\(\int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx\) [595]

Optimal result

Integrand size = 17, antiderivative size = 59 \[ \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (c d^2+a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {4 c d \sqrt {d+e x}}{e^3}+\frac {2 c (d+e x)^{3/2}}{3 e^3} \]

[Out]

2/3*c*(e*x+d)^(3/2)/e^3-2*(a*e^2+c*d^2)/e^3/(e*x+d)^(1/2)-4*c*d*(e*x+d)^(1/2)/e^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (a e^2+c d^2\right )}{e^3 \sqrt {d+e x}}+\frac {2 c (d+e x)^{3/2}}{3 e^3}-\frac {4 c d \sqrt {d+e x}}{e^3} \]

[In]

Int[(a + c*x^2)/(d + e*x)^(3/2),x]

[Out]

(-2*(c*d^2 + a*e^2))/(e^3*Sqrt[d + e*x]) - (4*c*d*Sqrt[d + e*x])/e^3 + (2*c*(d + e*x)^(3/2))/(3*e^3)

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx=\frac {2 \left (-3 a e^2+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt {d+e x}} \]

[In]

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

[Out]

(2*(-3*a*e^2 + c*(-8*d^2 - 4*d*e*x + e^2*x^2)))/(3*e^3*Sqrt[d + e*x])

Maple [A] (verified)

Time = 1.93 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(\frac {\frac {2 \left (c \,x^{2}-3 a \right ) e^{2}}{3}-\frac {8 x c d e}{3}-\frac {16 c \,d^{2}}{3}}{\sqrt {e x +d}\, e^{3}}\)	\(39\)
gosper	\(-\frac {2 \left (-c \,x^{2} e^{2}+4 x c d e +3 e^{2} a +8 c \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\)	\(41\)
trager	\(-\frac {2 \left (-c \,x^{2} e^{2}+4 x c d e +3 e^{2} a +8 c \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\)	\(41\)
risch	\(-\frac {2 c \left (-e x +5 d \right ) \sqrt {e x +d}}{3 e^{3}}-\frac {2 \left (e^{2} a +c \,d^{2}\right )}{e^{3} \sqrt {e x +d}}\)	\(46\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {3}{2}}}{3}-4 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a +c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(48\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {3}{2}}}{3}-4 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a +c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(48\)

[In]

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

[Out]

2/3*((c*x^2-3*a)*e^2-4*x*c*d*e-8*c*d^2)/(e*x+d)^(1/2)/e^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (c e^{2} x^{2} - 4 \, c d e x - 8 \, c d^{2} - 3 \, a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \]

[In]

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

[Out]

2/3*(c*e^2*x^2 - 4*c*d*e*x - 8*c*d^2 - 3*a*e^2)*sqrt(e*x + d)/(e^4*x + d*e^3)

Sympy [A] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {2 c d \sqrt {d + e x}}{e^{2}} + \frac {c \left (d + e x\right )^{\frac {3}{2}}}{3 e^{2}} - \frac {a e^{2} + c d^{2}}{e^{2} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a x + \frac {c x^{3}}{3}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(-2*c*d*sqrt(d + e*x)/e**2 + c*(d + e*x)**(3/2)/(3*e**2) - (a*e**2 + c*d**2)/(e**2*sqrt(d + e*x))
)/e, Ne(e, 0)), ((a*x + c*x**3/3)/d**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} c - 6 \, \sqrt {e x + d} c d}{e^{2}} - \frac {3 \, {\left (c d^{2} + a e^{2}\right )}}{\sqrt {e x + d} e^{2}}\right )}}{3 \, e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (c d^{2} + a e^{2}\right )}}{\sqrt {e x + d} e^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c e^{6} - 6 \, \sqrt {e x + d} c d e^{6}\right )}}{3 \, e^{9}} \]

[In]

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

[Out]

-2*(c*d^2 + a*e^2)/(sqrt(e*x + d)*e^3) + 2/3*((e*x + d)^(3/2)*c*e^6 - 6*sqrt(e*x + d)*c*d*e^6)/e^9

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.75 \[ \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx=-\frac {6\,a\,e^2-2\,c\,{\left (d+e\,x\right )}^2+6\,c\,d^2+12\,c\,d\,\left (d+e\,x\right )}{3\,e^3\,\sqrt {d+e\,x}} \]

[In]

int((a + c*x^2)/(d + e*x)^(3/2),x)

[Out]

-(6*a*e^2 - 2*c*(d + e*x)^2 + 6*c*d^2 + 12*c*d*(d + e*x))/(3*e^3*(d + e*x)^(1/2))