\(\int \frac {(d+e x)^{3/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx\) [889]

Optimal result

Integrand size = 29, antiderivative size = 36 \[ \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {663} \[ \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}} \]

[In]

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

[Out]

(2*Sqrt[d + e*x])/(c*e*Sqrt[c*d^2 - c*e^2*x^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*} \text {integral}& = \frac {2 \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {d+e x}}{c e \sqrt {c \left (d^2-e^2 x^2\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00

method	result	size
gosper	\(\frac {2 \left (-e x +d \right ) \left (e x +d \right )^{\frac {3}{2}}}{e \left (-c \,x^{2} e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}\)	\(36\)
default	\(\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}}{\sqrt {e x +d}\, c^{2} \left (-e x +d \right ) e}\)	\(40\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{c^{2} e^{3} x^{2} - c^{2} d^{2} e} \]

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.44 \[ \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2}{\sqrt {-e x + d} c^{\frac {3}{2}} e} \]

[In]

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

[Out]

2/(sqrt(-e*x + d)*c^(3/2)*e)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {2}}{\sqrt {c d} c e} + \frac {2}{\sqrt {-{\left (e x + d\right )} c + 2 \, c d} c e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.92 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2\,\sqrt {c\,d^2-c\,e^2\,x^2}\,\sqrt {d+e\,x}}{e\,\left (c^2\,d^2-c^2\,e^2\,x^2\right )} \]

[In]

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

[Out]

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