3.9.68 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=32 \[ \frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 32, 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} \begin {gather*} \frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))/(3*e)

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 )^{5/2}}{(d+e x)^3} \, dx &=c^2 \int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx\\ &=\frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 21, normalized size = 0.66 \begin {gather*} \frac {c \left (c (d+e x)^2\right )^{3/2}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(c*(d + e*x)^2)^(3/2))/(3*e)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.05, size = 21, normalized size = 0.66 \begin {gather*} \frac {c \left (c (d+e x)^2\right )^{3/2}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(c*(d + e*x)^2)^(3/2))/(3*e)

________________________________________________________________________________________

fricas [B]  time = 0.39, size = 61, normalized size = 1.91 \begin {gather*} \frac {{\left (c^{2} e^{2} x^{3} + 3 \, c^{2} d e x^{2} + 3 \, c^{2} d^{2} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \, {\left (e x + d\right )}} \end {gather*}

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

[Out]

1/3*(c^2*e^2*x^3 + 3*c^2*d*e*x^2 + 3*c^2*d^2*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((16*c^2*exp(1)^4*1/96/exp(1)^3*x+32*c
^2*exp(1)^3*d*1/96/exp(1)^3)*x+16*c^2*exp(1)^2*d^2*1/96/exp(1)^3)*sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))+2*(-
(-9*c^3*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^4*exp(1)^7+28*c^3*exp(2)*(sqrt(c*d^2+2*
c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)^3*d^4*exp(1)^5-29*c^3*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*e
xp(2))-sqrt(c*exp(2))*x)^3*d^4*exp(1)^3+10*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)^3*d^4*exp(1)+15*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)^2*d^5*exp(1
)^6-48*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)^2*d^5*exp(1)^4+51*
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)^2*d^5*exp(1)^2-18*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)^2*d^5-7*c^4*(sqrt(c*d^2+2*c*
d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^6*exp(1)^7+24*c^4*exp(2)*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))
-sqrt(c*exp(2))*x)*d^6*exp(1)^5-27*c^4*exp(2)^2*(sqrt(c*d^2+2*c*d*x*exp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d^6
*exp(1)^3+10*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)*d^6*exp(1)-c^4*sqrt(c*exp
(2))*d^7*exp(1)^6+2*c^4*exp(2)*sqrt(c*exp(2))*d^7*exp(1)^4-c^4*exp(2)^2*sqrt(c*exp(2))*d^7*exp(1)^2)/2/exp(1)^
6/(-(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*e
xp(1)+c*x^2*exp(2))-sqrt(c*exp(2))*x)*d-c*d^2*exp(1))^2+(-15*c^3*d^4*exp(1)^6+50*c^3*exp(2)*d^4*exp(1)^4-55*c^
3*exp(2)^2*d^4*exp(1)^2+20*c^3*exp(2)^3*d^4)/2/exp(1)^6/d/sqrt(c*exp(1)^2-c*exp(2))*atan((-d*sqrt(c*exp(2))+(s
qrt(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))))

________________________________________________________________________________________

maple [A]  time = 0.04, size = 51, normalized size = 1.59 \begin {gather*} \frac {\left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} x}{3 \left (e x +d \right )^{5}} \end {gather*}

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)^(5/2)/(e*x+d)^3,x)

[Out]

1/3*x*(e^2*x^2+3*d*e*x+3*d^2)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^5

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

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)^(5/2)/(d + e*x)^3,x)

[Out]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^3, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \end {gather*}

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

[Out]

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

________________________________________________________________________________________