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

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

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Rubi [A]  time = 0.03, 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 = {698} \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 698

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 (-a e+2 b d+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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IntegrateAlgebraic [A]  time = 0.05, size = 61, normalized size = 0.86 \begin {gather*} \frac {2 \left (-3 a e^2+3 b e (d+e x)+3 b d e-3 c d^2-6 c d (d+e x)+c (d+e x)^2\right )}{3 e^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 63, normalized size = 0.89 \begin {gather*} \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 )}} \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*(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)

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giac [A]  time = 0.16, 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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maple [A]  time = 0.05, size = 53, normalized size = 0.75 \begin {gather*} -\frac {2 \left (-c \,e^{2} x^{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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(e*x+d)^(1/2)*(-c*e^2*x^2-3*b*e^2*x+4*c*d*e*x+3*a*e^2-6*b*d*e+8*c*d^2)/e^3

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maxima [A]  time = 0.92, size = 66, normalized size = 0.93 \begin {gather*} \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} \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*(((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

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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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sympy [A]  time = 13.07, 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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