### 3.1944 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^9} \, dx$$

Optimal. Leaf size=171 $\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^7 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^8 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )}$

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^9) + (8*c*d*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(99*(c*d^2 - a*e^2)^2*(d + e*x)^8) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(7/2))/(693*(c*d^2 - a*e^2)^3*(d + e*x)^7)

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Rubi [A]  time = 0.0804385, 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} $\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{693 (d+e x)^7 \left (c d^2-a e^2\right )^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{99 (d+e x)^8 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^9 \left (c d^2-a e^2\right )}$

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)^9,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*(c*d^2 - a*e^2)*(d + e*x)^9) + (8*c*d*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(99*(c*d^2 - a*e^2)^2*(d + e*x)^8) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(7/2))/(693*(c*d^2 - a*e^2)^3*(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)^9} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac{(4 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)^8} \, dx}{11 \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}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac{8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 \left (c d^2-a e^2\right )^2 (d+e x)^8}+\frac{\left (8 c^2 d^2\right ) \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}{99 \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 )^{7/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^9}+\frac{8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 \left (c d^2-a e^2\right )^2 (d+e x)^8}+\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 )^{7/2}}{693 \left (c d^2-a e^2\right )^3 (d+e x)^7}\\ \end{align*}

Mathematica [A]  time = 0.0606825, size = 104, normalized size = 0.61 $\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (63 a^2 e^4-14 a c d e^2 (11 d+2 e x)+c^2 d^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )\right )}{693 (d+e x)^6 \left (c d^2-a e^2\right )^3}$

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)^9,x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(63*a^2*e^4 - 14*a*c*d*e^2*(11*d + 2*e*x) + c^2*d^2*(99*d^2 +
44*d*e*x + 8*e^2*x^2)))/(693*(c*d^2 - a*e^2)^3*(d + e*x)^6)

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Maple [A]  time = 0.046, 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}-28\,acd{e}^{3}x+44\,{c}^{2}{d}^{3}ex+63\,{a}^{2}{e}^{4}-154\,ac{d}^{2}{e}^{2}+99\,{c}^{2}{d}^{4} \right ) }{693\, \left ( ex+d \right ) ^{8} \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 ) } \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)^9,x)

[Out]

-2/693*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-28*a*c*d*e^3*x+44*c^2*d^3*e*x+63*a^2*e^4-154*a*c*d^2*e^2+99*c^2*d^4)*(c*
d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(e*x+d)^8/(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)^(5/2)/(e*x+d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)^9,x, algorithm="fricas")

[Out]

Timed out

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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)**9,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)^9,x, algorithm="giac")

[Out]

Timed out