3.8.48 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx\) [748]

Optimal. Leaf size=129 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{5/2}} \]

[Out]

2/7*(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)^(7/2)+4/35*c*d*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e*g+c*d*f)^2/(e*x+d)^(5/2)/(g*x+f)^(5/2)

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Rubi [A]
time = 0.10, antiderivative size = 129, 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 = {886, 874} \begin {gather*} \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^{5/2} (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)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(9/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(7/2)) + (4*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*(c*d*f - a*e*g)^2*(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]

Rule 886

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

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)^{9/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {(2 c d) \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}{7 (c d f-a e g)}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 69, normalized size = 0.53 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2} (-5 a e g+c d (7 f+2 g x))}{35 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{7/2}} \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)^(9/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-5*a*e*g + c*d*(7*f + 2*g*x)))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f +
g*x)^(7/2))

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Maple [A]
time = 0.14, size = 100, normalized size = 0.78

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +5 a e g -7 c d f \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{35 \left (g x +f \right )^{\frac {7}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}\)	\(99\)
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-2 g \,x^{2} c^{2} d^{2}+3 a c d e g x -7 c^{2} d^{2} f x +5 a^{2} e^{2} g -7 a c d e f \right ) \left (c d x +a e \right )}{35 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right )^{2}}\)	\(100\)

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)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/35*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(7/2)*(-2*c^2*d^2*g*x^2+3*a*c*d*e*g*x-7*c^2*d^2*f*x+5*
a^2*e^2*g-7*a*c*d*e*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((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)^(9/2),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)^(9/2)*(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. 553 vs. \(2 (119) = 238\).
time = 0.72, size = 553, normalized size = 4.29 \begin {gather*} \frac {2 \, {\left (2 \, c^{3} d^{3} g x^{3} + 7 \, c^{3} d^{3} f x^{2} - 5 \, a^{3} g e^{3} - {\left (8 \, a^{2} c d g x - 7 \, a^{2} c d f\right )} e^{2} - {\left (a c^{2} d^{2} g x^{2} - 14 \, a c^{2} d^{2} f x\right )} e\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}}{35 \, {\left (c^{2} d^{3} f^{2} g^{4} x^{4} + 4 \, c^{2} d^{3} f^{3} g^{3} x^{3} + 6 \, c^{2} d^{3} f^{4} g^{2} x^{2} + 4 \, c^{2} d^{3} f^{5} g x + c^{2} d^{3} f^{6} + {\left (a^{2} g^{6} x^{5} + 4 \, a^{2} f g^{5} x^{4} + 6 \, a^{2} f^{2} g^{4} x^{3} + 4 \, a^{2} f^{3} g^{3} x^{2} + a^{2} f^{4} g^{2} x\right )} e^{3} - {\left (2 \, a c d f g^{5} x^{5} - a^{2} d f^{4} g^{2} + {\left (8 \, a c d f^{2} g^{4} - a^{2} d g^{6}\right )} x^{4} + 4 \, {\left (3 \, a c d f^{3} g^{3} - a^{2} d f g^{5}\right )} x^{3} + 2 \, {\left (4 \, a c d f^{4} g^{2} - 3 \, a^{2} d f^{2} g^{4}\right )} x^{2} + 2 \, {\left (a c d f^{5} g - 2 \, a^{2} d f^{3} g^{3}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{4} x^{5} - 2 \, a c d^{2} f^{5} g + 2 \, {\left (2 \, c^{2} d^{2} f^{3} g^{3} - a c d^{2} f g^{5}\right )} x^{4} + 2 \, {\left (3 \, c^{2} d^{2} f^{4} g^{2} - 4 \, a c d^{2} f^{2} g^{4}\right )} x^{3} + 4 \, {\left (c^{2} d^{2} f^{5} g - 3 \, a c d^{2} f^{3} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} f^{6} - 8 \, a c d^{2} f^{4} 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)^(9/2),x, algorithm="fricas")

[Out]

2/35*(2*c^3*d^3*g*x^3 + 7*c^3*d^3*f*x^2 - 5*a^3*g*e^3 - (8*a^2*c*d*g*x - 7*a^2*c*d*f)*e^2 - (a*c^2*d^2*g*x^2 -
14*a*c^2*d^2*f*x)*e)*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^2*g^4
*x^4 + 4*c^2*d^3*f^3*g^3*x^3 + 6*c^2*d^3*f^4*g^2*x^2 + 4*c^2*d^3*f^5*g*x + c^2*d^3*f^6 + (a^2*g^6*x^5 + 4*a^2*
f*g^5*x^4 + 6*a^2*f^2*g^4*x^3 + 4*a^2*f^3*g^3*x^2 + a^2*f^4*g^2*x)*e^3 - (2*a*c*d*f*g^5*x^5 - a^2*d*f^4*g^2 +
(8*a*c*d*f^2*g^4 - a^2*d*g^6)*x^4 + 4*(3*a*c*d*f^3*g^3 - a^2*d*f*g^5)*x^3 + 2*(4*a*c*d*f^4*g^2 - 3*a^2*d*f^2*g
^4)*x^2 + 2*(a*c*d*f^5*g - 2*a^2*d*f^3*g^3)*x)*e^2 + (c^2*d^2*f^2*g^4*x^5 - 2*a*c*d^2*f^5*g + 2*(2*c^2*d^2*f^3
*g^3 - a*c*d^2*f*g^5)*x^4 + 2*(3*c^2*d^2*f^4*g^2 - 4*a*c*d^2*f^2*g^4)*x^3 + 4*(c^2*d^2*f^5*g - 3*a*c*d^2*f^3*g
^3)*x^2 + (c^2*d^2*f^6 - 8*a*c*d^2*f^4*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)**(9/2),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)^(9/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 4.31, size = 247, normalized size = 1.91 \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^2\,e^2\,\left (5\,a\,e\,g-7\,c\,d\,f\right )}{35\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {4\,c^3\,d^3\,x^3}{35\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {2\,c^2\,d^2\,x^2\,\left (a\,e\,g-7\,c\,d\,f\right )}{35\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {4\,a\,c\,d\,e\,x\,\left (4\,a\,e\,g-7\,c\,d\,f\right )}{35\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )}{x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {3\,f\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {3\,f^2\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{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)^(9/2)*(d + e*x)^(3/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*a^2*e^2*(5*a*e*g - 7*c*d*f))/(35*g^3*(a*e*g - c*d*f)^2) -
(4*c^3*d^3*x^3)/(35*g^2*(a*e*g - c*d*f)^2) + (2*c^2*d^2*x^2*(a*e*g - 7*c*d*f))/(35*g^3*(a*e*g - c*d*f)^2) + (4
*a*c*d*e*x*(4*a*e*g - 7*c*d*f))/(35*g^3*(a*e*g - c*d*f)^2)))/(x^3*(f + g*x)^(1/2)*(d + e*x)^(1/2) + (f^3*(f +
g*x)^(1/2)*(d + e*x)^(1/2))/g^3 + (3*f*x^2*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g + (3*f^2*x*(f + g*x)^(1/2)*(d +
e*x)^(1/2))/g^2)

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