### 3.495 $$\int \frac{(a+c x^2)^4}{d+e x} \, dx$$

Optimal. Leaf size=264 $\frac{c^2 (d+e x)^4 \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{2 e^9}+\frac{2 c^3 (d+e x)^6 \left (a e^2+7 c d^2\right )}{3 e^9}-\frac{8 c^3 d (d+e x)^5 \left (3 a e^2+7 c d^2\right )}{5 e^9}-\frac{8 c^2 d (d+e x)^3 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{3 e^9}+\frac{2 c (d+e x)^2 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{e^9}-\frac{8 c d x \left (a e^2+c d^2\right )^3}{e^8}+\frac{\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9}+\frac{c^4 (d+e x)^8}{8 e^9}-\frac{8 c^4 d (d+e x)^7}{7 e^9}$

[Out]

(-8*c*d*(c*d^2 + a*e^2)^3*x)/e^8 + (2*c*(c*d^2 + a*e^2)^2*(7*c*d^2 + a*e^2)*(d + e*x)^2)/e^9 - (8*c^2*d*(c*d^2
+ a*e^2)*(7*c*d^2 + 3*a*e^2)*(d + e*x)^3)/(3*e^9) + (c^2*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)*(d + e*x)^
4)/(2*e^9) - (8*c^3*d*(7*c*d^2 + 3*a*e^2)*(d + e*x)^5)/(5*e^9) + (2*c^3*(7*c*d^2 + a*e^2)*(d + e*x)^6)/(3*e^9)
- (8*c^4*d*(d + e*x)^7)/(7*e^9) + (c^4*(d + e*x)^8)/(8*e^9) + ((c*d^2 + a*e^2)^4*Log[d + e*x])/e^9

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Rubi [A]  time = 0.280778, antiderivative size = 264, 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} $\frac{c^2 (d+e x)^4 \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{2 e^9}+\frac{2 c^3 (d+e x)^6 \left (a e^2+7 c d^2\right )}{3 e^9}-\frac{8 c^3 d (d+e x)^5 \left (3 a e^2+7 c d^2\right )}{5 e^9}-\frac{8 c^2 d (d+e x)^3 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{3 e^9}+\frac{2 c (d+e x)^2 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{e^9}-\frac{8 c d x \left (a e^2+c d^2\right )^3}{e^8}+\frac{\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9}+\frac{c^4 (d+e x)^8}{8 e^9}-\frac{8 c^4 d (d+e x)^7}{7 e^9}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^4/(d + e*x),x]

[Out]

(-8*c*d*(c*d^2 + a*e^2)^3*x)/e^8 + (2*c*(c*d^2 + a*e^2)^2*(7*c*d^2 + a*e^2)*(d + e*x)^2)/e^9 - (8*c^2*d*(c*d^2
+ a*e^2)*(7*c*d^2 + 3*a*e^2)*(d + e*x)^3)/(3*e^9) + (c^2*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)*(d + e*x)^
4)/(2*e^9) - (8*c^3*d*(7*c*d^2 + 3*a*e^2)*(d + e*x)^5)/(5*e^9) + (2*c^3*(7*c*d^2 + a*e^2)*(d + e*x)^6)/(3*e^9)
- (8*c^4*d*(d + e*x)^7)/(7*e^9) + (c^4*(d + e*x)^8)/(8*e^9) + ((c*d^2 + a*e^2)^4*Log[d + e*x])/e^9

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 )^4}{d+e x} \, dx &=\int \left (-\frac{8 c d \left (c d^2+a e^2\right )^3}{e^8}+\frac{\left (c d^2+a e^2\right )^4}{e^8 (d+e x)}+\frac{4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)}{e^8}+\frac{8 c^2 d \left (-7 c d^2-3 a e^2\right ) \left (c d^2+a e^2\right ) (d+e x)^2}{e^8}+\frac{2 c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^3}{e^8}-\frac{8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^4}{e^8}+\frac{4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^5}{e^8}-\frac{8 c^4 d (d+e x)^6}{e^8}+\frac{c^4 (d+e x)^7}{e^8}\right ) \, dx\\ &=-\frac{8 c d \left (c d^2+a e^2\right )^3 x}{e^8}+\frac{2 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^2}{e^9}-\frac{8 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^3}{3 e^9}+\frac{c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^4}{2 e^9}-\frac{8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^5}{5 e^9}+\frac{2 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^6}{3 e^9}-\frac{8 c^4 d (d+e x)^7}{7 e^9}+\frac{c^4 (d+e x)^8}{8 e^9}+\frac{\left (c d^2+a e^2\right )^4 \log (d+e x)}{e^9}\\ \end{align*}

