3.5.82 \(\int \frac {(a+c x^2)^3}{(d+e x)^5} \, dx\) [482]

Optimal. Leaf size=171 \[ -\frac {5 c^3 d x}{e^6}+\frac {c^3 x^2}{2 e^5}-\frac {\left (c d^2+a e^2\right )^3}{4 e^7 (d+e x)^4}+\frac {2 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^3}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^2}+\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^7 (d+e x)}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) \log (d+e x)}{e^7} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 171, 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 {4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac {3 c^2 \left (a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac {2 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^3}-\frac {\left (a e^2+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac {5 c^3 d x}{e^6}+\frac {c^3 x^2}{2 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/(d + e*x)^5,x]

[Out]

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

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Mathematica [A]
time = 0.04, size = 185, normalized size = 1.08 \begin {gather*} \frac {-a^3 e^6-a^2 c e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+a c^2 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+c^3 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )+12 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^4 \log (d+e x)}{4 e^7 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^3/(d + e*x)^5,x]

[Out]

(-(a^3*e^6) - a^2*c*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + a*c^2*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^
3*x^3) + c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*
x^6) + 12*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^4*Log[d + e*x])/(4*e^7*(d + e*x)^4)

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Maple [A]
time = 0.42, size = 205, normalized size = 1.20

method	result	size
risch	\(\frac {c^{3} x^{2}}{2 e^{5}}-\frac {5 c^{3} d x}{e^{6}}+\frac {\left (12 d \,e^{4} c^{2} a +20 d^{3} e^{2} c^{3}\right ) x^{3}-\frac {3 e c \left (a^{2} e^{4}-18 a c \,d^{2} e^{2}-35 c^{2} d^{4}\right ) x^{2}}{2}+\left (-d \,e^{4} a^{2} c +22 d^{3} e^{2} c^{2} a +47 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}+e^{4} d^{2} a^{2} c -25 d^{4} e^{2} c^{2} a -57 d^{6} c^{3}}{4 e}}{e^{6} \left (e x +d \right )^{4}}+\frac {3 c^{2} \ln \left (e x +d \right ) a}{e^{5}}+\frac {15 c^{3} \ln \left (e x +d \right ) d^{2}}{e^{7}}\)	\(203\)
norman	\(\frac {-\frac {e^{6} a^{3}+e^{4} d^{2} a^{2} c -25 d^{4} e^{2} c^{2} a -125 d^{6} c^{3}}{4 e^{7}}+\frac {c^{3} x^{6}}{2 e}-\frac {3 \left (e^{4} a^{2} c -18 d^{2} e^{2} c^{2} a -90 d^{4} c^{3}\right ) x^{2}}{2 e^{5}}-\frac {3 c^{3} d \,x^{5}}{e^{2}}+\frac {4 d \left (3 e^{2} c^{2} a +15 c^{3} d^{2}\right ) x^{3}}{e^{4}}-\frac {d \left (e^{4} a^{2} c -22 d^{2} e^{2} c^{2} a -110 d^{4} c^{3}\right ) x}{e^{6}}}{\left (e x +d \right )^{4}}+\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(204\)
default	\(-\frac {c^{3} \left (-\frac {1}{2} e \,x^{2}+5 d x \right )}{e^{6}}+\frac {2 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{7} \left (e x +d \right )^{3}}+\frac {4 c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{e^{7} \left (e x +d \right )}+\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{4 e^{7} \left (e x +d \right )^{4}}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^3/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 224, normalized size = 1.31 \begin {gather*} 3 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{2} \, {\left (c^{3} x^{2} e - 10 \, c^{3} d x\right )} e^{\left (-6\right )} + \frac {57 \, c^{3} d^{6} + 25 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + 16 \, {\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} - a^{3} e^{6} + 6 \, {\left (35 \, c^{3} d^{4} e^{2} + 18 \, a c^{2} d^{2} e^{4} - a^{2} c e^{6}\right )} x^{2} + 4 \, {\left (47 \, c^{3} d^{5} e + 22 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x}{4 \, {\left (x^{4} e^{11} + 4 \, d x^{3} e^{10} + 6 \, d^{2} x^{2} e^{9} + 4 \, d^{3} x e^{8} + d^{4} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^5,x, algorithm="maxima")

[Out]

3*(5*c^3*d^2 + a*c^2*e^2)*e^(-7)*log(x*e + d) + 1/2*(c^3*x^2*e - 10*c^3*d*x)*e^(-6) + 1/4*(57*c^3*d^6 + 25*a*c
^2*d^4*e^2 - a^2*c*d^2*e^4 + 16*(5*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 - a^3*e^6 + 6*(35*c^3*d^4*e^2 + 18*a*c^2*d
^2*e^4 - a^2*c*e^6)*x^2 + 4*(47*c^3*d^5*e + 22*a*c^2*d^3*e^3 - a^2*c*d*e^5)*x)/(x^4*e^11 + 4*d*x^3*e^10 + 6*d^
2*x^2*e^9 + 4*d^3*x*e^8 + d^4*e^7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (157) = 314\).
time = 1.85, size = 332, normalized size = 1.94 \begin {gather*} \frac {168 \, c^{3} d^{5} x e + 57 \, c^{3} d^{6} + {\left (2 \, c^{3} x^{6} - 6 \, a^{2} c x^{2} - a^{3}\right )} e^{6} - 4 \, {\left (3 \, c^{3} d x^{5} - 12 \, a c^{2} d x^{3} + a^{2} c d x\right )} e^{5} - {\left (68 \, c^{3} d^{2} x^{4} - 108 \, a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{4} - 8 \, {\left (4 \, c^{3} d^{3} x^{3} - 11 \, a c^{2} d^{3} x\right )} e^{3} + {\left (132 \, c^{3} d^{4} x^{2} + 25 \, a c^{2} d^{4}\right )} e^{2} + 12 \, {\left (20 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + a c^{2} x^{4} e^{6} + 4 \, a c^{2} d x^{3} e^{5} + {\left (5 \, c^{3} d^{2} x^{4} + 6 \, a c^{2} d^{2} x^{2}\right )} e^{4} + 4 \, {\left (5 \, c^{3} d^{3} x^{3} + a c^{2} d^{3} x\right )} e^{3} + {\left (30 \, c^{3} d^{4} x^{2} + a c^{2} d^{4}\right )} e^{2}\right )} \log \left (x e + d\right )}{4 \, {\left (x^{4} e^{11} + 4 \, d x^{3} e^{10} + 6 \, d^{2} x^{2} e^{9} + 4 \, d^{3} x e^{8} + d^{4} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/4*(168*c^3*d^5*x*e + 57*c^3*d^6 + (2*c^3*x^6 - 6*a^2*c*x^2 - a^3)*e^6 - 4*(3*c^3*d*x^5 - 12*a*c^2*d*x^3 + a^
2*c*d*x)*e^5 - (68*c^3*d^2*x^4 - 108*a*c^2*d^2*x^2 + a^2*c*d^2)*e^4 - 8*(4*c^3*d^3*x^3 - 11*a*c^2*d^3*x)*e^3 +
(132*c^3*d^4*x^2 + 25*a*c^2*d^4)*e^2 + 12*(20*c^3*d^5*x*e + 5*c^3*d^6 + a*c^2*x^4*e^6 + 4*a*c^2*d*x^3*e^5 + (
5*c^3*d^2*x^4 + 6*a*c^2*d^2*x^2)*e^4 + 4*(5*c^3*d^3*x^3 + a*c^2*d^3*x)*e^3 + (30*c^3*d^4*x^2 + a*c^2*d^4)*e^2)
*log(x*e + d))/(x^4*e^11 + 4*d*x^3*e^10 + 6*d^2*x^2*e^9 + 4*d^3*x*e^8 + d^4*e^7)

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Sympy [A]
time = 1.93, size = 243, normalized size = 1.42 \begin {gather*} - \frac {5 c^{3} d x}{e^{6}} + \frac {c^{3} x^{2}}{2 e^{5}} + \frac {3 c^{2} \left (a e^{2} + 5 c d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + \frac {- a^{3} e^{6} - a^{2} c d^{2} e^{4} + 25 a c^{2} d^{4} e^{2} + 57 c^{3} d^{6} + x^{3} \cdot \left (48 a c^{2} d e^{5} + 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 6 a^{2} c e^{6} + 108 a c^{2} d^{2} e^{4} + 210 c^{3} d^{4} e^{2}\right ) + x \left (- 4 a^{2} c d e^{5} + 88 a c^{2} d^{3} e^{3} + 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**3/(e*x+d)**5,x)

[Out]

