3.4.42 \(\int \frac {b x+c x^2}{(d+e x)^{3/2}} \, dx\) [342]

Optimal. Leaf size=64 \[ -\frac {2 d (c d-b e)}{e^3 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \sqrt {d+e x}}{e^3}+\frac {2 c (d+e x)^{3/2}}{3 e^3} \]

[Out]

2/3*c*(e*x+d)^(3/2)/e^3-2*d*(-b*e+c*d)/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 = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \begin {gather*} -\frac {2 \sqrt {d+e x} (2 c d-b e)}{e^3}-\frac {2 d (c d-b e)}{e^3 \sqrt {d+e x}}+\frac {2 c (d+e x)^{3/2}}{3 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)/(d + e*x)^(3/2),x]

[Out]

(-2*d*(c*d - b*e))/(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 {b x+c x^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {d (c d-b e)}{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 d (c d-b e)}{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 = 0.04, size = 48, normalized size = 0.75 \begin {gather*} \frac {2 \left (3 b e (2 d+e x)+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3*b*e*(2*d + e*x) + 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 = 0.42, size = 55, normalized size = 0.86

method	result	size
gosper	\(\frac {\frac {2}{3} c \,x^{2} e^{2}+2 b \,e^{2} x -\frac {8}{3} c d e x +4 b d e -\frac {16}{3} c \,d^{2}}{\sqrt {e x +d}\, e^{3}}\)	\(46\)
trager	\(\frac {\frac {2}{3} c \,x^{2} e^{2}+2 b \,e^{2} x -\frac {8}{3} c d e x +4 b d e -\frac {16}{3} c \,d^{2}}{\sqrt {e x +d}\, e^{3}}\)	\(46\)
risch	\(\frac {2 \left (c e x +3 b e -5 c d \right ) \sqrt {e x +d}}{3 e^{3}}+\frac {2 d \left (b e -c d \right )}{e^{3} \sqrt {e x +d}}\)	\(48\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b e \sqrt {e x +d}-4 c d \sqrt {e x +d}+\frac {2 d \left (b e -c d \right )}{\sqrt {e x +d}}}{e^{3}}\)	\(55\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b e \sqrt {e x +d}-4 c d \sqrt {e x +d}+\frac {2 d \left (b e -c d \right )}{\sqrt {e x +d}}}{e^{3}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 63, normalized size = 0.98 \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\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)/(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)*e^(-2)/sqrt(x*e + d))*e^(-
1)

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Fricas [A]
time = 1.74, size = 56, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (8 \, c d^{2} - {\left (c x^{2} + 3 \, b x\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)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

-2/3*(8*c*d^2 - (c*x^2 + 3*b*x)*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 = 4.28, size = 60, normalized size = 0.94 \begin {gather*} \frac {2 c \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} + \frac {2 d \left (b e - c d\right )}{e^{3} \sqrt {d + e x}} + \frac {\sqrt {d + e x} \left (2 b e - 4 c d\right )}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.78, size = 69, normalized size = 1.08 \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\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)/(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)*e^(-3
)/sqrt(x*e + d)

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Mupad [B]
time = 0.06, size = 52, normalized size = 0.81 \begin {gather*} \frac {2\,c\,{\left (d+e\,x\right )}^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((b*x + c*x^2)/(d + e*x)^(3/2),x)

[Out]

(2*c*(d + e*x)^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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