\(\int \frac {a+b x+c x^2}{(d+e x)^{3/2}} \, dx\) [2272]

Optimal result

Integrand size = 20, antiderivative size = 71 \[ \int \frac {a+b x+c x^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (c d^2-b d e+a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \sqrt {d+e x}}{e^3}+\frac {2 c (d+e x)^{3/2}}{3 e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int \frac {a+b x+c x^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (a e^2-b d e+c d^2\right )}{e^3 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} (2 c d-b e)}{e^3}+\frac {2 c (d+e x)^{3/2}}{3 e^3} \]

[In]

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

[Out]

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

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c d^2-b d e+a e^2}{e^2 (d+e x)^{3/2}}+\frac {-2 c d+b e}{e^2 \sqrt {d+e x}}+\frac {c \sqrt {d+e x}}{e^2}\right ) \, dx \\ & = -\frac {2 \left (c d^2-b d e+a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \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 = 54, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x+c x^2}{(d+e x)^{3/2}} \, dx=\frac {6 e (2 b d-a e+b e x)+2 c \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 \sqrt {d+e x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(\frac {\frac {2 \left (c \,x^{2}+3 b x -3 a \right ) e^{2}}{3}+4 \left (-\frac {2 c x}{3}+b \right ) d e -\frac {16 c \,d^{2}}{3}}{\sqrt {e x +d}\, e^{3}}\)	\(47\)
gosper	\(-\frac {2 \left (-c \,x^{2} e^{2}-3 b \,e^{2} x +4 c d e x +3 a \,e^{2}-6 b d e +8 c \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\)	\(53\)
trager	\(-\frac {2 \left (-c \,x^{2} e^{2}-3 b \,e^{2} x +4 c d e x +3 a \,e^{2}-6 b d e +8 c \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\)	\(53\)
risch	\(\frac {2 \left (c x e +3 b e -5 c d \right ) \sqrt {e x +d}}{3 e^{3}}-\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{3} \sqrt {e x +d}}\)	\(55\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b e \sqrt {e x +d}-4 c d \sqrt {e x +d}-\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(63\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b e \sqrt {e x +d}-4 c d \sqrt {e x +d}-\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(63\)

[In]

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

[Out]

2/3*((c*x^2+3*b*x-3*a)*e^2+6*(-2/3*c*x+b)*d*e-8*c*d^2)/(e*x+d)^(1/2)/e^3

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x+c x^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (c e^{2} x^{2} - 8 \, c d^{2} + 6 \, b d e - 3 \, a e^{2} - {\left (4 \, c d e - 3 \, b e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \]

[In]

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

[Out]

2/3*(c*e^2*x^2 - 8*c*d^2 + 6*b*d*e - 3*a*e^2 - (4*c*d*e - 3*b*e^2)*x)*sqrt(e*x + d)/(e^4*x + d*e^3)

Sympy [A] (verification not implemented)

Time = 1.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.32 \[ \int \frac {a+b x+c x^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c \left (d + e x\right )^{\frac {3}{2}}}{3 e^{2}} + \frac {\sqrt {d + e x} \left (b e - 2 c d\right )}{e^{2}} - \frac {a e^{2} - b d e + c d^{2}}{e^{2} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a x + \frac {b x^{2}}{2} + \frac {c x^{3}}{3}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x+c x^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} c - 3 \, {\left (2 \, c d - b e\right )} \sqrt {e x + d}}{e^{2}} - \frac {3 \, {\left (c d^{2} - b d e + a e^{2}\right )}}{\sqrt {e x + d} e^{2}}\right )}}{3 \, e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04 \[ \int \frac {a+b x+c x^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (c d^{2} - b d e + a e^{2}\right )}}{\sqrt {e x + d} e^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c e^{6} - 6 \, \sqrt {e x + d} c d e^{6} + 3 \, \sqrt {e x + d} b e^{7}\right )}}{3 \, e^{9}} \]

[In]

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

[Out]

-2*(c*d^2 - b*d*e + 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 + 3*sqrt
(e*x + d)*b*e^7)/e^9

Mupad [B] (verification not implemented)

Time = 9.77 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x+c x^2}{(d+e x)^{3/2}} \, dx=\frac {2\,c\,{\left (d+e\,x\right )}^2-6\,a\,e^2-6\,c\,d^2+6\,b\,e\,\left (d+e\,x\right )-12\,c\,d\,\left (d+e\,x\right )+6\,b\,d\,e}{3\,e^3\,\sqrt {d+e\,x}} \]

[In]

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

[Out]

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