Mathematica [A]  time = 0.105041, size = 227, normalized size = 0.86 $\frac{c x \left (420 a^2 c e^4 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+1680 a^3 e^6 (e x-2 d)+56 a c^2 e^2 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )+c^3 \left (-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5+420 d^6 e x-840 d^7-120 d e^6 x^6+105 e^7 x^7\right )\right )}{840 e^8}+\frac{\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^4/(d + e*x),x]

[Out]

(c*x*(1680*a^3*e^6*(-2*d + e*x) + 420*a^2*c*e^4*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 56*a*c^2*e^2
*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + c^3*(-840*d^7 + 420*d^
6*e*x - 280*d^5*e^2*x^2 + 210*d^4*e^3*x^3 - 168*d^3*e^4*x^4 + 140*d^2*e^5*x^5 - 120*d*e^6*x^6 + 105*e^7*x^7)))
/(840*e^8) + ((c*d^2 + a*e^2)^4*Log[d + e*x])/e^9

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Maple [A]  time = 0.046, size = 358, normalized size = 1.4 \begin{align*} -{\frac{4\,{c}^{3}{x}^{5}ad}{5\,{e}^{2}}}+4\,{\frac{\ln \left ( ex+d \right ){a}^{3}c{d}^{2}}{{e}^{3}}}+{\frac{3\,{c}^{2}{x}^{4}{a}^{2}}{2\,e}}+{\frac{{c}^{4}{x}^{4}{d}^{4}}{4\,{e}^{5}}}-{\frac{{c}^{4}{x}^{3}{d}^{5}}{3\,{e}^{6}}}+{\frac{\ln \left ( ex+d \right ){c}^{4}{d}^{8}}{{e}^{9}}}+2\,{\frac{c{x}^{2}{a}^{3}}{e}}+{\frac{{c}^{4}{x}^{2}{d}^{6}}{2\,{e}^{7}}}-{\frac{{c}^{4}{d}^{7}x}{{e}^{8}}}-{\frac{{c}^{4}d{x}^{7}}{7\,{e}^{2}}}+{\frac{2\,{c}^{3}{x}^{6}a}{3\,e}}+{\frac{{c}^{4}{x}^{6}{d}^{2}}{6\,{e}^{3}}}+{\frac{{c}^{3}{x}^{4}a{d}^{2}}{{e}^{3}}}+{\frac{{c}^{4}{x}^{8}}{8\,e}}+{\frac{\ln \left ( ex+d \right ){a}^{4}}{e}}-4\,{\frac{{d}^{5}a{c}^{3}x}{{e}^{6}}}-2\,{\frac{{c}^{2}{x}^{3}{a}^{2}d}{{e}^{2}}}-{\frac{4\,{x}^{3}{c}^{3}a{d}^{3}}{3\,{e}^{4}}}+3\,{\frac{{c}^{2}{x}^{2}{a}^{2}{d}^{2}}{{e}^{3}}}+2\,{\frac{{c}^{3}{x}^{2}a{d}^{4}}{{e}^{5}}}-4\,{\frac{{a}^{3}cdx}{{e}^{2}}}-6\,{\frac{{a}^{2}{c}^{2}{d}^{3}x}{{e}^{4}}}+6\,{\frac{\ln \left ( ex+d \right ){a}^{2}{c}^{2}{d}^{4}}{{e}^{5}}}+4\,{\frac{\ln \left ( ex+d \right ) a{c}^{3}{d}^{6}}{{e}^{7}}}-{\frac{{c}^{4}{x}^{5}{d}^{3}}{5\,{e}^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^4/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 1.18096, size = 431, normalized size = 1.63 \begin{align*} \frac{105 \, c^{4} e^{7} x^{8} - 120 \, c^{4} d e^{6} x^{7} + 140 \,{\left (c^{4} d^{2} e^{5} + 4 \, a c^{3} e^{7}\right )} x^{6} - 168 \,{\left (c^{4} d^{3} e^{4} + 4 \, a c^{3} d e^{6}\right )} x^{5} + 210 \,{\left (c^{4} d^{4} e^{3} + 4 \, a c^{3} d^{2} e^{5} + 6 \, a^{2} c^{2} e^{7}\right )} x^{4} - 280 \,{\left (c^{4} d^{5} e^{2} + 4 \, a c^{3} d^{3} e^{4} + 6 \, a^{2} c^{2} d e^{6}\right )} x^{3} + 420 \,{\left (c^{4} d^{6} e + 4 \, a c^{3} d^{4} e^{3} + 6 \, a^{2} c^{2} d^{2} e^{5} + 4 \, a^{3} c e^{7}\right )} x^{2} - 840 \,{\left (c^{4} d^{7} + 4 \, a c^{3} d^{5} e^{2} + 6 \, a^{2} c^{2} d^{3} e^{4} + 4 \, a^{3} c d e^{6}\right )} x}{840 \, e^{8}} + \frac{{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (e x + d\right )}{e^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^4/(e*x+d),x, algorithm="maxima")

