\(\int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx\) [881]

Optimal result

Integrand size = 29, antiderivative size = 78 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e} \]

[Out]

-8/3*d*(-c*e^2*x^2+c*d^2)^(1/2)/c/e/(e*x+d)^(1/2)-2/3*(e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(1/2)/c/e

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663} \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e} \]

[In]

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

[Out]

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

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]

Rule 671

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e}+\frac {1}{3} (4 d) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx \\ & = -\frac {8 d \sqrt {c d^2-c e^2 x^2}}{3 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{3 c e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.56 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {2 (5 d+e x) \sqrt {c \left (d^2-e^2 x^2\right )}}{3 c e \sqrt {d+e x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.50

method	result	size
default	\(-\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, \left (e x +5 d \right )}{3 \sqrt {e x +d}\, c e}\)	\(39\)
gosper	\(-\frac {2 \left (-e x +d \right ) \left (e x +5 d \right ) \sqrt {e x +d}}{3 e \sqrt {-c \,x^{2} e^{2}+c \,d^{2}}}\)	\(43\)
risch	\(-\frac {2 \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, \left (e x +5 d \right ) \left (-e x +d \right )}{3 \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\)	\(81\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.59 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (e x + 5 \, d\right )} \sqrt {e x + d}}{3 \, {\left (c e^{2} x + c d e\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.44 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=\frac {2 \, {\left (e^{2} x^{2} + 4 \, d e x - 5 \, d^{2}\right )}}{3 \, \sqrt {-e x + d} \sqrt {c} e} \]

[In]

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

[Out]

2/3*(e^2*x^2 + 4*d*e*x - 5*d^2)/(sqrt(-e*x + d)*sqrt(c)*e)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=\frac {2 \, {\left (\frac {4 \, \sqrt {2} \sqrt {c d} d}{c} - \frac {6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} d}{c} + \frac {{\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}}}{c^{2}}\right )}}{3 \, e} \]

[In]

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

[Out]

2/3*(4*sqrt(2)*sqrt(c*d)*d/c - 6*sqrt(-(e*x + d)*c + 2*c*d)*d/c + (-(e*x + d)*c + 2*c*d)^(3/2)/c^2)/e

Mupad [B] (verification not implemented)

Time = 10.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {10\,d\,\sqrt {d+e\,x}}{3\,c\,e^2}+\frac {2\,x\,\sqrt {d+e\,x}}{3\,c\,e}\right )}{x+\frac {d}{e}} \]

[In]

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

[Out]

-((c*d^2 - c*e^2*x^2)^(1/2)*((10*d*(d + e*x)^(1/2))/(3*c*e^2) + (2*x*(d + e*x)^(1/2))/(3*c*e)))/(x + d/e)