3.9.73 \(\int \frac {(c d^2-c e^2 x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [873]

Optimal. Leaf size=38 \[ -\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {663} \begin {gather*} -\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 663

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}}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (c d^2-c e^2 x^2\right )^{5/2}}{5 c e (d+e x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 43, normalized size = 1.13 \begin {gather*} -\frac {2 c (d-e x)^2 \sqrt {c \left (d^2-e^2 x^2\right )}}{5 e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.48, size = 38, normalized size = 1.00

method	result	size
gosper	\(-\frac {2 \left (-e x +d \right ) \left (-x^{2} c \,e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{5 e \left (e x +d \right )^{\frac {3}{2}}}\)	\(36\)
default	\(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, c \left (-e x +d \right )^{2}}{5 \sqrt {e x +d}\, e}\)	\(38\)
risch	\(-\frac {2 \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2} \left (e^{2} x^{2}-2 d x e +d^{2}\right ) \left (-e x +d \right )}{5 \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\)	\(93\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*(c*(-e^2*x^2+d^2))^(1/2)/(e*x+d)^(1/2)*c*(-e*x+d)^2/e

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Maxima [A]
time = 0.27, size = 39, normalized size = 1.03 \begin {gather*} -\frac {2}{5} \, {\left (c^{\frac {3}{2}} x^{2} e^{2} - 2 \, c^{\frac {3}{2}} d x e + c^{\frac {3}{2}} d^{2}\right )} \sqrt {-x e + d} e^{\left (-1\right )} \end {gather*}

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)^(3/2),x, algorithm="maxima")

[Out]

-2/5*(c^(3/2)*x^2*e^2 - 2*c^(3/2)*d*x*e + c^(3/2)*d^2)*sqrt(-x*e + d)*e^(-1)

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Fricas [A]
time = 3.19, size = 57, normalized size = 1.50 \begin {gather*} -\frac {2 \, {\left (c x^{2} e^{2} - 2 \, c d x e + c d^{2}\right )} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d}}{5 \, {\left (x e^{2} + d e\right )}} \end {gather*}

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)^(3/2),x, algorithm="fricas")

[Out]

-2/5*(c*x^2*e^2 - 2*c*d*x*e + c*d^2)*sqrt(-c*x^2*e^2 + c*d^2)*sqrt(x*e + d)/(x*e^2 + d*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

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)**(3/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (31) = 62\).
time = 1.61, size = 116, normalized size = 3.05 \begin {gather*} \frac {2}{15} \, {\left (5 \, {\left (2 \, \sqrt {2} \sqrt {c d} d - \frac {{\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c}\right )} c d + {\left (2 \, \sqrt {2} \sqrt {c d} d^{2} + \frac {5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c d - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c^{2}}\right )} c\right )} e^{\left (-1\right )} \end {gather*}

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)^(3/2),x, algorithm="giac")

[Out]

2/15*(5*(2*sqrt(2)*sqrt(c*d)*d - (-(x*e + d)*c + 2*c*d)^(3/2)/c)*c*d + (2*sqrt(2)*sqrt(c*d)*d^2 + (5*(-(x*e +
d)*c + 2*c*d)^(3/2)*c*d - 3*((x*e + d)*c - 2*c*d)^2*sqrt(-(x*e + d)*c + 2*c*d))/c^2)*c)*e^(-1)

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Mupad [B]
time = 0.55, size = 48, normalized size = 1.26 \begin {gather*} -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,c\,d^2}{5\,e}-\frac {4\,c\,d\,x}{5}+\frac {2\,c\,e\,x^2}{5}\right )}{\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c*d^2 - c*e^2*x^2)^(1/2)*((2*c*d^2)/(5*e) - (4*c*d*x)/5 + (2*c*e*x^2)/5))/(d + e*x)^(1/2)

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