\(\int \frac {(d+e x)^{5/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx\) [888]

Optimal result

Integrand size = 29, antiderivative size = 74 \[ \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {8 d \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, 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} \[ \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {8 d \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}} \]

[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 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 (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] (verified)

Time = 0.47 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.58 \[ \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (3 d-e x) \sqrt {d+e x}}{c e \sqrt {c \left (d^2-e^2 x^2\right )}} \]

[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)])

Maple [A] (verified)

Time = 2.47 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.59

method	result	size
gosper	\(\frac {2 \left (-e x +d \right ) \left (-e x +3 d \right ) \left (e x +d \right )^{\frac {3}{2}}}{e \left (-c \,x^{2} e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}\)	\(44\)
default	\(\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, \left (-e x +3 d \right )}{\sqrt {e x +d}\, c^{2} \left (-e x +d \right ) e}\)	\(48\)
risch	\(\frac {2 \left (-e x +d \right ) \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{e \sqrt {-c \left (e x -d \right )}\, \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, c}+\frac {4 d \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{e \sqrt {c \left (-e x +d \right )}\, \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, c}\)	\(147\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.74 \[ \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} {\left (e x - 3 \, d\right )}}{c^{2} e^{3} x^{2} - c^{2} d^{2} e} \]

[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)

Sympy [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.31 \[ \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (e x - 3 \, d\right )}}{\sqrt {-e x + d} c^{\frac {3}{2}} e} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {2 \, d}{\sqrt {-{\left (e x + d\right )} c + 2 \, c d} e} + \frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c e}\right )}}{c} - \frac {4 \, \sqrt {2} d}{\sqrt {c d} c e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {6\,d\,\sqrt {d+e\,x}}{c^2\,e^3}-\frac {2\,x\,\sqrt {d+e\,x}}{c^2\,e^2}\right )}{x^2-\frac {d^2}{e^2}} \]

[In]

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

[Out]

-((c*d^2 - c*e^2*x^2)^(1/2)*((6*d*(d + e*x)^(1/2))/(c^2*e^3) - (2*x*(d + e*x)^(1/2))/(c^2*e^2)))/(x^2 - d^2/e^
2)