### 3.886 $$\int \frac{(d+e x)^{9/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=160 $-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}}$

[Out]

(256*d^3*Sqrt[d + e*x])/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (64*d^2*(d + e*x)^(3/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x
^2]) - (8*d*(d + e*x)^(5/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (2*(d + e*x)^(7/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x
^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0726321, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {657, 649} $-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(256*d^3*Sqrt[d + e*x])/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (64*d^2*(d + e*x)^(3/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x
^2]) - (8*d*(d + e*x)^(5/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (2*(d + e*x)^(7/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x
^2])

Rule 657

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

Rule 649

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

Rubi steps

\begin{align*} \int \frac{(d+e x)^{9/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{5} (12 d) \int \frac{(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{5} \left (32 d^2\right ) \int \frac{(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{5} \left (128 d^3\right ) \int \frac{(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0834965, size = 66, normalized size = 0.41 $-\frac{2 \sqrt{d+e x} \left (43 d^2 e x-91 d^3+7 d e^2 x^2+e^3 x^3\right )}{5 c e \sqrt{c \left (d^2-e^2 x^2\right )}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(-91*d^3 + 43*d^2*e*x + 7*d*e^2*x^2 + e^3*x^3))/(5*c*e*Sqrt[c*(d^2 - e^2*x^2)])

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 66, normalized size = 0.4 \begin{align*}{\frac{ \left ( -2\,ex+2\,d \right ) \left ( -{e}^{3}{x}^{3}-7\,d{e}^{2}{x}^{2}-43\,{d}^{2}xe+91\,{d}^{3} \right ) }{5\,e} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

2/5*(-e*x+d)*(-e^3*x^3-7*d*e^2*x^2-43*d^2*e*x+91*d^3)*(e*x+d)^(3/2)/e/(-c*e^2*x^2+c*d^2)^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.11729, size = 61, normalized size = 0.38 \begin{align*} -\frac{2 \,{\left (e^{3} x^{3} + 7 \, d e^{2} x^{2} + 43 \, d^{2} e x - 91 \, d^{3}\right )}}{5 \, \sqrt{-e x + d} c^{\frac{3}{2}} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/5*(e^3*x^3 + 7*d*e^2*x^2 + 43*d^2*e*x - 91*d^3)/(sqrt(-e*x + d)*c^(3/2)*e)

________________________________________________________________________________________

Fricas [A]  time = 2.18276, size = 157, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (e^{3} x^{3} + 7 \, d e^{2} x^{2} + 43 \, d^{2} e x - 91 \, d^{3}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{5 \,{\left (c^{2} e^{3} x^{2} - c^{2} d^{2} e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2/5*(e^3*x^3 + 7*d*e^2*x^2 + 43*d^2*e*x - 91*d^3)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(c^2*e^3*x^2 - c^2*d^
2*e)

________________________________________________________________________________________

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((e*x+d)**(9/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x