-5*c**3*d*x/e**6 + c**3*x**2/(2*e**5) + 3*c**2*(a*e**2 + 5*c*d**2)*log(d + e*x)/e**7 + (-a**3*e**6 - a**2*c*d*
*2*e**4 + 25*a*c**2*d**4*e**2 + 57*c**3*d**6 + x**3*(48*a*c**2*d*e**5 + 80*c**3*d**3*e**3) + x**2*(-6*a**2*c*e
**6 + 108*a*c**2*d**2*e**4 + 210*c**3*d**4*e**2) + x*(-4*a**2*c*d*e**5 + 88*a*c**2*d**3*e**3 + 188*c**3*d**5*e
))/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4)

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Giac [A]
time = 0.88, size = 288, normalized size = 1.68 \begin {gather*} \frac {1}{2} \, {\left (c^{3} - \frac {12 \, c^{3} d}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-7\right )} - 3 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac {1}{4} \, {\left (\frac {80 \, c^{3} d^{3} e^{29}}{x e + d} - \frac {30 \, c^{3} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac {8 \, c^{3} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac {c^{3} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} + \frac {48 \, a c^{2} d e^{31}}{x e + d} - \frac {36 \, a c^{2} d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac {16 \, a c^{2} d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac {3 \, a c^{2} d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac {6 \, a^{2} c e^{33}}{{\left (x e + d\right )}^{2}} + \frac {8 \, a^{2} c d e^{33}}{{\left (x e + d\right )}^{3}} - \frac {3 \, a^{2} c d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac {a^{3} e^{35}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^3/(e*x+d)^5,x, algorithm="giac")

[Out]

1/2*(c^3 - 12*c^3*d/(x*e + d))*(x*e + d)^2*e^(-7) - 3*(5*c^3*d^2 + a*c^2*e^2)*e^(-7)*log(abs(x*e + d)*e^(-1)/(
x*e + d)^2) + 1/4*(80*c^3*d^3*e^29/(x*e + d) - 30*c^3*d^4*e^29/(x*e + d)^2 + 8*c^3*d^5*e^29/(x*e + d)^3 - c^3*
d^6*e^29/(x*e + d)^4 + 48*a*c^2*d*e^31/(x*e + d) - 36*a*c^2*d^2*e^31/(x*e + d)^2 + 16*a*c^2*d^3*e^31/(x*e + d)
^3 - 3*a*c^2*d^4*e^31/(x*e + d)^4 - 6*a^2*c*e^33/(x*e + d)^2 + 8*a^2*c*d*e^33/(x*e + d)^3 - 3*a^2*c*d^2*e^33/(
x*e + d)^4 - a^3*e^35/(x*e + d)^4)*e^(-36)

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Mupad [B]
time = 0.09, size = 236, normalized size = 1.38 \begin {gather*} \frac {x^2\,\left (-\frac {3\,a^2\,c\,e^5}{2}+27\,a\,c^2\,d^2\,e^3+\frac {105\,c^3\,d^4\,e}{2}\right )-\frac {a^3\,e^6+a^2\,c\,d^2\,e^4-25\,a\,c^2\,d^4\,e^2-57\,c^3\,d^6}{4\,e}+x\,\left (-a^2\,c\,d\,e^4+22\,a\,c^2\,d^3\,e^2+47\,c^3\,d^5\right )+x^3\,\left (20\,c^3\,d^3\,e^2+12\,a\,c^2\,d\,e^4\right )}{d^4\,e^6+4\,d^3\,e^7\,x+6\,d^2\,e^8\,x^2+4\,d\,e^9\,x^3+e^{10}\,x^4}+\frac {\ln \left (d+e\,x\right )\,\left (15\,c^3\,d^2+3\,a\,c^2\,e^2\right )}{e^7}+\frac {c^3\,x^2}{2\,e^5}-\frac {5\,c^3\,d\,x}{e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3/(d + e*x)^5,x)

[Out]

(x^2*((105*c^3*d^4*e)/2 - (3*a^2*c*e^5)/2 + 27*a*c^2*d^2*e^3) - (a^3*e^6 - 57*c^3*d^6 - 25*a*c^2*d^4*e^2 + a^2
*c*d^2*e^4)/(4*e) + x*(47*c^3*d^5 + 22*a*c^2*d^3*e^2 - a^2*c*d*e^4) + x^3*(20*c^3*d^3*e^2 + 12*a*c^2*d*e^4))/(
d^4*e^6 + e^10*x^4 + 4*d^3*e^7*x + 4*d*e^9*x^3 + 6*d^2*e^8*x^2) + (log(d + e*x)*(15*c^3*d^2 + 3*a*c^2*e^2))/e^
7 + (c^3*x^2)/(2*e^5) - (5*c^3*d*x)/e^6

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