3.6.95 \(\int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx\) [595]

Optimal. Leaf size=59 \[ -\frac {2 \left (c d^2+a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {4 c d \sqrt {d+e x}}{e^3}+\frac {2 c (d+e x)^{3/2}}{3 e^3} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \begin {gather*} -\frac {2 \left (a e^2+c d^2\right )}{e^3 \sqrt {d+e x}}+\frac {2 c (d+e x)^{3/2}}{3 e^3}-\frac {4 c d \sqrt {d+e x}}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)/(d + e*x)^(3/2),x]

[Out]

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

Rule 711

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

Rubi steps

\begin {align*} \int \frac {a+c x^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac {c d^2+a e^2}{e^2 (d+e x)^{3/2}}-\frac {2 c d}{e^2 \sqrt {d+e x}}+\frac {c \sqrt {d+e x}}{e^2}\right ) \, dx\\ &=-\frac {2 \left (c d^2+a e^2\right )}{e^3 \sqrt {d+e x}}-\frac {4 c d \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 = 0.03, size = 43, normalized size = 0.73 \begin {gather*} \frac {2 \left (-3 a e^2+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[(a + c*x^2)/(d + e*x)^(3/2),x]

[Out]

(2*(-3*a*e^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 = 0.40, size = 48, normalized size = 0.81

method	result	size
gosper	\(-\frac {2 \left (-c \,e^{2} x^{2}+4 c d e x +3 e^{2} a +8 c \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\)	\(41\)
trager	\(-\frac {2 \left (-c \,e^{2} x^{2}+4 c d e x +3 e^{2} a +8 c \,d^{2}\right )}{3 \sqrt {e x +d}\, e^{3}}\)	\(41\)
risch	\(-\frac {2 c \left (-e x +5 d \right ) \sqrt {e x +d}}{3 e^{3}}-\frac {2 \left (e^{2} a +c \,d^{2}\right )}{e^{3} \sqrt {e x +d}}\)	\(46\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {3}{2}}}{3}-4 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a +c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(48\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {3}{2}}}{3}-4 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a +c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{3}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 53, normalized size = 0.90 \begin {gather*} \frac {2}{3} \, {\left ({\left ({\left (x e + d\right )}^{\frac {3}{2}} c - 6 \, \sqrt {x e + d} c d\right )} e^{\left (-2\right )} - \frac {3 \, {\left (c d^{2} + 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+a)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.77, size = 48, normalized size = 0.81 \begin {gather*} -\frac {2 \, {\left (4 \, c d x e + 8 \, c d^{2} - {\left (c x^{2} - 3 \, a\right )} e^{2}\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+a)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 3.49, size = 58, normalized size = 0.98 \begin {gather*} - \frac {4 c d \sqrt {d + e x}}{e^{3}} + \frac {2 c \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} - \frac {2 \left (a e^{2} + 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+a)/(e*x+d)**(3/2),x)

[Out]

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

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Giac [A]
time = 3.78, size = 54, normalized size = 0.92 \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}\right )} e^{\left (-9\right )} - \frac {2 \, {\left (c d^{2} + 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+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)*e^(-9) - 2*(c*d^2 + a*e^2)*e^(-3)/sqrt(x*e + d)

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Mupad [B]
time = 0.05, size = 44, normalized size = 0.75 \begin {gather*} -\frac {6\,a\,e^2-2\,c\,{\left (d+e\,x\right )}^2+6\,c\,d^2+12\,c\,d\,\left (d+e\,x\right )}{3\,e^3\,\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)/(d + e*x)^(3/2),x)

[Out]

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

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