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

Optimal. Leaf size=78 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}} \]

[Out]

(-8*d*(c*d^2 - c*e^2*x^2)^(5/2))/(35*c*e*(d + e*x)^(5/2)) - (2*(c*d^2 - c*e^2*x^2)^(5/2))/(7*c*e*(d + e*x)^(3/
2))

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Rubi [A]  time = 0.0306673, antiderivative size = 78, 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} \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*d*(c*d^2 - c*e^2*x^2)^(5/2))/(35*c*e*(d + e*x)^(5/2)) - (2*(c*d^2 - c*e^2*x^2)^(5/2))/(7*c*e*(d + e*x)^(3/
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{\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx &=-\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}+\frac{1}{7} (4 d) \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\\ &=-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}}-\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0487574, size = 51, normalized size = 0.65 \[ -\frac{2 c (d-e x)^2 (9 d+5 e x) \sqrt{c \left (d^2-e^2 x^2\right )}}{35 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*(d - e*x)^2*(9*d + 5*e*x)*Sqrt[c*(d^2 - e^2*x^2)])/(35*e*Sqrt[d + e*x])

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Maple [A]  time = 0.041, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 5\,ex+9\,d \right ) }{35\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(1/2),x)

[Out]

-2/35*(-e*x+d)*(5*e*x+9*d)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d)^(3/2)

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Maxima [A]  time = 1.08443, size = 74, normalized size = 0.95 \begin{align*} -\frac{2 \,{\left (5 \, c^{\frac{3}{2}} e^{3} x^{3} - c^{\frac{3}{2}} d e^{2} x^{2} - 13 \, c^{\frac{3}{2}} d^{2} e x + 9 \, c^{\frac{3}{2}} d^{3}\right )} \sqrt{-e x + d}}{35 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-2/35*(5*c^(3/2)*e^3*x^3 - c^(3/2)*d*e^2*x^2 - 13*c^(3/2)*d^2*e*x + 9*c^(3/2)*d^3)*sqrt(-e*x + d)/e

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Fricas [A]  time = 1.99887, size = 153, normalized size = 1.96 \begin{align*} -\frac{2 \,{\left (5 \, c e^{3} x^{3} - c d e^{2} x^{2} - 13 \, c d^{2} e x + 9 \, c d^{3}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{35 \,{\left (e^{2} x + d e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-2/35*(5*c*e^3*x^3 - c*d*e^2*x^2 - 13*c*d^2*e*x + 9*c*d^3)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(e^2*x + d*e
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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