3.8.23 \(\int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [723]

Optimal. Leaf size=61 \[ -\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{(c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {874} \begin {gather*} -\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

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

Rule 874

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))),
x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d
*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{(c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 50, normalized size = 0.82 \begin {gather*} -\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{(c d f-a e g) \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*Sqrt[d + e*x]*Sqrt[f + g*x])/((c*d*f - a*e*g)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]
time = 0.14, size = 55, normalized size = 0.90

method	result	size
default	\(\frac {2 \sqrt {g x +f}\, \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{\sqrt {e x +d}\, \left (c d x +a e \right ) \left (a e g -c d f \right )}\)	\(55\)
gosper	\(\frac {2 \sqrt {g x +f}\, \left (c d x +a e \right ) \left (e x +d \right )^{\frac {3}{2}}}{\left (a e g -c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((x*e + d)^(3/2)/((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*sqrt(g*x + f)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (58) = 116\).
time = 4.93, size = 122, normalized size = 2.00 \begin {gather*} -\frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{c^{2} d^{3} f x - a^{2} g x e^{3} - {\left (a c d g x^{2} - a c d f x + a^{2} d g\right )} e^{2} + {\left (c^{2} d^{2} f x^{2} - a c d^{2} g x + a c d^{2} f\right )} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g*x + f)*sqrt(x*e + d)/(c^2*d^3*f*x - a^2*g*x*e^3 - (a*c*d
*g*x^2 - a*c*d*f*x + a^2*d*g)*e^2 + (c^2*d^2*f*x^2 - a*c*d^2*g*x + a*c*d^2*f)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \sqrt {f + g x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(g*x+f)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Integral((d + e*x)**(3/2)/(((d + e*x)*(a*e + c*d*x))**(3/2)*sqrt(f + g*x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 4.68, size = 147, normalized size = 2.41 \begin {gather*} \frac {\left (\frac {2\,f\,\sqrt {d+e\,x}}{c\,d\,e\,\left (a\,e\,g-c\,d\,f\right )}+\frac {2\,g\,x\,\sqrt {d+e\,x}}{c\,d\,e\,\left (a\,e\,g-c\,d\,f\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}+\frac {a\,\sqrt {f+g\,x}}{c}+\frac {x\,\sqrt {f+g\,x}\,\left (c\,d^2+a\,e^2\right )}{c\,d\,e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/((f + g*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)

[Out]

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

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