3.5.68 \(\int \frac {(a+c x^2)^2}{(d+e x)^5} \, dx\) [468]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 109, 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 = {711} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x)^5,x]

[Out]

-1/4*(c*d^2 + a*e^2)^2/(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 711

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*}

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Mathematica [A]
time = 0.02, size = 100, normalized size = 0.92 \begin {gather*} \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 (25 d^3+88 d^2 e x+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} \end {gather*}

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 = 0.42, size = 118, normalized size = 1.08

method	result	size
risch	\(\frac {\frac {4 c^{2} d \,x^{3}}{e^{2}}-\frac {c \left (e^{2} a -9 c \,d^{2}\right ) x^{2}}{e^{3}}-\frac {2 c d \left (e^{2} a -11 c \,d^{2}\right ) x}{3 e^{4}}-\frac {3 a^{2} e^{4}+2 a c \,d^{2} e^{2}-25 c^{2} d^{4}}{12 e^{5}}}{\left (e x +d \right )^{4}}+\frac {c^{2} \ln \left (e x +d \right )}{e^{5}}\)	\(109\)
norman	\(\frac {-\frac {3 a^{2} e^{4}+2 a c \,d^{2} e^{2}-25 c^{2} d^{4}}{12 e^{5}}-\frac {\left (a c \,e^{2}-9 d^{2} c^{2}\right ) x^{2}}{e^{3}}+\frac {4 c^{2} d \,x^{3}}{e^{2}}-\frac {2 d \left (a c \,e^{2}-11 d^{2} c^{2}\right ) x}{3 e^{4}}}{\left (e x +d \right )^{4}}+\frac {c^{2} \ln \left (e x +d \right )}{e^{5}}\)	\(113\)
default	\(\frac {4 c d \left (e^{2} a +c \,d^{2}\right )}{3 e^{5} \left (e x +d \right )^{3}}+\frac {4 c^{2} d}{e^{5} \left (e x +d \right )}+\frac {c^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {c \left (e^{2} a +3 c \,d^{2}\right )}{e^{5} \left (e x +d \right )^{2}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{4 e^{5} \left (e x +d \right )^{4}}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 136, normalized size = 1.25 \begin {gather*} c^{2} e^{\left (-5\right )} \log \left (x e + d\right ) + \frac {48 \, c^{2} d x^{3} e^{3} + 25 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} + 12 \, {\left (9 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} - 3 \, a^{2} e^{4} + 8 \, {\left (11 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{12 \, {\left (x^{4} e^{9} + 4 \, d x^{3} e^{8} + 6 \, d^{2} x^{2} e^{7} + 4 \, d^{3} x e^{6} + d^{4} 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)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.00, size = 182, normalized size = 1.67 \begin {gather*} \frac {88 \, c^{2} d^{3} x e + 25 \, c^{2} d^{4} - 3 \, {\left (4 \, a c x^{2} + a^{2}\right )} e^{4} + 8 \, {\left (6 \, c^{2} d x^{3} - a c d x\right )} e^{3} + 2 \, {\left (54 \, c^{2} d^{2} x^{2} - a c d^{2}\right )} e^{2} + 12 \, {\left (c^{2} x^{4} e^{4} + 4 \, c^{2} d x^{3} e^{3} + 6 \, c^{2} d^{2} x^{2} e^{2} + 4 \, c^{2} d^{3} x e + c^{2} d^{4}\right )} \log \left (x e + d\right )}{12 \, {\left (x^{4} e^{9} + 4 \, d x^{3} e^{8} + 6 \, d^{2} x^{2} e^{7} + 4 \, d^{3} x e^{6} + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/12*(88*c^2*d^3*x*e + 25*c^2*d^4 - 3*(4*a*c*x^2 + a^2)*e^4 + 8*(6*c^2*d*x^3 - a*c*d*x)*e^3 + 2*(54*c^2*d^2*x^
2 - a*c*d^2)*e^2 + 12*(c^2*x^4*e^4 + 4*c^2*d*x^3*e^3 + 6*c^2*d^2*x^2*e^2 + 4*c^2*d^3*x*e + c^2*d^4)*log(x*e +
d))/(x^4*e^9 + 4*d*x^3*e^8 + 6*d^2*x^2*e^7 + 4*d^3*x*e^6 + d^4*e^5)

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Sympy [A]
time = 0.76, size = 150, normalized size = 1.38 \begin {gather*} \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 {gather*}

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 = 0.71, size = 163, normalized size = 1.50 \begin {gather*} -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 {gather*}

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)

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Mupad [B]
time = 0.32, size = 144, normalized size = 1.32 \begin {gather*} \frac {c^2\,\ln \left (d+e\,x\right )}{e^5}-\frac {\frac {3\,a^2\,e^4+2\,a\,c\,d^2\,e^2-25\,c^2\,d^4}{12\,e^5}-\frac {2\,x\,\left (11\,c^2\,d^3-a\,c\,d\,e^2\right )}{3\,e^4}-\frac {4\,c^2\,d\,x^3}{e^2}+\frac {c\,x^2\,\left (a\,e^2-9\,c\,d^2\right )}{e^3}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2/(d + e*x)^5,x)

[Out]

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

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