### 3.1945 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{10}} \, dx$$

Optimal. Leaf size=231 $\frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac{12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )}$

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d^2 - a*e^2)*(d + e*x)^10) + (12*c*d*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(7/2))/(143*(c*d^2 - a*e^2)^2*(d + e*x)^9) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2)^(7/2))/(429*(c*d^2 - a*e^2)^3*(d + e*x)^8) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7
/2))/(3003*(c*d^2 - a*e^2)^4*(d + e*x)^7)

________________________________________________________________________________________

Rubi [A]  time = 0.12055, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.054, Rules used = {658, 650} $\frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac{12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{10} \left (c d^2-a e^2\right )}$

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)^10,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d^2 - a*e^2)*(d + e*x)^10) + (12*c*d*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(7/2))/(143*(c*d^2 - a*e^2)^2*(d + e*x)^9) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2)^(7/2))/(429*(c*d^2 - a*e^2)^3*(d + e*x)^8) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7
/2))/(3003*(c*d^2 - a*e^2)^4*(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)^{10}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac{(6 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)^9} \, dx}{13 \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}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac{12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac{\left (24 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)^8} \, dx}{143 \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}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac{12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\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}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\frac{\left (16 c^3 d^3\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}{429 \left (c d^2-a e^2\right )^3}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac{12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\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}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\frac{32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^7}\\ \end{align*}

Mathematica [A]  time = 0.0797922, size = 148, normalized size = 0.64 $\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (63 a^2 c d e^4 (13 d+2 e x)-231 a^3 e^6-7 a c^2 d^2 e^2 \left (143 d^2+52 d e x+8 e^2 x^2\right )+c^3 d^3 \left (286 d^2 e x+429 d^3+104 d e^2 x^2+16 e^3 x^3\right )\right )}{3003 (d+e x)^7 \left (c d^2-a e^2\right )^4}$

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)^10,x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-231*a^3*e^6 + 63*a^2*c*d*e^4*(13*d + 2*e*x) - 7*a*c^2*d^2*e
^2*(143*d^2 + 52*d*e*x + 8*e^2*x^2) + c^3*d^3*(429*d^3 + 286*d^2*e*x + 104*d*e^2*x^2 + 16*e^3*x^3)))/(3003*(c*
d^2 - a*e^2)^4*(d + e*x)^7)

________________________________________________________________________________________

Maple [A]  time = 0.05, size = 217, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+56\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-104\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-126\,{a}^{2}cd{e}^{5}x+364\,a{c}^{2}{d}^{3}{e}^{3}x-286\,{c}^{3}{d}^{5}ex+231\,{a}^{3}{e}^{6}-819\,{a}^{2}c{d}^{2}{e}^{4}+1001\,a{c}^{2}{d}^{4}{e}^{2}-429\,{c}^{3}{d}^{6} \right ) }{3003\, \left ( ex+d \right ) ^{9} \left ({a}^{4}{e}^{8}-4\,{a}^{3}c{d}^{2}{e}^{6}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}+{c}^{4}{d}^{8} \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)^10,x)

[Out]

-2/3003*(c*d*x+a*e)*(-16*c^3*d^3*e^3*x^3+56*a*c^2*d^2*e^4*x^2-104*c^3*d^4*e^2*x^2-126*a^2*c*d*e^5*x+364*a*c^2*
d^3*e^3*x-286*c^3*d^5*e*x+231*a^3*e^6-819*a^2*c*d^2*e^4+1001*a*c^2*d^4*e^2-429*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d
^2*x+a*d*e)^(5/2)/(e*x+d)^9/(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^8)

________________________________________________________________________________________

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)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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)^10,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)**10,x)

[Out]

Timed out

________________________________________________________________________________________

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)^10,x, algorithm="giac")

[Out]

Timed out