3.1916 \(\int \frac{\sqrt{a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=171 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^3 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^5 \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^5) + (8*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(3/2))/(35*(c*d^2 - a*e^2)^2*(d + e*x)^4) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2))/(105*(c*d^2 - a*e^2)^3*(d + e*x)^3)

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Rubi [A]  time = 0.080947, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {658, 650} \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^3 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^5 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^5) + (8*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(3/2))/(35*(c*d^2 - a*e^2)^2*(d + e*x)^4) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2))/(105*(c*d^2 - a*e^2)^3*(d + e*x)^3)

Rule 658

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac{(4 c d) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx}{7 \left (c d^2-a e^2\right )}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac{8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac{\left (8 c^2 d^2\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^3} \, dx}{35 \left (c d^2-a e^2\right )^2}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac{8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac{16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0441684, size = 124, normalized size = 0.73 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (3 a^2 c d e^3 (e x-14 d)+15 a^3 e^5+a c^2 d^2 e \left (35 d^2-14 d e x-4 e^2 x^2\right )+c^3 d^3 x \left (35 d^2+28 d e x+8 e^2 x^2\right )\right )}{105 (d+e x)^4 \left (c d^2-a e^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(15*a^3*e^5 + 3*a^2*c*d*e^3*(-14*d + e*x) + a*c^2*d^2*e*(35*d^2 - 14*d*e*x -
4*e^2*x^2) + c^3*d^3*x*(35*d^2 + 28*d*e*x + 8*e^2*x^2)))/(105*(c*d^2 - a*e^2)^3*(d + e*x)^4)

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Maple [A]  time = 0.045, size = 146, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-12\,acd{e}^{3}x+28\,{c}^{2}{d}^{3}ex+15\,{a}^{2}{e}^{4}-42\,ac{d}^{2}{e}^{2}+35\,{c}^{2}{d}^{4} \right ) }{105\, \left ( ex+d \right ) ^{4} \left ({a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,a{c}^{2}{d}^{4}{e}^{2}-{c}^{3}{d}^{6} \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^5,x)

[Out]

-2/105*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-12*a*c*d*e^3*x+28*c^2*d^3*e*x+15*a^2*e^4-42*a*c*d^2*e^2+35*c^2*d^4)*(c*d
*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(e*x+d)^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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 17.0962, size = 740, normalized size = 4.33 \begin{align*} \frac{2 \,{\left (8 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 42 \, a^{2} c d^{2} e^{3} + 15 \, a^{3} e^{5} + 4 \,{\left (7 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} +{\left (35 \, c^{3} d^{5} - 14 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{105 \,{\left (c^{3} d^{10} - 3 \, a c^{2} d^{8} e^{2} + 3 \, a^{2} c d^{6} e^{4} - a^{3} d^{4} e^{6} +{\left (c^{3} d^{6} e^{4} - 3 \, a c^{2} d^{4} e^{6} + 3 \, a^{2} c d^{2} e^{8} - a^{3} e^{10}\right )} x^{4} + 4 \,{\left (c^{3} d^{7} e^{3} - 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} - a^{3} d e^{9}\right )} x^{3} + 6 \,{\left (c^{3} d^{8} e^{2} - 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} - a^{3} d^{2} e^{8}\right )} x^{2} + 4 \,{\left (c^{3} d^{9} e - 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} - a^{3} d^{3} e^{7}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

2/105*(8*c^3*d^3*e^2*x^3 + 35*a*c^2*d^4*e - 42*a^2*c*d^2*e^3 + 15*a^3*e^5 + 4*(7*c^3*d^4*e - a*c^2*d^2*e^3)*x^
2 + (35*c^3*d^5 - 14*a*c^2*d^3*e^2 + 3*a^2*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^10 -
3*a*c^2*d^8*e^2 + 3*a^2*c*d^6*e^4 - a^3*d^4*e^6 + (c^3*d^6*e^4 - 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 - a^3*e^10
)*x^4 + 4*(c^3*d^7*e^3 - 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7 - a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 - 3*a*c^2*d^6*e^4
+ 3*a^2*c*d^4*e^6 - a^3*d^2*e^8)*x^2 + 4*(c^3*d^9*e - 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 - a^3*d^3*e^7)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**5,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError