### 3.484 $$\int \frac{(a+c x^2)^3}{(d+e x)^7} \, dx$$

Optimal. Leaf size=184 $-\frac{3 c^2 \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{6 c d \left (a e^2+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{6 c^3 d}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7}$

[Out]

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

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Rubi [A]  time = 0.135293, antiderivative size = 184, 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} $-\frac{3 c^2 \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac{4 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac{6 c d \left (a e^2+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{6 c^3 d}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^3/(d + e*x)^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0623787, size = 172, normalized size = 0.93 $\frac{-3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )-10 a^3 e^6-6 a c^2 e^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+c^3 d \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^7,x]

[Out]

(-10*a^3*e^6 - 3*a^2*c*e^4*(d^2 + 6*d*e*x + 15*e^2*x^2) - 6*a*c^2*e^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) + c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 +
360*e^5*x^5) + 60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)

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Maple [A]  time = 0.049, size = 278, normalized size = 1.5 \begin{align*} -{\frac{3\,{a}^{2}c}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{9\,a{c}^{2}{d}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{c}^{3}{d}^{4}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{a}^{3}}{6\,e \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{2}c{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{a{c}^{2}{d}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{d}^{6}{c}^{3}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}+4\,{\frac{a{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{20\,{c}^{3}{d}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{3}\ln \left ( ex+d \right ) }{{e}^{7}}}+{\frac{6\,cd{a}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}+{\frac{12\,a{c}^{2}{d}^{3}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}+{\frac{6\,{c}^{3}{d}^{5}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}+6\,{\frac{{c}^{3}d}{{e}^{7} \left ( ex+d \right ) }}-{\frac{3\,a{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{c}^{3}{d}^{2}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^7,x)

[Out]

-3/4*c/e^3/(e*x+d)^4*a^2-9/2*c^2/e^5/(e*x+d)^4*a*d^2-15/4*c^3/e^7/(e*x+d)^4*d^4-1/6/e/(e*x+d)^6*a^3-1/2/e^3/(e
*x+d)^6*a^2*c*d^2-1/2/e^5/(e*x+d)^6*d^4*a*c^2-1/6*d^6/e^7/(e*x+d)^6*c^3+4*c^2*d/e^5/(e*x+d)^3*a+20/3/e^7/(e*x+
d)^3*c^3*d^3+c^3*ln(e*x+d)/e^7+6/5*c*d/e^3/(e*x+d)^5*a^2+12/5*c^2*d^3/e^5/(e*x+d)^5*a+6/5*c^3*d^5/e^7/(e*x+d)^
5+6*c^3*d/e^7/(e*x+d)-3/2*c^2/e^5/(e*x+d)^2*a-15/2*c^3/e^7/(e*x+d)^2*d^2

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Maxima [A]  time = 1.25224, size = 355, normalized size = 1.93 \begin{align*} \frac{360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \,{\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{c^{3} \log \left (e x + d\right )}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/60*(360*c^3*d*e^5*x^5 + 147*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 10*a^3*e^6 + 90*(15*c^3*d^2*e^4 -
a*c^2*e^6)*x^4 + 40*(55*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 15*(125*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 - 3*a^2*c*e^6
)*x^2 + 6*(137*c^3*d^5*e - 6*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20
*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + c^3*log(e*x + d)/e^7

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Fricas [A]  time = 1.98179, size = 710, normalized size = 3.86 \begin{align*} \frac{360 \, c^{3} d e^{5} x^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \,{\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x + 60 \,{\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/60*(360*c^3*d*e^5*x^5 + 147*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 10*a^3*e^6 + 90*(15*c^3*d^2*e^4 -
a*c^2*e^6)*x^4 + 40*(55*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 15*(125*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 - 3*a^2*c*e^6
)*x^2 + 6*(137*c^3*d^5*e - 6*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*x + 60*(c^3*e^6*x^6 + 6*c^3*d*e^5*x^5 + 15*c^3*d^2
*e^4*x^4 + 20*c^3*d^3*e^3*x^3 + 15*c^3*d^4*e^2*x^2 + 6*c^3*d^5*e*x + c^3*d^6)*log(e*x + d))/(e^13*x^6 + 6*d*e^
12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

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Sympy [A]  time = 11.546, size = 272, normalized size = 1.48 \begin{align*} \frac{c^{3} \log{\left (d + e x \right )}}{e^{7}} + \frac{- 10 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 6 a c^{2} d^{4} e^{2} + 147 c^{3} d^{6} + 360 c^{3} d e^{5} x^{5} + x^{4} \left (- 90 a c^{2} e^{6} + 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 120 a c^{2} d e^{5} + 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 45 a^{2} c e^{6} - 90 a c^{2} d^{2} e^{4} + 1875 c^{3} d^{4} e^{2}\right ) + x \left (- 18 a^{2} c d e^{5} - 36 a c^{2} d^{3} e^{3} + 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**7,x)

[Out]

c**3*log(d + e*x)/e**7 + (-10*a**3*e**6 - 3*a**2*c*d**2*e**4 - 6*a*c**2*d**4*e**2 + 147*c**3*d**6 + 360*c**3*d
*e**5*x**5 + x**4*(-90*a*c**2*e**6 + 1350*c**3*d**2*e**4) + x**3*(-120*a*c**2*d*e**5 + 2200*c**3*d**3*e**3) +
x**2*(-45*a**2*c*e**6 - 90*a*c**2*d**2*e**4 + 1875*c**3*d**4*e**2) + x*(-18*a**2*c*d*e**5 - 36*a*c**2*d**3*e**
3 + 822*c**3*d**5*e))/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e
**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6)

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Giac [A]  time = 1.29102, size = 265, normalized size = 1.44 \begin{align*} c^{3} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (360 \, c^{3} d x^{5} e^{4} + 90 \,{\left (15 \, c^{3} d^{2} e^{3} - a c^{2} e^{5}\right )} x^{4} + 40 \,{\left (55 \, c^{3} d^{3} e^{2} - 3 \, a c^{2} d e^{4}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e - 6 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} x +{\left (147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^7,x, algorithm="giac")

[Out]

c^3*e^(-7)*log(abs(x*e + d)) + 1/60*(360*c^3*d*x^5*e^4 + 90*(15*c^3*d^2*e^3 - a*c^2*e^5)*x^4 + 40*(55*c^3*d^3*
e^2 - 3*a*c^2*d*e^4)*x^3 + 15*(125*c^3*d^4*e - 6*a*c^2*d^2*e^3 - 3*a^2*c*e^5)*x^2 + 6*(137*c^3*d^5 - 6*a*c^2*d
^3*e^2 - 3*a^2*c*d*e^4)*x + (147*c^3*d^6 - 6*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 - 10*a^3*e^6)*e^(-1))*e^(-6)/(x*e
+ d)^6