3.23.71 \(\int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx\) [2271]

Optimal. Leaf size=73 \[ \frac {2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^3}-\frac {2 (2 c d-b e) (d+e x)^{3/2}}{3 e^3}+\frac {2 c (d+e x)^{5/2}}{5 e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \begin {gather*} \frac {2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}{e^3}-\frac {2 (d+e x)^{3/2} (2 c d-b e)}{3 e^3}+\frac {2 c (d+e x)^{5/2}}{5 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x])/e^3 - (2*(2*c*d - b*e)*(d + e*x)^(3/2))/(3*e^3) + (2*c*(d + e*x)^(5/
2))/(5*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*} \int \frac {a+b x+c x^2}{\sqrt {d+e x}} \, dx &=\int \left (\frac {c d^2-b d e+a e^2}{e^2 \sqrt {d+e x}}+\frac {(-2 c d+b e) \sqrt {d+e x}}{e^2}+\frac {c (d+e x)^{3/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^3}-\frac {2 (2 c d-b e) (d+e x)^{3/2}}{3 e^3}+\frac {2 c (d+e x)^{5/2}}{5 e^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 54, normalized size = 0.74 \begin {gather*} \frac {2 \sqrt {d+e x} \left (5 e (-2 b d+3 a e+b e x)+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(5*e*(-2*b*d + 3*a*e + b*e*x) + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)

________________________________________________________________________________________

Maple [A]
time = 0.46, size = 75, normalized size = 1.03

method	result	size
gosper	\(\frac {2 \sqrt {e x +d}\, \left (3 x^{2} c \,e^{2}+5 b \,e^{2} x -4 c d e x +15 e^{2} a -10 b d e +8 c \,d^{2}\right )}{15 e^{3}}\)	\(53\)
trager	\(\frac {2 \sqrt {e x +d}\, \left (3 x^{2} c \,e^{2}+5 b \,e^{2} x -4 c d e x +15 e^{2} a -10 b d e +8 c \,d^{2}\right )}{15 e^{3}}\)	\(53\)
risch	\(\frac {2 \sqrt {e x +d}\, \left (3 x^{2} c \,e^{2}+5 b \,e^{2} x -4 c d e x +15 e^{2} a -10 b d e +8 c \,d^{2}\right )}{15 e^{3}}\)	\(53\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 b e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {4 c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}-2 b d e \sqrt {e x +d}+2 c \,d^{2} \sqrt {e x +d}}{e^{3}}\)	\(75\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 b e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {4 c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}-2 b d e \sqrt {e x +d}+2 c \,d^{2} \sqrt {e x +d}}{e^{3}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 80, normalized size = 1.10 \begin {gather*} \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b e^{\left (-1\right )} + {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c e^{\left (-2\right )} + 15 \, \sqrt {x e + d} a\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(5*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*b*e^(-1) + (3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x
*e + d)*d^2)*c*e^(-2) + 15*sqrt(x*e + d)*a)*e^(-1)

________________________________________________________________________________________

Fricas [A]
time = 2.86, size = 50, normalized size = 0.68 \begin {gather*} \frac {2}{15} \, {\left (8 \, c d^{2} + {\left (3 \, c x^{2} + 5 \, b x + 15 \, a\right )} e^{2} - 2 \, {\left (2 \, c d x + 5 \, b d\right )} e\right )} \sqrt {x e + d} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(8*c*d^2 + (3*c*x^2 + 5*b*x + 15*a)*e^2 - 2*(2*c*d*x + 5*b*d)*e)*sqrt(x*e + d)*e^(-3)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (73) = 146\).
time = 5.60, size = 223, normalized size = 3.05 \begin {gather*} \begin {cases} \frac {- \frac {2 a d}{\sqrt {d + e x}} - 2 a \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {2 b d \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 b \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {2 c d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {2 c \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {a x + \frac {b x^{2}}{2} + \frac {c x^{3}}{3}}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**(1/2),x)

[Out]

Piecewise(((-2*a*d/sqrt(d + e*x) - 2*a*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 2*b*d*(-d/sqrt(d + e*x) - sqrt(d +
e*x))/e - 2*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 2*c*d*(d**2/sqrt(d + e*x) + 2
*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 2*c*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(
3/2) - (d + e*x)**(5/2)/5)/e**2)/e, Ne(e, 0)), ((a*x + b*x**2/2 + c*x**3/3)/sqrt(d), True))

________________________________________________________________________________________

Giac [A]
time = 0.79, size = 80, normalized size = 1.10 \begin {gather*} \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b e^{\left (-1\right )} + {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c e^{\left (-2\right )} + 15 \, \sqrt {x e + d} a\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(5*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*b*e^(-1) + (3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x
*e + d)*d^2)*c*e^(-2) + 15*sqrt(x*e + d)*a)*e^(-1)

________________________________________________________________________________________

Mupad [B]
time = 0.81, size = 58, normalized size = 0.79 \begin {gather*} \frac {2\,\sqrt {d+e\,x}\,\left (3\,c\,{\left (d+e\,x\right )}^2+15\,a\,e^2+15\,c\,d^2+5\,b\,e\,\left (d+e\,x\right )-10\,c\,d\,\left (d+e\,x\right )-15\,b\,d\,e\right )}{15\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)/(d + e*x)^(1/2),x)

[Out]

(2*(d + e*x)^(1/2)*(3*c*(d + e*x)^2 + 15*a*e^2 + 15*c*d^2 + 5*b*e*(d + e*x) - 10*c*d*(d + e*x) - 15*b*d*e))/(1
5*e^3)

________________________________________________________________________________________