\(\int \frac {(a+c x^2)^3}{(d+e x)^3} \, dx\) [480]

Optimal result

Integrand size = 17, antiderivative size = 163 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {c^2 d \left (10 c d^2+9 a e^2\right ) x}{e^6}+\frac {3 c^2 \left (2 c d^2+a e^2\right ) x^2}{2 e^5}-\frac {c^3 d x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {\left (c d^2+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac {6 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \log (d+e x)}{e^7} \]

[Out]

-c^2*d*(9*a*e^2+10*c*d^2)*x/e^6+3/2*c^2*(a*e^2+2*c*d^2)*x^2/e^5-c^3*d*x^3/e^4+1/4*c^3*x^4/e^3-1/2*(a*e^2+c*d^2
)^3/e^7/(e*x+d)^2+6*c*d*(a*e^2+c*d^2)^2/e^7/(e*x+d)+3*c*(a*e^2+c*d^2)*(a*e^2+5*c*d^2)*ln(e*x+d)/e^7

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {c^2 d x \left (9 a e^2+10 c d^2\right )}{e^6}+\frac {3 c^2 x^2 \left (a e^2+2 c d^2\right )}{2 e^5}+\frac {6 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)}-\frac {\left (a e^2+c d^2\right )^3}{2 e^7 (d+e x)^2}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac {c^3 d x^3}{e^4}+\frac {c^3 x^4}{4 e^3} \]

[In]

Int[(a + c*x^2)^3/(d + e*x)^3,x]

[Out]

-((c^2*d*(10*c*d^2 + 9*a*e^2)*x)/e^6) + (3*c^2*(2*c*d^2 + a*e^2)*x^2)/(2*e^5) - (c^3*d*x^3)/e^4 + (c^3*x^4)/(4
*e^3) - (c*d^2 + a*e^2)^3/(2*e^7*(d + e*x)^2) + (6*c*d*(c*d^2 + a*e^2)^2)/(e^7*(d + e*x)) + (3*c*(c*d^2 + a*e^
2)*(5*c*d^2 + a*e^2)*Log[d + e*x])/e^7

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*} \text {integral}& = \int \left (-\frac {c^2 d \left (10 c d^2+9 a e^2\right )}{e^6}+\frac {3 c^2 \left (2 c d^2+a e^2\right ) x}{e^5}-\frac {3 c^3 d x^2}{e^4}+\frac {c^3 x^3}{e^3}+\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^3}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^2}+\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)}\right ) \, dx \\ & = -\frac {c^2 d \left (10 c d^2+9 a e^2\right ) x}{e^6}+\frac {3 c^2 \left (2 c d^2+a e^2\right ) x^2}{2 e^5}-\frac {c^3 d x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {\left (c d^2+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac {6 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \log (d+e x)}{e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {-2 a^3 e^6+6 a^2 c d e^4 (3 d+4 e x)+6 a c^2 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+c^3 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )+12 c \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right ) (d+e x)^2 \log (d+e x)}{4 e^7 (d+e x)^2} \]

[In]

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

[Out]

(-2*a^3*e^6 + 6*a^2*c*d*e^4*(3*d + 4*e*x) + 6*a*c^2*e^2*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^
4*x^4) + c^3*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6) +
12*c*(5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*(d + e*x)^2*Log[d + e*x])/(4*e^7*(d + e*x)^2)

Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.17

method	result	size
default	\(-\frac {c^{2} \left (-\frac {1}{4} c \,x^{4} e^{3}+c d \,e^{2} x^{3}-\frac {3}{2} a \,e^{3} x^{2}-3 c \,d^{2} e \,x^{2}+9 a d \,e^{2} x +10 c \,d^{3} x \right )}{e^{6}}+\frac {6 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{7} \left (e x +d \right )}-\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{2 e^{7} \left (e x +d \right )^{2}}+\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(191\)
norman	\(\frac {-\frac {e^{6} a^{3}-9 d^{2} e^{4} a^{2} c -54 d^{4} e^{2} c^{2} a -45 c^{3} d^{6}}{2 e^{7}}+\frac {c^{3} x^{6}}{4 e}+\frac {c^{2} \left (6 e^{2} a +5 c \,d^{2}\right ) x^{4}}{4 e^{3}}-\frac {c^{3} d \,x^{5}}{2 e^{2}}+\frac {2 d \left (3 e^{4} a^{2} c +18 d^{2} e^{2} c^{2} a +15 d^{4} c^{3}\right ) x}{e^{6}}-\frac {c^{2} d \left (6 e^{2} a +5 c \,d^{2}\right ) x^{3}}{e^{4}}}{\left (e x +d \right )^{2}}+\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(203\)
risch	\(\frac {c^{3} x^{4}}{4 e^{3}}-\frac {c^{3} d \,x^{3}}{e^{4}}+\frac {3 c^{2} a \,x^{2}}{2 e^{3}}+\frac {3 c^{3} d^{2} x^{2}}{e^{5}}-\frac {9 c^{2} a d x}{e^{4}}-\frac {10 c^{3} d^{3} x}{e^{6}}+\frac {\left (6 d \,e^{4} a^{2} c +12 d^{3} e^{2} c^{2} a +6 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}-9 d^{2} e^{4} a^{2} c -21 d^{4} e^{2} c^{2} a -11 c^{3} d^{6}}{2 e}}{e^{6} \left (e x +d \right )^{2}}+\frac {3 c \ln \left (e x +d \right ) a^{2}}{e^{3}}+\frac {18 c^{2} \ln \left (e x +d \right ) a \,d^{2}}{e^{5}}+\frac {15 c^{3} \ln \left (e x +d \right ) d^{4}}{e^{7}}\)	\(214\)
parallelrisch	\(\frac {144 \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{3}+24 \ln \left (e x +d \right ) x \,a^{2} c d \,e^{5}+108 d^{4} e^{2} c^{2} a +18 d^{2} e^{4} a^{2} c +5 x^{4} c^{3} d^{2} e^{4}-20 x^{3} c^{3} d^{3} e^{3}+120 x \,c^{3} d^{5} e +90 c^{3} d^{6}+24 x \,a^{2} c d \,e^{5}+144 x a \,c^{2} d^{3} e^{3}+120 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +x^{6} c^{3} e^{6}+72 \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{2} e^{4}-2 e^{6} a^{3}+12 \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{4}+72 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}+60 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}-24 x^{3} a \,c^{2} d \,e^{5}-2 x^{5} c^{3} d \,e^{5}+12 \ln \left (e x +d \right ) x^{2} a^{2} c \,e^{6}+6 x^{4} a \,c^{2} e^{6}+60 \ln \left (e x +d \right ) c^{3} d^{6}}{4 e^{7} \left (e x +d \right )^{2}}\)	\(324\)

[In]

int((c*x^2+a)^3/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

-c^2/e^6*(-1/4*c*x^4*e^3+c*d*e^2*x^3-3/2*a*e^3*x^2-3*c*d^2*e*x^2+9*a*d*e^2*x+10*c*d^3*x)+6*c*d/e^7*(a^2*e^4+2*
a*c*d^2*e^2+c^2*d^4)/(e*x+d)-1/2*(a^3*e^6+3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2+c^3*d^6)/e^7/(e*x+d)^2+3/e^7*c*(a^2*
e^4+6*a*c*d^2*e^2+5*c^2*d^4)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (157) = 314\).

Time = 0.25 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {c^{3} e^{6} x^{6} - 2 \, c^{3} d e^{5} x^{5} + 22 \, c^{3} d^{6} + 42 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + {\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} - 4 \, {\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} - 2 \, {\left (34 \, c^{3} d^{4} e^{2} + 33 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 4 \, {\left (4 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} - 6 \, a^{2} c d e^{5}\right )} x + 12 \, {\left (5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + {\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 2 \, {\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

[In]

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

[Out]

1/4*(c^3*e^6*x^6 - 2*c^3*d*e^5*x^5 + 22*c^3*d^6 + 42*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 - 2*a^3*e^6 + (5*c^3*d^2
*e^4 + 6*a*c^2*e^6)*x^4 - 4*(5*c^3*d^3*e^3 + 6*a*c^2*d*e^5)*x^3 - 2*(34*c^3*d^4*e^2 + 33*a*c^2*d^2*e^4)*x^2 -
4*(4*c^3*d^5*e - 3*a*c^2*d^3*e^3 - 6*a^2*c*d*e^5)*x + 12*(5*c^3*d^6 + 6*a*c^2*d^4*e^2 + a^2*c*d^2*e^4 + (5*c^3
*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)*x^2 + 2*(5*c^3*d^5*e + 6*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)*log(e*x + d))
/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

Sympy [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^3} \, dx=- \frac {c^{3} d x^{3}}{e^{4}} + \frac {c^{3} x^{4}}{4 e^{3}} + \frac {3 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + x^{2} \cdot \left (\frac {3 a c^{2}}{2 e^{3}} + \frac {3 c^{3} d^{2}}{e^{5}}\right ) + x \left (- \frac {9 a c^{2} d}{e^{4}} - \frac {10 c^{3} d^{3}}{e^{6}}\right ) + \frac {- a^{3} e^{6} + 9 a^{2} c d^{2} e^{4} + 21 a c^{2} d^{4} e^{2} + 11 c^{3} d^{6} + x \left (12 a^{2} c d e^{5} + 24 a c^{2} d^{3} e^{3} + 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} \]

