3.8.47 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx\) [747]

Optimal. Leaf size=63 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.05, 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 = {874} \begin {gather*} \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^{5/2} (f+g x)^{5/2} (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)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(7/2)),x]

[Out]

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

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 {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2}}{5 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(7/2)),x]

[Out]

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

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Maple [A]
time = 0.16, size = 55, normalized size = 0.87

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{2}}{5 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {5}{2}} \left (a e g -c d f \right )}\)	\(55\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{5 \left (g x +f \right )^{\frac {5}{2}} \left (a e g -c d f \right ) \left (e x +d \right )^{\frac {3}{2}}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(5/2)*(c*d*x+a*e)^2/(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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((g*x + f)^(7/2)*(x*e + d)^(3/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (58) = 116\).
time = 0.86, size = 244, normalized size = 3.87 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d x e + a^{2} e^{2}\right )} \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}}{5 \, {\left (c d^{2} f g^{3} x^{3} + 3 \, c d^{2} f^{2} g^{2} x^{2} + 3 \, c d^{2} f^{3} g x + c d^{2} f^{4} - {\left (a g^{4} x^{4} + 3 \, a f g^{3} x^{3} + 3 \, a f^{2} g^{2} x^{2} + a f^{3} g x\right )} e^{2} + {\left (c d f g^{3} x^{4} - a d f^{3} g + {\left (3 \, c d f^{2} g^{2} - a d g^{4}\right )} x^{3} + 3 \, {\left (c d f^{3} g - a d f g^{3}\right )} x^{2} + {\left (c d f^{4} - 3 \, a d f^{2} g^{2}\right )} x\right )} e\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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(7/2),x, algorithm="fricas")

[Out]

2/5*(c^2*d^2*x^2 + 2*a*c*d*x*e + a^2*e^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*d^2*f*g^3*x^3 + 3*c*d^2*f^2*g^2*x^2 + 3*c*d^2*f^3*g*x + c*d^2*f^4 - (a*g^4*x^4 + 3*a*f*g^3*x^3 + 3*a*f^
2*g^2*x^2 + a*f^3*g*x)*e^2 + (c*d*f*g^3*x^4 - a*d*f^3*g + (3*c*d*f^2*g^2 - a*d*g^4)*x^3 + 3*(c*d*f^3*g - a*d*f
*g^3)*x^2 + (c*d*f^4 - 3*a*d*f^2*g^2)*x)*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**(7/2),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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(7/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 4.07, size = 232, normalized size = 3.68 \begin {gather*} -\frac {\left (\frac {2\,a^2\,e^2}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {2\,c^2\,d^2\,x^2}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {4\,a\,c\,d\,e\,x}{5\,a\,e\,g^3-5\,c\,d\,f\,g^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}\,\sqrt {d+e\,x}-\frac {\sqrt {f+g\,x}\,\left (5\,c\,d\,f^3-5\,a\,e\,f^2\,g\right )\,\sqrt {d+e\,x}}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}+\frac {x\,\sqrt {f+g\,x}\,\left (10\,a\,e\,f\,g^2-10\,c\,d\,f^2\,g\right )\,\sqrt {d+e\,x}}{5\,a\,e\,g^3-5\,c\,d\,f\,g^2}} \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)^(3/2)/((f + g*x)^(7/2)*(d + e*x)^(3/2)),x)

[Out]

-(((2*a^2*e^2)/(5*a*e*g^3 - 5*c*d*f*g^2) + (2*c^2*d^2*x^2)/(5*a*e*g^3 - 5*c*d*f*g^2) + (4*a*c*d*e*x)/(5*a*e*g^
3 - 5*c*d*f*g^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)*(d + e*x)^(1/2) - ((f +
g*x)^(1/2)*(5*c*d*f^3 - 5*a*e*f^2*g)*(d + e*x)^(1/2))/(5*a*e*g^3 - 5*c*d*f*g^2) + (x*(f + g*x)^(1/2)*(10*a*e*f
*g^2 - 10*c*d*f^2*g)*(d + e*x)^(1/2))/(5*a*e*g^3 - 5*c*d*f*g^2))

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