3.1307 \(\int \frac{(A+B x) (a+c x^2)^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=189 \[ -\frac{2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)}+\frac{c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (d+e x)^2}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6 (d+e x)^4}-\frac{c^2 (5 B d-A e) \log (d+e x)}{e^6}+\frac{B c^2 x}{e^5} \]

[Out]

(B*c^2*x)/e^5 + ((B*d - A*e)*(c*d^2 + a*e^2)^2)/(4*e^6*(d + e*x)^4) - ((c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e
+ a*B*e^2))/(3*e^6*(d + e*x)^3) + (c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(e^6*(d + e*x)^2) - (2
*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^6*(d + e*x)) - (c^2*(5*B*d - A*e)*Log[d + e*x])/e^6

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Rubi [A]  time = 0.167643, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ -\frac{2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)}+\frac{c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (d+e x)^2}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6 (d+e x)^4}-\frac{c^2 (5 B d-A e) \log (d+e x)}{e^6}+\frac{B c^2 x}{e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*c^2*x)/e^5 + ((B*d - A*e)*(c*d^2 + a*e^2)^2)/(4*e^6*(d + e*x)^4) - ((c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e
+ a*B*e^2))/(3*e^6*(d + e*x)^3) + (c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(e^6*(d + e*x)^2) - (2
*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^6*(d + e*x)) - (c^2*(5*B*d - A*e)*Log[d + e*x])/e^6

Rule 772

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

Rubi steps

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

Mathematica [A]  time = 0.117007, size = 221, normalized size = 1.17 \[ \frac{A e \left (-3 a^2 e^4-2 a c e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^2 d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )\right )-B \left (a^2 e^4 (d+4 e x)+6 a c e^2 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+c^2 \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )\right )-12 c^2 (d+e x)^4 (5 B d-A e) \log (d+e x)}{12 e^6 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*e*(-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)) - B*(a^2*e^4*(d + 4*e*x) + 6*a*c*e^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + c^2*(77*d^5 + 248*d^
4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) - 12*c^2*(5*B*d - A*e)*(d + e*x)^4*Log[
d + e*x])/(12*e^6*(d + e*x)^4)

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Maple [A]  time = 0.01, size = 356, normalized size = 1.9 \begin{align*}{\frac{B{c}^{2}x}{{e}^{5}}}+4\,{\frac{A{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }}-2\,{\frac{aBc}{{e}^{4} \left ( ex+d \right ) }}-10\,{\frac{B{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{aAc}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{A{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+3\,{\frac{aBcd}{{e}^{4} \left ( ex+d \right ) ^{2}}}+5\,{\frac{B{c}^{2}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{4\,Adac}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{4\,A{d}^{3}{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{B{a}^{2}}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-2\,{\frac{aBc{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{5\,B{c}^{2}{d}^{4}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{2}\ln \left ( ex+d \right ) A}{{e}^{5}}}-5\,{\frac{{c}^{2}\ln \left ( ex+d \right ) Bd}{{e}^{6}}}-{\frac{A{a}^{2}}{4\,e \left ( ex+d \right ) ^{4}}}-{\frac{A{d}^{2}ac}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{A{d}^{4}{c}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{Bd{a}^{2}}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}+{\frac{B{d}^{3}ac}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}+{\frac{B{c}^{2}{d}^{5}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*c^2*x/e^5+4/e^5/(e*x+d)*A*c^2*d-2*c/e^4/(e*x+d)*a*B-10/e^6/(e*x+d)*B*c^2*d^2-c/e^3/(e*x+d)^2*a*A-3/e^5/(e*x+
d)^2*A*c^2*d^2+3*c/e^4/(e*x+d)^2*a*B*d+5/e^6/(e*x+d)^2*B*c^2*d^3+4/3/e^3/(e*x+d)^3*A*a*c*d+4/3*d^3/e^5/(e*x+d)
^3*A*c^2-1/3/e^2/(e*x+d)^3*B*a^2-2/e^4/(e*x+d)^3*B*a*c*d^2-5/3*d^4/e^6/(e*x+d)^3*B*c^2+c^2/e^5*ln(e*x+d)*A-5*c
^2/e^6*ln(e*x+d)*B*d-1/4/e/(e*x+d)^4*A*a^2-1/2/e^3/(e*x+d)^4*A*d^2*a*c-1/4*d^4/e^5/(e*x+d)^4*A*c^2+1/4/e^2/(e*
x+d)^4*B*d*a^2+1/2/e^4/(e*x+d)^4*B*d^3*a*c+1/4*d^5/e^6/(e*x+d)^4*B*c^2

