3.6.24 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)} \]

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Rubi [A]  time = 0.07, 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 )^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(9/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(7/2))

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 {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(9/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(7/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(7/2))

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IntegrateAlgebraic [B]  time = 1.88, size = 248, normalized size = 3.94 \begin {gather*} -\frac {2 \left (\frac {(d g+e g x) (a e g+c d g x)}{g^2}\right )^{5/2} \sqrt {a e g+c d (f+g x)-c d f} \left (a^3 e^3 g^3+3 a^2 c d e^2 g^2 (f+g x)-3 a^2 c d e^2 f g^2+3 a c^2 d^2 e f^2 g+3 a c^2 d^2 e g (f+g x)^2-6 a c^2 d^2 e f g (f+g x)-c^3 d^3 f^3+3 c^3 d^3 f^2 (f+g x)+c^3 d^3 (f+g x)^3-3 c^3 d^3 f (f+g x)^2\right )}{7 g (d+e x)^{5/2} (f+g x)^{7/2} (a e g-c d f) (a e g+c d g x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(9/2)),x]

[Out]

(-2*(((a*e*g + c*d*g*x)*(d*g + e*g*x))/g^2)^(5/2)*Sqrt[-(c*d*f) + a*e*g + c*d*(f + g*x)]*(-(c^3*d^3*f^3) + 3*a
*c^2*d^2*e*f^2*g - 3*a^2*c*d*e^2*f*g^2 + a^3*e^3*g^3 + 3*c^3*d^3*f^2*(f + g*x) - 6*a*c^2*d^2*e*f*g*(f + g*x) +
3*a^2*c*d*e^2*g^2*(f + g*x) - 3*c^3*d^3*f*(f + g*x)^2 + 3*a*c^2*d^2*e*g*(f + g*x)^2 + c^3*d^3*(f + g*x)^3))/(
7*g*(-(c*d*f) + a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(7/2)*(a*e*g + c*d*g*x)^(5/2))

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fricas [B]  time = 0.44, size = 299, normalized size = 4.75 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{7 \, {\left (c d^{2} f^{5} - a d e f^{4} g + {\left (c d e f g^{4} - a e^{2} g^{5}\right )} x^{5} + {\left (4 \, c d e f^{2} g^{3} - a d e g^{5} + {\left (c d^{2} - 4 \, a e^{2}\right )} f g^{4}\right )} x^{4} + 2 \, {\left (3 \, c d e f^{3} g^{2} - 2 \, a d e f g^{4} + {\left (2 \, c d^{2} - 3 \, a e^{2}\right )} f^{2} g^{3}\right )} x^{3} + 2 \, {\left (2 \, c d e f^{4} g - 3 \, a d e f^{2} g^{3} + {\left (3 \, c d^{2} - 2 \, a e^{2}\right )} f^{3} g^{2}\right )} x^{2} + {\left (c d e f^{5} - 4 \, a d e f^{3} g^{2} + {\left (4 \, c d^{2} - a e^{2}\right )} f^{4} 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)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(9/2),x, algorithm="fricas")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*
sqrt(e*x + d)*sqrt(g*x + f)/(c*d^2*f^5 - a*d*e*f^4*g + (c*d*e*f*g^4 - a*e^2*g^5)*x^5 + (4*c*d*e*f^2*g^3 - a*d*
e*g^5 + (c*d^2 - 4*a*e^2)*f*g^4)*x^4 + 2*(3*c*d*e*f^3*g^2 - 2*a*d*e*f*g^4 + (2*c*d^2 - 3*a*e^2)*f^2*g^3)*x^3 +
2*(2*c*d*e*f^4*g - 3*a*d*e*f^2*g^3 + (3*c*d^2 - 2*a*e^2)*f^3*g^2)*x^2 + (c*d*e*f^5 - 4*a*d*e*f^3*g^2 + (4*c*d
^2 - a*e^2)*f^4*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)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(9/2),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 ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{7 \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right ) \left (e x +d \right )^{\frac {5}{2}}} \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)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(9/2),x)

[Out]

-2/7/(g*x+f)^(7/2)*(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)^(5/2)/(e*x+d)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {9}{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)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^(9/2)), x)

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mupad [B]  time = 4.34, size = 325, normalized size = 5.16 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,a^3\,e^3}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {2\,c^3\,d^3\,x^3}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {6\,a^2\,c\,d\,e^2\,x}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {6\,a\,c^2\,d^2\,e\,x^2}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}\right )}{x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (7\,c\,d\,f^4-7\,a\,e\,f^3\,g\right )\,\sqrt {d+e\,x}}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}+\frac {x^2\,\sqrt {f+g\,x}\,\left (21\,a\,e\,f\,g^3-21\,c\,d\,f^2\,g^2\right )\,\sqrt {d+e\,x}}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}-\frac {x\,\sqrt {f+g\,x}\,\left (21\,c\,d\,f^3\,g-21\,a\,e\,f^2\,g^2\right )\,\sqrt {d+e\,x}}{7\,a\,e\,g^4-7\,c\,d\,f\,g^3}} \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)^(5/2)/((f + g*x)^(9/2)*(d + e*x)^(5/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*a^3*e^3)/(7*a*e*g^4 - 7*c*d*f*g^3) + (2*c^3*d^3*x^3)/(7*a*
e*g^4 - 7*c*d*f*g^3) + (6*a^2*c*d*e^2*x)/(7*a*e*g^4 - 7*c*d*f*g^3) + (6*a*c^2*d^2*e*x^2)/(7*a*e*g^4 - 7*c*d*f*
g^3)))/(x^3*(f + g*x)^(1/2)*(d + e*x)^(1/2) - ((f + g*x)^(1/2)*(7*c*d*f^4 - 7*a*e*f^3*g)*(d + e*x)^(1/2))/(7*a
*e*g^4 - 7*c*d*f*g^3) + (x^2*(f + g*x)^(1/2)*(21*a*e*f*g^3 - 21*c*d*f^2*g^2)*(d + e*x)^(1/2))/(7*a*e*g^4 - 7*c
*d*f*g^3) - (x*(f + g*x)^(1/2)*(21*c*d*f^3*g - 21*a*e*f^2*g^2)*(d + e*x)^(1/2))/(7*a*e*g^4 - 7*c*d*f*g^3))

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sympy [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)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**(9/2),x)

[Out]

Timed out

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