### 3.887 $$\int \frac{(d+e x)^{7/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=119 $-\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{64 d^2 \sqrt{d+e x}}{3 c e \sqrt{c d^2-c e^2 x^2}}$

[Out]

(64*d^2*Sqrt[d + e*x])/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (16*d*(d + e*x)^(3/2))/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2]
) - (2*(d + e*x)^(5/2))/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2])

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Rubi [A]  time = 0.0486483, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {657, 649} $-\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{64 d^2 \sqrt{d+e x}}{3 c e \sqrt{c d^2-c e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(64*d^2*Sqrt[d + e*x])/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (16*d*(d + e*x)^(3/2))/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2]
) - (2*(d + e*x)^(5/2))/(3*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)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{3} (8 d) \int \frac{(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{1}{3} \left (32 d^2\right ) \int \frac{(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{64 d^2 \sqrt{d+e x}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0589753, size = 55, normalized size = 0.46 $-\frac{2 \sqrt{d+e x} \left (-23 d^2+10 d e x+e^2 x^2\right )}{3 c e \sqrt{c \left (d^2-e^2 x^2\right )}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(-23*d^2 + 10*d*e*x + e^2*x^2))/(3*c*e*Sqrt[c*(d^2 - e^2*x^2)])

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Maple [A]  time = 0.042, size = 55, normalized size = 0.5 \begin{align*}{\frac{ \left ( -2\,ex+2\,d \right ) \left ( -{e}^{2}{x}^{2}-10\,dxe+23\,{d}^{2} \right ) }{3\,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)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

2/3*(-e*x+d)*(-e^2*x^2-10*d*e*x+23*d^2)*(e*x+d)^(3/2)/e/(-c*e^2*x^2+c*d^2)^(3/2)

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Maxima [A]  time = 1.13826, size = 58, normalized size = 0.49 \begin{align*} -\frac{2 \,{\left (\sqrt{c} e^{2} x^{2} + 10 \, \sqrt{c} d e x - 23 \, \sqrt{c} d^{2}\right )}}{3 \, \sqrt{-e x + d} c^{2} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/3*(sqrt(c)*e^2*x^2 + 10*sqrt(c)*d*e*x - 23*sqrt(c)*d^2)/(sqrt(-e*x + d)*c^2*e)

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Fricas [A]  time = 2.11554, size = 135, normalized size = 1.13 \begin{align*} \frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (e^{2} x^{2} + 10 \, d e x - 23 \, d^{2}\right )} \sqrt{e x + d}}{3 \,{\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)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2/3*sqrt(-c*e^2*x^2 + c*d^2)*(e^2*x^2 + 10*d*e*x - 23*d^2)*sqrt(e*x + d)/(c^2*e^3*x^2 - c^2*d^2*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{\frac{7}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral((d + e*x)**(7/2)/(-c*(-d + e*x)*(d + e*x))**(3/2), x)

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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)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x