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

Optimal. Leaf size=71 \[ -\frac {2 \left (c d^2-b d e+a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \sqrt {d+e x}}{e^3}+\frac {2 c (d+e x)^{3/2}}{3 e^3} \]

[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

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Rubi [A]
time = 0.02, antiderivative size = 71, 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 \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} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.05, size = 54, normalized size = 0.76 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [A]
time = 0.54, size = 63, normalized size = 0.89

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 67, normalized size = 0.94 \begin {gather*} \frac {2}{3} \, {\left ({\left ({\left (x e + d\right )}^{\frac {3}{2}} c - 3 \, {\left (2 \, c d - b e\right )} \sqrt {x e + d}\right )} e^{\left (-2\right )} - \frac {3 \, {\left (c d^{2} - b d e + a e^{2}\right )} e^{\left (-2\right )}}{\sqrt {x e + d}}\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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.45, size = 59, normalized size = 0.83 \begin {gather*} -\frac {2 \, {\left (8 \, c d^{2} - {\left (c x^{2} + 3 \, b x - 3 \, a\right )} e^{2} + 2 \, {\left (2 \, c d x - 3 \, b d\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (x e^{4} + d e^{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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 6.16, size = 70, normalized size = 0.99 \begin {gather*} \frac {2 c \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} + \frac {\sqrt {d + e x} \left (2 b e - 4 c d\right )}{e^{3}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right )}{e^{3} \sqrt {d + e x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.83, size = 73, normalized size = 1.03 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c e^{6} - 6 \, \sqrt {x e + d} c d e^{6} + 3 \, \sqrt {x e + d} b e^{7}\right )} e^{\left (-9\right )} - \frac {2 \, {\left (c d^{2} - b d e + a e^{2}\right )} e^{\left (-3\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((x*e + d)^(3/2)*c*e^6 - 6*sqrt(x*e + d)*c*d*e^6 + 3*sqrt(x*e + d)*b*e^7)*e^(-9) - 2*(c*d^2 - b*d*e + a*e^
2)*e^(-3)/sqrt(x*e + d)

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Mupad [B]
time = 0.81, size = 58, normalized size = 0.82 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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