3.4.90 \(\int \frac {(a+c x^2)^2}{(d+e x)^7} \, dx\)

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

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Rubi [A]  time = 0.07, antiderivative size = 117, 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 {c \left (a e^2+3 c d^2\right )}{2 e^5 (d+e x)^4}+\frac {4 c d \left (a e^2+c d^2\right )}{5 e^5 (d+e x)^5}-\frac {\left (a e^2+c d^2\right )^2}{6 e^5 (d+e x)^6}-\frac {c^2}{2 e^5 (d+e x)^2}+\frac {4 c^2 d}{3 e^5 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x)^7,x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 89, normalized size = 0.76 \begin {gather*} -\frac {5 a^2 e^4+a c e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )}{30 e^5 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^2/(d + e*x)^7,x]

[Out]

-1/30*(5*a^2*e^4 + a*c*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + c^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3
+ 15*e^4*x^4))/(e^5*(d + e*x)^6)

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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)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + c*x^2)^2/(d + e*x)^7,x]

[Out]

IntegrateAlgebraic[(a + c*x^2)^2/(d + e*x)^7, x]

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fricas [A]  time = 0.38, size = 159, normalized size = 1.36 \begin {gather*} -\frac {15 \, c^{2} e^{4} x^{4} + 20 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + a c d^{2} e^{2} + 5 \, a^{2} e^{4} + 15 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{30 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

-1/30*(15*c^2*e^4*x^4 + 20*c^2*d*e^3*x^3 + c^2*d^4 + a*c*d^2*e^2 + 5*a^2*e^4 + 15*(c^2*d^2*e^2 + a*c*e^4)*x^2
+ 6*(c^2*d^3*e + a*c*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 + 6
*d^5*e^6*x + d^6*e^5)

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giac [A]  time = 0.15, size = 97, normalized size = 0.83 \begin {gather*} -\frac {{\left (15 \, c^{2} x^{4} e^{4} + 20 \, c^{2} d x^{3} e^{3} + 15 \, c^{2} d^{2} x^{2} e^{2} + 6 \, c^{2} d^{3} x e + c^{2} d^{4} + 15 \, a c x^{2} e^{4} + 6 \, a c d x e^{3} + a c d^{2} e^{2} + 5 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{30 \, {\left (x e + d\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/30*(15*c^2*x^4*e^4 + 20*c^2*d*x^3*e^3 + 15*c^2*d^2*x^2*e^2 + 6*c^2*d^3*x*e + c^2*d^4 + 15*a*c*x^2*e^4 + 6*a
*c*d*x*e^3 + a*c*d^2*e^2 + 5*a^2*e^4)*e^(-5)/(x*e + d)^6

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maple [A]  time = 0.04, size = 120, normalized size = 1.03 \begin {gather*} \frac {4 c^{2} d}{3 \left (e x +d \right )^{3} e^{5}}-\frac {c^{2}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {4 \left (a \,e^{2}+c \,d^{2}\right ) c d}{5 \left (e x +d \right )^{5} e^{5}}-\frac {\left (a \,e^{2}+3 c \,d^{2}\right ) c}{2 \left (e x +d \right )^{4} e^{5}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{6 \left (e x +d \right )^{6} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^2/(e*x+d)^7,x)

[Out]

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

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maxima [A]  time = 1.50, size = 159, normalized size = 1.36 \begin {gather*} -\frac {15 \, c^{2} e^{4} x^{4} + 20 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + a c d^{2} e^{2} + 5 \, a^{2} e^{4} + 15 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{30 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} 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)^7,x, algorithm="maxima")

[Out]

-1/30*(15*c^2*e^4*x^4 + 20*c^2*d*e^3*x^3 + c^2*d^4 + a*c*d^2*e^2 + 5*a^2*e^4 + 15*(c^2*d^2*e^2 + a*c*e^4)*x^2
+ 6*(c^2*d^3*e + a*c*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 + 6
*d^5*e^6*x + d^6*e^5)

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mupad [B]  time = 0.31, size = 157, normalized size = 1.34 \begin {gather*} -\frac {\frac {5\,a^2\,e^4+a\,c\,d^2\,e^2+c^2\,d^4}{30\,e^5}+\frac {c^2\,x^4}{2\,e}+\frac {2\,c^2\,d\,x^3}{3\,e^2}+\frac {c\,x^2\,\left (c\,d^2+a\,e^2\right )}{2\,e^3}+\frac {c\,d\,x\,\left (c\,d^2+a\,e^2\right )}{5\,e^4}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2/(d + e*x)^7,x)

[Out]

-((5*a^2*e^4 + c^2*d^4 + a*c*d^2*e^2)/(30*e^5) + (c^2*x^4)/(2*e) + (2*c^2*d*x^3)/(3*e^2) + (c*x^2*(a*e^2 + c*d
^2))/(2*e^3) + (c*d*x*(a*e^2 + c*d^2))/(5*e^4))/(d^6 + e^6*x^6 + 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3
+ 15*d^2*e^4*x^4 + 6*d^5*e*x)

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sympy [A]  time = 2.32, size = 172, normalized size = 1.47 \begin {gather*} \frac {- 5 a^{2} e^{4} - a c d^{2} e^{2} - c^{2} d^{4} - 20 c^{2} d e^{3} x^{3} - 15 c^{2} e^{4} x^{4} + x^{2} \left (- 15 a c e^{4} - 15 c^{2} d^{2} e^{2}\right ) + x \left (- 6 a c d e^{3} - 6 c^{2} d^{3} e\right )}{30 d^{6} e^{5} + 180 d^{5} e^{6} x + 450 d^{4} e^{7} x^{2} + 600 d^{3} e^{8} x^{3} + 450 d^{2} e^{9} x^{4} + 180 d e^{10} x^{5} + 30 e^{11} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**2/(e*x+d)**7,x)

[Out]

(-5*a**2*e**4 - a*c*d**2*e**2 - c**2*d**4 - 20*c**2*d*e**3*x**3 - 15*c**2*e**4*x**4 + x**2*(-15*a*c*e**4 - 15*
c**2*d**2*e**2) + x*(-6*a*c*d*e**3 - 6*c**2*d**3*e))/(30*d**6*e**5 + 180*d**5*e**6*x + 450*d**4*e**7*x**2 + 60
0*d**3*e**8*x**3 + 450*d**2*e**9*x**4 + 180*d*e**10*x**5 + 30*e**11*x**6)

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