[Out]

1/840*(105*c^4*e^7*x^8 - 120*c^4*d*e^6*x^7 + 140*(c^4*d^2*e^5 + 4*a*c^3*e^7)*x^6 - 168*(c^4*d^3*e^4 + 4*a*c^3*
d*e^6)*x^5 + 210*(c^4*d^4*e^3 + 4*a*c^3*d^2*e^5 + 6*a^2*c^2*e^7)*x^4 - 280*(c^4*d^5*e^2 + 4*a*c^3*d^3*e^4 + 6*
a^2*c^2*d*e^6)*x^3 + 420*(c^4*d^6*e + 4*a*c^3*d^4*e^3 + 6*a^2*c^2*d^2*e^5 + 4*a^3*c*e^7)*x^2 - 840*(c^4*d^7 +
4*a*c^3*d^5*e^2 + 6*a^2*c^2*d^3*e^4 + 4*a^3*c*d*e^6)*x)/e^8 + (c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 +
4*a^3*c*d^2*e^6 + a^4*e^8)*log(e*x + d)/e^9

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Fricas [A]  time = 1.88995, size = 662, normalized size = 2.51 \begin{align*} \frac{105 \, c^{4} e^{8} x^{8} - 120 \, c^{4} d e^{7} x^{7} + 140 \,{\left (c^{4} d^{2} e^{6} + 4 \, a c^{3} e^{8}\right )} x^{6} - 168 \,{\left (c^{4} d^{3} e^{5} + 4 \, a c^{3} d e^{7}\right )} x^{5} + 210 \,{\left (c^{4} d^{4} e^{4} + 4 \, a c^{3} d^{2} e^{6} + 6 \, a^{2} c^{2} e^{8}\right )} x^{4} - 280 \,{\left (c^{4} d^{5} e^{3} + 4 \, a c^{3} d^{3} e^{5} + 6 \, a^{2} c^{2} d e^{7}\right )} x^{3} + 420 \,{\left (c^{4} d^{6} e^{2} + 4 \, a c^{3} d^{4} e^{4} + 6 \, a^{2} c^{2} d^{2} e^{6} + 4 \, a^{3} c e^{8}\right )} x^{2} - 840 \,{\left (c^{4} d^{7} e + 4 \, a c^{3} d^{5} e^{3} + 6 \, a^{2} c^{2} d^{3} e^{5} + 4 \, a^{3} c d e^{7}\right )} x + 840 \,{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (e x + d\right )}{840 \, e^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^4/(e*x+d),x, algorithm="fricas")

[Out]

1/840*(105*c^4*e^8*x^8 - 120*c^4*d*e^7*x^7 + 140*(c^4*d^2*e^6 + 4*a*c^3*e^8)*x^6 - 168*(c^4*d^3*e^5 + 4*a*c^3*
d*e^7)*x^5 + 210*(c^4*d^4*e^4 + 4*a*c^3*d^2*e^6 + 6*a^2*c^2*e^8)*x^4 - 280*(c^4*d^5*e^3 + 4*a*c^3*d^3*e^5 + 6*
a^2*c^2*d*e^7)*x^3 + 420*(c^4*d^6*e^2 + 4*a*c^3*d^4*e^4 + 6*a^2*c^2*d^2*e^6 + 4*a^3*c*e^8)*x^2 - 840*(c^4*d^7*
e + 4*a*c^3*d^5*e^3 + 6*a^2*c^2*d^3*e^5 + 4*a^3*c*d*e^7)*x + 840*(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^
4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*log(e*x + d))/e^9