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Maxima [A]  time = 1.0402, size = 377, normalized size = 1.99 \begin{align*} -\frac{77 \, B c^{2} d^{5} - 25 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} + 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + 3 \, A a^{2} e^{5} + 24 \,{\left (5 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 12 \,{\left (25 \, B c^{2} d^{3} e^{2} - 9 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} + 4 \,{\left (65 \, B c^{2} d^{4} e - 22 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 2 \, A a c d e^{4} + B a^{2} e^{5}\right )} x}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac{B c^{2} x}{e^{5}} - \frac{{\left (5 \, B c^{2} d - A c^{2} e\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(77*B*c^2*d^5 - 25*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 + 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 + 3*A*a^2*e^5 + 24*(5*B
*c^2*d^2*e^3 - 2*A*c^2*d*e^4 + B*a*c*e^5)*x^3 + 12*(25*B*c^2*d^3*e^2 - 9*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 + A*a*c
*e^5)*x^2 + 4*(65*B*c^2*d^4*e - 22*A*c^2*d^3*e^2 + 6*B*a*c*d^2*e^3 + 2*A*a*c*d*e^4 + B*a^2*e^5)*x)/(e^10*x^4 +
4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6) + B*c^2*x/e^5 - (5*B*c^2*d - A*c^2*e)*log(e*x + d)/e^6

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Fricas [B]  time = 1.83531, size = 842, normalized size = 4.46 \begin{align*} \frac{12 \, B c^{2} e^{5} x^{5} + 48 \, B c^{2} d e^{4} x^{4} - 77 \, B c^{2} d^{5} + 25 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - 3 \, A a^{2} e^{5} - 24 \,{\left (2 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} - 12 \,{\left (21 \, B c^{2} d^{3} e^{2} - 9 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} - 4 \,{\left (62 \, B c^{2} d^{4} e - 22 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 2 \, A a c d e^{4} + B a^{2} e^{5}\right )} x - 12 \,{\left (5 \, B c^{2} d^{5} - A c^{2} d^{4} e +{\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 4 \,{\left (5 \, B c^{2} d^{2} e^{3} - A c^{2} d e^{4}\right )} x^{3} + 6 \,{\left (5 \, B c^{2} d^{3} e^{2} - A c^{2} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (5 \, B c^{2} d^{4} e - A c^{2} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*B*c^2*e^5*x^5 + 48*B*c^2*d*e^4*x^4 - 77*B*c^2*d^5 + 25*A*c^2*d^4*e - 6*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^
3 - B*a^2*d*e^4 - 3*A*a^2*e^5 - 24*(2*B*c^2*d^2*e^3 - 2*A*c^2*d*e^4 + B*a*c*e^5)*x^3 - 12*(21*B*c^2*d^3*e^2 -
9*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 + A*a*c*e^5)*x^2 - 4*(62*B*c^2*d^4*e - 22*A*c^2*d^3*e^2 + 6*B*a*c*d^2*e^3 + 2*
A*a*c*d*e^4 + B*a^2*e^5)*x - 12*(5*B*c^2*d^5 - A*c^2*d^4*e + (5*B*c^2*d*e^4 - A*c^2*e^5)*x^4 + 4*(5*B*c^2*d^2*
e^3 - A*c^2*d*e^4)*x^3 + 6*(5*B*c^2*d^3*e^2 - A*c^2*d^2*e^3)*x^2 + 4*(5*B*c^2*d^4*e - A*c^2*d^3*e^2)*x)*log(e*
x + d))/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)

