) + x \left (- 5 a c d e^{3} - 15 c^{2} d^{3} e\right )}{15 d^{5} e^{5} + 75 d^{4} e^{6} x + 150 d^{3} e^{7} x^{2} + 150 d^{2} e^{8} x^{3} + 75 d e^{9} x^{4} + 15 e^{10} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**2/(e*x+d)**6,x)

[Out]

(-3*a**2*e**4 - a*c*d**2*e**2 - 3*c**2*d**4 - 30*c**2*d*e**3*x**3 - 15*c**2*e**4*x**4 + x**2*(-10*a*c*e**4 - 3
0*c**2*d**2*e**2) + x*(-5*a*c*d*e**3 - 15*c**2*d**3*e))/(15*d**5*e**5 + 75*d**4*e**6*x + 150*d**3*e**7*x**2 +
150*d**2*e**8*x**3 + 75*d*e**9*x**4 + 15*e**10*x**5)

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