[In]

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

[Out]

-c**3*d*x**3/e**4 + c**3*x**4/(4*e**3) + 3*c*(a*e**2 + c*d**2)*(a*e**2 + 5*c*d**2)*log(d + e*x)/e**7 + x**2*(3
*a*c**2/(2*e**3) + 3*c**3*d**2/e**5) + x*(-9*a*c**2*d/e**4 - 10*c**3*d**3/e**6) + (-a**3*e**6 + 9*a**2*c*d**2*
e**4 + 21*a*c**2*d**4*e**2 + 11*c**3*d**6 + x*(12*a**2*c*d*e**5 + 24*a*c**2*d**3*e**3 + 12*c**3*d**5*e))/(2*d*
*2*e**7 + 4*d*e**8*x + 2*e**9*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {11 \, c^{3} d^{6} + 21 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 12 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac {c^{3} e^{3} x^{4} - 4 \, c^{3} d e^{2} x^{3} + 6 \, {\left (2 \, c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{2} - 4 \, {\left (10 \, c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x}{4 \, e^{6}} + \frac {3 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

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

[Out]

1/2*(11*c^3*d^6 + 21*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - a^3*e^6 + 12*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + a^2*c*d*e^5
)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(c^3*e^3*x^4 - 4*c^3*d*e^2*x^3 + 6*(2*c^3*d^2*e + a*c^2*e^3)*x^2 -
4*(10*c^3*d^3 + 9*a*c^2*d*e^2)*x)/e^6 + 3*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*log(e*x + d)/e^7

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {3 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {11 \, c^{3} d^{6} + 21 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 12 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{7}} + \frac {c^{3} e^{9} x^{4} - 4 \, c^{3} d e^{8} x^{3} + 12 \, c^{3} d^{2} e^{7} x^{2} + 6 \, a c^{2} e^{9} x^{2} - 40 \, c^{3} d^{3} e^{6} x - 36 \, a c^{2} d e^{8} x}{4 \, e^{12}} \]

[In]

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

[Out]

3*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*log(abs(e*x + d))/e^7 + 1/2*(11*c^3*d^6 + 21*a*c^2*d^4*e^2 + 9*a^2
*c*d^2*e^4 - a^3*e^6 + 12*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)/((e*x + d)^2*e^7) + 1/4*(c^3*e^9*x^4
- 4*c^3*d*e^8*x^3 + 12*c^3*d^2*e^7*x^2 + 6*a*c^2*e^9*x^2 - 40*c^3*d^3*e^6*x - 36*a*c^2*d*e^8*x)/e^12

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^3} \, dx=x^2\,\left (\frac {3\,a\,c^2}{2\,e^3}+\frac {3\,c^3\,d^2}{e^5}\right )+\frac {\frac {-a^3\,e^6+9\,a^2\,c\,d^2\,e^4+21\,a\,c^2\,d^4\,e^2+11\,c^3\,d^6}{2\,e}+x\,\left (6\,a^2\,c\,d\,e^4+12\,a\,c^2\,d^3\,e^2+6\,c^3\,d^5\right )}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x\,\left (\frac {8\,c^3\,d^3}{e^6}-\frac {3\,d\,\left (\frac {3\,a\,c^2}{e^3}+\frac {6\,c^3\,d^2}{e^5}\right )}{e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,a^2\,c\,e^4+18\,a\,c^2\,d^2\,e^2+15\,c^3\,d^4\right )}{e^7}+\frac {c^3\,x^4}{4\,e^3}-\frac {c^3\,d\,x^3}{e^4} \]

[In]

int((a + c*x^2)^3/(d + e*x)^3,x)

[Out]

x^2*((3*a*c^2)/(2*e^3) + (3*c^3*d^2)/e^5) + ((11*c^3*d^6 - a^3*e^6 + 21*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4)/(2*e)
+ x*(6*c^3*d^5 + 12*a*c^2*d^3*e^2 + 6*a^2*c*d*e^4))/(d^2*e^6 + e^8*x^2 + 2*d*e^7*x) + x*((8*c^3*d^3)/e^6 - (3
*d*((3*a*c^2)/e^3 + (6*c^3*d^2)/e^5))/e) + (log(d + e*x)*(15*c^3*d^4 + 3*a^2*c*e^4 + 18*a*c^2*d^2*e^2))/e^7 +
(c^3*x^4)/(4*e^3) - (c^3*d*x^3)/e^4