3.6.5 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \]

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Rubi [A]  time = 0.06, antiderivative size = 63, 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 = {860} \begin {gather*} \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(5/2)),x]

[Out]

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

Rule 860

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 {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{3/2}}{3 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(5/2)),x]

[Out]

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

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IntegrateAlgebraic [B]  time = 2.66, size = 145, normalized size = 2.30 \begin {gather*} \frac {2 \sqrt {d+e x} (e f+e g x)^{5/2} \sqrt {a e^2+c d e x} \left (a e^2-c d^2+c d (d+e x)\right )^{3/2}}{3 e^3 \sqrt {\frac {(d+e x) \left (a e^2+c d e x\right )}{e}} (g (d+e x)-d g+e f)^{3/2} \left (\frac {g (d+e x)-d g+e f}{e}\right )^{5/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^(5/2)),x]

[Out]

(2*Sqrt[d + e*x]*Sqrt[a*e^2 + c*d*e*x]*(e*f + e*g*x)^(5/2)*(-(c*d^2) + a*e^2 + c*d*(d + e*x))^(3/2))/(3*e^3*(c
*d*f - a*e*g)*Sqrt[((d + e*x)*(a*e^2 + c*d*e*x))/e]*(e*f - d*g + g*(d + e*x))^(3/2)*((e*f - d*g + g*(d + e*x))
/e)^(5/2))

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fricas [B]  time = 0.43, size = 169, normalized size = 2.68 \begin {gather*} \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d x + a e\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (c d^{2} f^{3} - a d e f^{2} g + {\left (c d e f g^{2} - a e^{2} g^{3}\right )} x^{3} + {\left (2 \, c d e f^{2} g - a d e g^{3} + {\left (c d^{2} - 2 \, a e^{2}\right )} f g^{2}\right )} x^{2} + {\left (c d e f^{3} - 2 \, a d e f g^{2} + {\left (2 \, c d^{2} - a e^{2}\right )} f^{2} g\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.01, size = 63, normalized size = 1.00 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a e g -c d f \right ) \sqrt {e x +d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)/(g*x+f)^(5/2)/(e*x+d)^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^(5/2)), x)

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mupad [B]  time = 3.92, size = 136, normalized size = 2.16 \begin {gather*} -\frac {\left (\frac {2\,a\,e}{3\,a\,e\,g^2-3\,c\,d\,f\,g}+\frac {2\,c\,d\,x}{3\,a\,e\,g^2-3\,c\,d\,f\,g}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (3\,c\,d\,f^2-3\,a\,e\,f\,g\right )\,\sqrt {d+e\,x}}{3\,a\,e\,g^2-3\,c\,d\,f\,g}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/((f + g*x)^(5/2)*(d + e*x)^(1/2)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\sqrt {d + e x} \left (f + g x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(sqrt(d + e*x)*(f + g*x)**(5/2)), x)

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