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Sympy [A]  time = 0.734259, size = 287, normalized size = 1.09 \begin{align*} - \frac{c^{4} d x^{7}}{7 e^{2}} + \frac{c^{4} x^{8}}{8 e} + \frac{x^{6} \left (4 a c^{3} e^{2} + c^{4} d^{2}\right )}{6 e^{3}} - \frac{x^{5} \left (4 a c^{3} d e^{2} + c^{4} d^{3}\right )}{5 e^{4}} + \frac{x^{4} \left (6 a^{2} c^{2} e^{4} + 4 a c^{3} d^{2} e^{2} + c^{4} d^{4}\right )}{4 e^{5}} - \frac{x^{3} \left (6 a^{2} c^{2} d e^{4} + 4 a c^{3} d^{3} e^{2} + c^{4} d^{5}\right )}{3 e^{6}} + \frac{x^{2} \left (4 a^{3} c e^{6} + 6 a^{2} c^{2} d^{2} e^{4} + 4 a c^{3} d^{4} e^{2} + c^{4} d^{6}\right )}{2 e^{7}} - \frac{x \left (4 a^{3} c d e^{6} + 6 a^{2} c^{2} d^{3} e^{4} + 4 a c^{3} d^{5} e^{2} + c^{4} d^{7}\right )}{e^{8}} + \frac{\left (a e^{2} + c d^{2}\right )^{4} \log{\left (d + e x \right )}}{e^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**4/(e*x+d),x)

[Out]

-c**4*d*x**7/(7*e**2) + c**4*x**8/(8*e) + x**6*(4*a*c**3*e**2 + c**4*d**2)/(6*e**3) - x**5*(4*a*c**3*d*e**2 +
c**4*d**3)/(5*e**4) + x**4*(6*a**2*c**2*e**4 + 4*a*c**3*d**2*e**2 + c**4*d**4)/(4*e**5) - x**3*(6*a**2*c**2*d*
e**4 + 4*a*c**3*d**3*e**2 + c**4*d**5)/(3*e**6) + x**2*(4*a**3*c*e**6 + 6*a**2*c**2*d**2*e**4 + 4*a*c**3*d**4*
e**2 + c**4*d**6)/(2*e**7) - x*(4*a**3*c*d*e**6 + 6*a**2*c**2*d**3*e**4 + 4*a*c**3*d**5*e**2 + c**4*d**7)/e**8
+ (a*e**2 + c*d**2)**4*log(d + e*x)/e**9

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Giac [A]  time = 1.33633, size = 427, normalized size = 1.62 \begin{align*}{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{840} \,{\left (105 \, c^{4} x^{8} e^{7} - 120 \, c^{4} d x^{7} e^{6} + 140 \, c^{4} d^{2} x^{6} e^{5} - 168 \, c^{4} d^{3} x^{5} e^{4} + 210 \, c^{4} d^{4} x^{4} e^{3} - 280 \, c^{4} d^{5} x^{3} e^{2} + 420 \, c^{4} d^{6} x^{2} e - 840 \, c^{4} d^{7} x + 560 \, a c^{3} x^{6} e^{7} - 672 \, a c^{3} d x^{5} e^{6} + 840 \, a c^{3} d^{2} x^{4} e^{5} - 1120 \, a c^{3} d^{3} x^{3} e^{4} + 1680 \, a c^{3} d^{4} x^{2} e^{3} - 3360 \, a c^{3} d^{5} x e^{2} + 1260 \, a^{2} c^{2} x^{4} e^{7} - 1680 \, a^{2} c^{2} d x^{3} e^{6} + 2520 \, a^{2} c^{2} d^{2} x^{2} e^{5} - 5040 \, a^{2} c^{2} d^{3} x e^{4} + 1680 \, a^{3} c x^{2} e^{7} - 3360 \, a^{3} c d x e^{6}\right )} e^{\left (-8\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^4/(e*x+d),x, algorithm="giac")

[Out]

(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*e^(-9)*log(abs(x*e + d)) + 1/840*(
105*c^4*x^8*e^7 - 120*c^4*d*x^7*e^6 + 140*c^4*d^2*x^6*e^5 - 168*c^4*d^3*x^5*e^4 + 210*c^4*d^4*x^4*e^3 - 280*c^
4*d^5*x^3*e^2 + 420*c^4*d^6*x^2*e - 840*c^4*d^7*x + 560*a*c^3*x^6*e^7 - 672*a*c^3*d*x^5*e^6 + 840*a*c^3*d^2*x^
4*e^5 - 1120*a*c^3*d^3*x^3*e^4 + 1680*a*c^3*d^4*x^2*e^3 - 3360*a*c^3*d^5*x*e^2 + 1260*a^2*c^2*x^4*e^7 - 1680*a
^2*c^2*d*x^3*e^6 + 2520*a^2*c^2*d^2*x^2*e^5 - 5040*a^2*c^2*d^3*x*e^4 + 1680*a^3*c*x^2*e^7 - 3360*a^3*c*d*x*e^6
)*e^(-8)