3.888 \(\int \frac{(d+e x)^{5/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{8 d \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{3/2}}{c e \sqrt{c d^2-c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0288803, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {657, 649} \[ \frac{8 d \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{3/2}}{c e \sqrt{c d^2-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(8*d*Sqrt[d + e*x])/(c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (2*(d + e*x)^(3/2))/(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)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^{3/2}}{c e \sqrt{c d^2-c e^2 x^2}}+(4 d) \int \frac{(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{8 d \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{3/2}}{c e \sqrt{c d^2-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0556384, size = 43, normalized size = 0.58 \[ \frac{2 (3 d-e x) \sqrt{d+e x}}{c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.039, size = 44, normalized size = 0.6 \begin{align*} 2\,{\frac{ \left ( -ex+d \right ) \left ( -ex+3\,d \right ) \left ( ex+d \right ) ^{3/2}}{e \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.14239, size = 31, normalized size = 0.42 \begin{align*} -\frac{2 \,{\left (e x - 3 \, d\right )}}{\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)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.04419, size = 108, normalized size = 1.46 \begin{align*} \frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}{\left (e x - 3 \, d\right )}}{c^{2} e^{3} x^{2} - c^{2} d^{2} e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*(e*x - 3*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{5}{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)**(5/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral((d + e*x)**(5/2)/(-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)^(5/2)/(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x