3.1049 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{3/2}}{(d+e x)^7} \, dx\)
Optimal. Leaf size=34 \[ -\frac {c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
[Out]
-1/3*c^3/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used =
{643, 629} \[ -\frac {c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^7,x]
[Out]
-c^3/(3*e*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))
Rule 629
Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]
Rule 643
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c,
0] && !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/2]
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx &=c^4 \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.79 \[ -\frac {\left (c (d+e x)^2\right )^{3/2}}{3 e (d+e x)^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^7,x]
[Out]
-1/3*(c*(d + e*x)^2)^(3/2)/(e*(d + e*x)^6)
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fricas [B] time = 1.05, size = 69, normalized size = 2.03 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c}{3 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^7,x, algorithm="fricas")
[Out]
-1/3*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*c/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x + d^4*e)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^7,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((-105*c^2*(sqrt(c*d^2+2*c*d*x*exp(1)+
c*x^2*exp(2))-sqrt(c*exp(2))*x)^11*exp(1)^11+90*c^2*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp
(2))*x)^11*exp(1)^9+1155*c^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^10*d*ex
p(1)^10-990*c^2*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^10*d*exp(1)^8
-595*c^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^2*exp(1)^11-4670*c^3*exp(2)*(sqrt(c*d^
2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^2*exp(1)^9+4440*c^3*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c
*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^2*exp(1)^7+320*c^3*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c
*exp(2))*x)^9*d^2*exp(1)^5-320*c^3*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^9*d^2*e
xp(1)^3+5355*c^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^8*d^3*exp(1)^10+738
0*c^3*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^8*d^3*exp(1)^8-12660*c^
3*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^8*d^3*exp(1)^6+1920*c^3*e
xp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^8*d^3*exp(1)^4+480*c^3*exp(2
)^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^8*d^3*exp(1)^2-1386*c^4*(sqrt(c*
d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^4*exp(1)^11-17460*c^4*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)
+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^4*exp(1)^9+9240*c^4*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqr
t(c*exp(2))*x)^7*d^4*exp(1)^7+7920*c^4*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d
^4*exp(1)^5-2880*c^4*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^4*exp(1)^3-384*c^
4*exp(2)^5*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^7*d^4*exp(1)+9702*c^4*sqrt(c*exp(2))*(sq
rt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5*exp(1)^10+8180*c^4*exp(2)*sqrt(c*exp(2))*(sqrt(c
*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5*exp(1)^8-14840*c^4*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d
^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5*exp(1)^6+2160*c^4*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+
2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5*exp(1)^4+1600*c^4*exp(2)^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c
*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5*exp(1)^2+128*c^4*exp(2)^5*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x
*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^6*d^5-1686*c^5*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))
*x)^5*d^6*exp(1)^11-8340*c^5*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^6*exp(1)^9+
6120*c^5*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^6*exp(1)^7+240*c^5*exp(2)^3*(
sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^6*exp(1)^5-2880*c^5*exp(2)^4*(sqrt(c*d^2+2*c*d*x
*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^5*d^6*exp(1)^3-384*c^5*exp(2)^5*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2
))-sqrt(c*exp(2))*x)^5*d^6*exp(1)+750*c^5*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2
))*x)^4*d^7*exp(1)^10-1560*c^5*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x
)^4*d^7*exp(1)^8+1440*c^5*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4
*d^7*exp(1)^6+3840*c^5*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^
7*exp(1)^4+480*c^5*exp(2)^4*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^4*d^7*ex
p(1)^2+595*c^6*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^8*exp(1)^11+3290*c^6*exp(2)*(sqr
t(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^8*exp(1)^9-1320*c^6*exp(2)^2*(sqrt(c*d^2+2*c*d*x*ex
p(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^8*exp(1)^7-4720*c^6*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))
-sqrt(c*exp(2))*x)^3*d^8*exp(1)^5-320*c^6*exp(2)^4*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^
3*d^8*exp(1)^3-1785*c^6*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d^9*exp(1)
^10-1350*c^6*exp(2)*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d^9*exp(1)^8+3
720*c^6*exp(2)^2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d^9*exp(1)^6+240*
c^6*exp(2)^3*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^2*d^9*exp(1)^4+105*c^7*
(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^10*exp(1)^11+1170*c^7*exp(2)*(sqrt(c*d^2+2*c*d*x*
exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^10*exp(1)^9-1200*c^7*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2)
)-sqrt(c*exp(2))*x)*d^10*exp(1)^7-240*c^7*exp(2)^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*
d^10*exp(1)^5-105*c^7*sqrt(c*exp(2))*d^11*exp(1)^10+20*c^7*exp(2)*sqrt(c*exp(2))*d^11*exp(1)^8+100*c^7*exp(2)^
2*sqrt(c*exp(2))*d^11*exp(1)^6)/(-240*d^2*exp(1)^6+240*exp(2)*d^2*exp(1)^4)/(-(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2
*exp(2))-sqrt(c*exp(2))*x)^2*exp(1)+2*sqrt(c*exp(2))*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x
)*d-c*d^2*exp(1))^6+(7*c^2*exp(1)^2-6*c^2*exp(2))/2/(8*d^2*exp(1)^2-8*exp(2)*d^2)/d/sqrt(c*exp(1)^2-c*exp(2))*
atan((-d*sqrt(c*exp(2))+(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*exp(1))/d/sqrt(c*exp(1)^2-c
*exp(2))))
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maple [A] time = 0.04, size = 35, normalized size = 1.03 \[ -\frac {\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{3 \left (e x +d \right )^{6} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^7,x)
[Out]
-1/3/(e*x+d)^6/e*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^7,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.
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mupad [B] time = 0.46, size = 35, normalized size = 1.03 \[ -\frac {c\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{3\,e\,{\left (d+e\,x\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(3/2)/(d + e*x)^7,x)
[Out]
-(c*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(1/2))/(3*e*(d + e*x)^4)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2)/(e*x+d)**7,x)
[Out]
Integral((c*(d + e*x)**2)**(3/2)/(d + e*x)**7, x)
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