### 3.1942 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^7} \, dx$$

Optimal. Leaf size=54 $\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^7 \left (c d^2-a e^2\right )}$

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^7)

________________________________________________________________________________________

Rubi [A]  time = 0.020905, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.027, Rules used = {650} $\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^7 \left (c d^2-a e^2\right )}$

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)^7,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^7)

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

Mathematica [A]  time = 0.0458011, size = 43, normalized size = 0.8 $\frac{2 ((d+e x) (a e+c d x))^{7/2}}{7 (d+e x)^7 \left (c d^2-a e^2\right )}$

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)^7,x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(7/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^7)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 58, normalized size = 1.1 \begin{align*} -{\frac{2\,cdx+2\,ae}{7\, \left ( ex+d \right ) ^{6} \left ( a{e}^{2}-c{d}^{2} \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)^7,x)

[Out]

-2/7*(c*d*x+a*e)/(e*x+d)^6/(a*e^2-c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)

________________________________________________________________________________________

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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 26.5348, size = 333, normalized size = 6.17 \begin{align*} \frac{2 \,{\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{7 \,{\left (c d^{6} - a d^{4} e^{2} +{\left (c d^{2} e^{4} - a e^{6}\right )} x^{4} + 4 \,{\left (c d^{3} e^{3} - a d e^{5}\right )} x^{3} + 6 \,{\left (c d^{4} e^{2} - a d^{2} e^{4}\right )} x^{2} + 4 \,{\left (c d^{5} e - a d^{3} e^{3}\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)^7,x, algorithm="fricas")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/
(c*d^6 - a*d^4*e^2 + (c*d^2*e^4 - a*e^6)*x^4 + 4*(c*d^3*e^3 - a*d*e^5)*x^3 + 6*(c*d^4*e^2 - a*d^2*e^4)*x^2 + 4
*(c*d^5*e - a*d^3*e^3)*x)

________________________________________________________________________________________

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)**7,x)

[Out]

Timed out

________________________________________________________________________________________

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)^(5/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError