### 3.468 $$\int \frac{(a+c x^2)^2}{(d+e x)^5} \, dx$$

Optimal. Leaf size=109 $-\frac{c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)^2}+\frac{4 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{4 c^2 d}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5}$

[Out]

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

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Rubi [A]  time = 0.0747606, antiderivative size = 109, normalized size of antiderivative = 1., 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} $-\frac{c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)^2}+\frac{4 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac{4 c^2 d}{e^5 (d+e x)}+\frac{c^2 \log (d+e x)}{e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^2/(d + e*x)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0357054, size = 100, normalized size = 0.92 $\frac{-3 a^2 e^4-2 a c e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^2 d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^2/(d + e*x)^5,x]

[Out]

(-3*a^2*e^4 - 2*a*c*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + c^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)
+ 12*c^2*(d + e*x)^4*Log[d + e*x])/(12*e^5*(d + e*x)^4)

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Maple [A]  time = 0.049, size = 146, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2}}{4\,e \left ( ex+d \right ) ^{4}}}-{\frac{ac{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{4\,acd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{c}^{2}{d}^{3}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{2}\ln \left ( ex+d \right ) }{{e}^{5}}}+4\,{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}-{\frac{ac}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^2/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 1.17105, size = 197, normalized size = 1.81 \begin{align*} \frac{48 \, c^{2} d e^{3} x^{3} + 25 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 12 \,{\left (9 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 8 \,{\left (11 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac{c^{2} \log \left (e x + d\right )}{e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^5,x, algorithm="maxima")

[Out]

1/12*(48*c^2*d*e^3*x^3 + 25*c^2*d^4 - 2*a*c*d^2*e^2 - 3*a^2*e^4 + 12*(9*c^2*d^2*e^2 - a*c*e^4)*x^2 + 8*(11*c^2
*d^3*e - a*c*d*e^3)*x)/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5) + c^2*log(e*x + d)/e^5

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Fricas [A]  time = 1.84814, size = 397, normalized size = 3.64 \begin{align*} \frac{48 \, c^{2} d e^{3} x^{3} + 25 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 12 \,{\left (9 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 8 \,{\left (11 \, c^{2} d^{3} e - a c d e^{3}\right )} x + 12 \,{\left (c^{2} e^{4} x^{4} + 4 \, c^{2} d e^{3} x^{3} + 6 \, c^{2} d^{2} e^{2} x^{2} + 4 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/12*(48*c^2*d*e^3*x^3 + 25*c^2*d^4 - 2*a*c*d^2*e^2 - 3*a^2*e^4 + 12*(9*c^2*d^2*e^2 - a*c*e^4)*x^2 + 8*(11*c^2
*d^3*e - a*c*d*e^3)*x + 12*(c^2*e^4*x^4 + 4*c^2*d*e^3*x^3 + 6*c^2*d^2*e^2*x^2 + 4*c^2*d^3*e*x + c^2*d^4)*log(e
*x + d))/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5)

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Sympy [A]  time = 1.84963, size = 150, normalized size = 1.38 \begin{align*} \frac{c^{2} \log{\left (d + e x \right )}}{e^{5}} + \frac{- 3 a^{2} e^{4} - 2 a c d^{2} e^{2} + 25 c^{2} d^{4} + 48 c^{2} d e^{3} x^{3} + x^{2} \left (- 12 a c e^{4} + 108 c^{2} d^{2} e^{2}\right ) + x \left (- 8 a c d e^{3} + 88 c^{2} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**2/(e*x+d)**5,x)

[Out]

c**2*log(d + e*x)/e**5 + (-3*a**2*e**4 - 2*a*c*d**2*e**2 + 25*c**2*d**4 + 48*c**2*d*e**3*x**3 + x**2*(-12*a*c*
e**4 + 108*c**2*d**2*e**2) + x*(-8*a*c*d*e**3 + 88*c**2*d**3*e))/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7
*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4)

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Giac [A]  time = 1.21402, size = 220, normalized size = 2.02 \begin{align*} -c^{2} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{12} \,{\left (\frac{48 \, c^{2} d e^{15}}{x e + d} - \frac{36 \, c^{2} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac{16 \, c^{2} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac{3 \, c^{2} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac{12 \, a c e^{17}}{{\left (x e + d\right )}^{2}} + \frac{16 \, a c d e^{17}}{{\left (x e + d\right )}^{3}} - \frac{6 \, a c d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac{3 \, a^{2} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^5,x, algorithm="giac")

[Out]

-c^2*e^(-5)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/12*(48*c^2*d*e^15/(x*e + d) - 36*c^2*d^2*e^15/(x*e + d)^2
+ 16*c^2*d^3*e^15/(x*e + d)^3 - 3*c^2*d^4*e^15/(x*e + d)^4 - 12*a*c*e^17/(x*e + d)^2 + 16*a*c*d*e^17/(x*e + d
)^3 - 6*a*c*d^2*e^17/(x*e + d)^4 - 3*a^2*e^19/(x*e + d)^4)*e^(-20)