3.8.24 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [724]

Optimal. Leaf size=124 \[ -\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {4 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {882, 874} \begin {gather*} -\frac {4 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}-\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)/((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

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

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]

Rule 882

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)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x]
+ Dist[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))), Int[(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*
x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, 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] && LtQ[p, -1] && RationalQ[n]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \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}}{(c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(2 g) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d f-a e g}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {4 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 64, normalized size = 0.52 \begin {gather*} -\frac {2 \sqrt {d+e x} (a e g+c d (f+2 g x))}{(c d f-a e g)^2 \sqrt {(a e+c d x) (d+e x)} \sqrt {f+g x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 70, normalized size = 0.56

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (2 c d g x +a e g +c d f \right )}{\sqrt {e x +d}\, \sqrt {g x +f}\, \left (c d x +a e \right ) \left (a e g -c d f \right )^{2}}\)	\(70\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (2 c d g x +a e g +c d f \right ) \left (e x +d \right )^{\frac {3}{2}}}{\sqrt {g x +f}\, \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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)^(3/2)/(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)*(g*x + f)^(3/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (118) = 236\).
time = 2.06, size = 336, normalized size = 2.71 \begin {gather*} -\frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d g x + c d f + a g e\right )} \sqrt {g x + f} \sqrt {x e + d}}{c^{3} d^{4} f^{2} g x^{2} + c^{3} d^{4} f^{3} x + {\left (a^{3} g^{3} x^{2} + a^{3} f g^{2} x\right )} e^{4} + {\left (a^{2} c d g^{3} x^{3} - a^{2} c d f g^{2} x^{2} + a^{3} d f g^{2} - {\left (2 \, a^{2} c d f^{2} g - a^{3} d g^{3}\right )} x\right )} e^{3} - {\left (2 \, a c^{2} d^{2} f g^{2} x^{3} + 2 \, a^{2} c d^{2} f^{2} g + {\left (a c^{2} d^{2} f^{2} g - a^{2} c d^{2} g^{3}\right )} x^{2} - {\left (a c^{2} d^{2} f^{3} - a^{2} c d^{2} f g^{2}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{2} g x^{3} - a c^{2} d^{3} f^{2} g x + a c^{2} d^{3} f^{3} + {\left (c^{3} d^{3} f^{3} - 2 \, a c^{2} d^{3} f g^{2}\right )} x^{2}\right )} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [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)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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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)^(3/2)/(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 = 4.98, size = 151, normalized size = 1.22 \begin {gather*} -\frac {\left (\frac {4\,g\,x\,\sqrt {d+e\,x}}{e\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {\left (2\,a\,e\,g+2\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^2}\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)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)

[Out]

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