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Sympy [A]  time = 114.915, size = 304, normalized size = 1.61 \begin{align*} \frac{B c^{2} x}{e^{5}} - \frac{c^{2} \left (- A e + 5 B d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{3 A a^{2} e^{5} + 2 A a c d^{2} e^{3} - 25 A c^{2} d^{4} e + B a^{2} d e^{4} + 6 B a c d^{3} e^{2} + 77 B c^{2} d^{5} + x^{3} \left (- 48 A c^{2} d e^{4} + 24 B a c e^{5} + 120 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (12 A a c e^{5} - 108 A c^{2} d^{2} e^{3} + 36 B a c d e^{4} + 300 B c^{2} d^{3} e^{2}\right ) + x \left (8 A a c d e^{4} - 88 A c^{2} d^{3} e^{2} + 4 B a^{2} e^{5} + 24 B a c d^{2} e^{3} + 260 B c^{2} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*c**2*x/e**5 - c**2*(-A*e + 5*B*d)*log(d + e*x)/e**6 - (3*A*a**2*e**5 + 2*A*a*c*d**2*e**3 - 25*A*c**2*d**4*e
+ B*a**2*d*e**4 + 6*B*a*c*d**3*e**2 + 77*B*c**2*d**5 + x**3*(-48*A*c**2*d*e**4 + 24*B*a*c*e**5 + 120*B*c**2*d*
*2*e**3) + x**2*(12*A*a*c*e**5 - 108*A*c**2*d**2*e**3 + 36*B*a*c*d*e**4 + 300*B*c**2*d**3*e**2) + x*(8*A*a*c*d
*e**4 - 88*A*c**2*d**3*e**2 + 4*B*a**2*e**5 + 24*B*a*c*d**2*e**3 + 260*B*c**2*d**4*e))/(12*d**4*e**6 + 48*d**3
*e**7*x + 72*d**2*e**8*x**2 + 48*d*e**9*x**3 + 12*e**10*x**4)

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Giac [B]  time = 1.4439, size = 502, normalized size = 2.66 \begin{align*}{\left (x e + d\right )} B c^{2} e^{\left (-6\right )} +{\left (5 \, B c^{2} d - A c^{2} e\right )} e^{\left (-6\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{120 \, B c^{2} d^{2} e^{22}}{x e + d} - \frac{60 \, B c^{2} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac{20 \, B c^{2} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B c^{2} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac{48 \, A c^{2} d e^{23}}{x e + d} + \frac{36 \, A c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac{16 \, A c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac{3 \, A c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac{24 \, B a c e^{24}}{x e + d} - \frac{36 \, B a c d e^{24}}{{\left (x e + d\right )}^{2}} + \frac{24 \, B a c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac{6 \, B a c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac{12 \, A a c e^{25}}{{\left (x e + d\right )}^{2}} - \frac{16 \, A a c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac{6 \, A a c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a^{2} e^{26}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a^{2} d e^{26}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a^{2} e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e + d)*B*c^2*e^(-6) + (5*B*c^2*d - A*c^2*e)*e^(-6)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(120*B*c^2*d
^2*e^22/(x*e + d) - 60*B*c^2*d^3*e^22/(x*e + d)^2 + 20*B*c^2*d^4*e^22/(x*e + d)^3 - 3*B*c^2*d^5*e^22/(x*e + d)
^4 - 48*A*c^2*d*e^23/(x*e + d) + 36*A*c^2*d^2*e^23/(x*e + d)^2 - 16*A*c^2*d^3*e^23/(x*e + d)^3 + 3*A*c^2*d^4*e
^23/(x*e + d)^4 + 24*B*a*c*e^24/(x*e + d) - 36*B*a*c*d*e^24/(x*e + d)^2 + 24*B*a*c*d^2*e^24/(x*e + d)^3 - 6*B*
a*c*d^3*e^24/(x*e + d)^4 + 12*A*a*c*e^25/(x*e + d)^2 - 16*A*a*c*d*e^25/(x*e + d)^3 + 6*A*a*c*d^2*e^25/(x*e + d
)^4 + 4*B*a^2*e^26/(x*e + d)^3 - 3*B*a^2*d*e^26/(x*e + d)^4 + 3*A*a^2*e^27/(x*e + d)^4)*e^(-28)