### 3.1943 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^8} \, dx$$

Optimal. Leaf size=111 $\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )}$

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^8) + (4*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(7/2))/(63*(c*d^2 - a*e^2)^2*(d + e*x)^7)

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Rubi [A]  time = 0.0488611, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.054, Rules used = {658, 650} $\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^8 \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^8) + (4*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(7/2))/(63*(c*d^2 - a*e^2)^2*(d + e*x)^7)

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{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^8} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac{(2 c d) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx}{9 \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 )^{7/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac{4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 \left (c d^2-a e^2\right )^2 (d+e x)^7}\\ \end{align*}

Mathematica [A]  time = 0.0408023, size = 71, normalized size = 0.64 $\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (c d (9 d+2 e x)-7 a e^2\right )}{63 (d+e x)^5 \left (c d^2-a e^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-7*a*e^2 + c*d*(9*d + 2*e*x)))/(63*(c*d^2 - a*e^2)^2*(d + e*
x)^5)

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Maple [A]  time = 0.045, size = 90, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,cdex+7\,a{e}^{2}-9\,c{d}^{2} \right ) }{63\, \left ( ex+d \right ) ^{7} \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}}} \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)^(5/2)/(e*x+d)^8,x)

[Out]

-2/63*(c*d*x+a*e)*(-2*c*d*e*x+7*a*e^2-9*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(e*x+d)^7/(a^2*e^4-2*a*
c*d^2*e^2+c^2*d^4)

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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)^(5/2)/(e*x+d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 96.0911, size = 686, normalized size = 6.18 \begin{align*} \frac{2 \,{\left (2 \, c^{4} d^{4} e x^{4} + 9 \, a^{3} c d^{2} e^{3} - 7 \, a^{4} e^{5} +{\left (9 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} + 3 \,{\left (9 \, a c^{3} d^{4} e - 5 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} +{\left (27 \, a^{2} c^{2} d^{3} e^{2} - 19 \, a^{3} 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}}{63 \,{\left (c^{2} d^{9} - 2 \, a c d^{7} e^{2} + a^{2} d^{5} e^{4} +{\left (c^{2} d^{4} e^{5} - 2 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} x^{5} + 5 \,{\left (c^{2} d^{5} e^{4} - 2 \, a c d^{3} e^{6} + a^{2} d e^{8}\right )} x^{4} + 10 \,{\left (c^{2} d^{6} e^{3} - 2 \, a c d^{4} e^{5} + a^{2} d^{2} e^{7}\right )} x^{3} + 10 \,{\left (c^{2} d^{7} e^{2} - 2 \, a c d^{5} e^{4} + a^{2} d^{3} e^{6}\right )} x^{2} + 5 \,{\left (c^{2} d^{8} e - 2 \, a c d^{6} e^{3} + a^{2} d^{4} e^{5}\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)^(5/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

2/63*(2*c^4*d^4*e*x^4 + 9*a^3*c*d^2*e^3 - 7*a^4*e^5 + (9*c^4*d^5 - a*c^3*d^3*e^2)*x^3 + 3*(9*a*c^3*d^4*e - 5*a
^2*c^2*d^2*e^3)*x^2 + (27*a^2*c^2*d^3*e^2 - 19*a^3*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^
2*d^9 - 2*a*c*d^7*e^2 + a^2*d^5*e^4 + (c^2*d^4*e^5 - 2*a*c*d^2*e^7 + a^2*e^9)*x^5 + 5*(c^2*d^5*e^4 - 2*a*c*d^3
*e^6 + a^2*d*e^8)*x^4 + 10*(c^2*d^6*e^3 - 2*a*c*d^4*e^5 + a^2*d^2*e^7)*x^3 + 10*(c^2*d^7*e^2 - 2*a*c*d^5*e^4 +
a^2*d^3*e^6)*x^2 + 5*(c^2*d^8*e - 2*a*c*d^6*e^3 + a^2*d^4*e^5)*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)**(5/2)/(e*x+d)**8,x)

[Out]

Timed out

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Giac [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)^(5/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

Timed out