3.8.40 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{9/2}} \, dx\) [740]

Optimal. Leaf size=198 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{3/2}} \]

[Out]

2/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e*g+c*d*f)/(e*x+d)^(3/2)/(g*x+f)^(7/2)+8/35*c*d*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e*g+c*d*f)^2/(e*x+d)^(3/2)/(g*x+f)^(5/2)+16/105*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(3/2)/(-a*e*g+c*d*f)^3/(e*x+d)^(3/2)/(g*x+f)^(3/2)

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Rubi [A]
time = 0.15, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^{3/2} (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 )^{3/2}}{7 (d+e x)^{3/2} (f+g x)^{7/2} (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)^(9/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(7/2)) + (8*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (16*c
^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(3/2))

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 {\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)^{9/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {(4 c d) \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)^{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 )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {\left (8 c^2 d^2\right ) \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}{35 (c d f-a e g)^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{7/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 105, normalized size = 0.53 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{3/2} \left (15 a^2 e^2 g^2-6 a c d e g (7 f+2 g x)+c^2 d^2 \left (35 f^2+28 f g x+8 g^2 x^2\right )\right )}{105 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{7/2}} \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)^(9/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(15*a^2*e^2*g^2 - 6*a*c*d*e*g*(7*f + 2*g*x) + c^2*d^2*(35*f^2 + 28*f*g*x +
8*g^2*x^2)))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(7/2))

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Maple [A]
time = 0.14, size = 119, normalized size = 0.60

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-12 a c d e \,g^{2} x +28 c^{2} d^{2} f g x +15 a^{2} e^{2} g^{2}-42 a c d e f g +35 f^{2} c^{2} d^{2}\right )}{105 \left (g x +f \right )^{\frac {7}{2}} \sqrt {e x +d}\, \left (a e g -c d f \right )^{3}}\)	\(119\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-12 a c d e \,g^{2} x +28 c^{2} d^{2} f g x +15 a^{2} e^{2} g^{2}-42 a c d e f g +35 f^{2} c^{2} d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (g x +f \right )^{\frac {7}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \sqrt {e x +d}}\)	\(169\)

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)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/105*((c*d*x+a*e)*(e*x+d))^(1/2)/(g*x+f)^(7/2)/(e*x+d)^(1/2)*(c*d*x+a*e)*(8*c^2*d^2*g^2*x^2-12*a*c*d*e*g^2*x
+28*c^2*d^2*f*g*x+15*a^2*e^2*g^2-42*a*c*d*e*f*g+35*c^2*d^2*f^2)/(a*e*g-c*d*f)^3

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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)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 783 vs. \(2 (183) = 366\).
time = 1.51, size = 783, normalized size = 3.95 \begin {gather*} \frac {2 \, {\left (8 \, c^{3} d^{3} g^{2} x^{3} + 28 \, c^{3} d^{3} f g x^{2} + 35 \, c^{3} d^{3} f^{2} x + 15 \, a^{3} g^{2} e^{3} + 3 \, {\left (a^{2} c d g^{2} x - 14 \, a^{2} c d f g\right )} e^{2} - {\left (4 \, a c^{2} d^{2} g^{2} x^{2} + 14 \, a c^{2} d^{2} f g x - 35 \, a c^{2} d^{2} f^{2}\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}}{105 \, {\left (c^{3} d^{4} f^{3} g^{4} x^{4} + 4 \, c^{3} d^{4} f^{4} g^{3} x^{3} + 6 \, c^{3} d^{4} f^{5} g^{2} x^{2} + 4 \, c^{3} d^{4} f^{6} g x + c^{3} d^{4} f^{7} - {\left (a^{3} g^{7} x^{5} + 4 \, a^{3} f g^{6} x^{4} + 6 \, a^{3} f^{2} g^{5} x^{3} + 4 \, a^{3} f^{3} g^{4} x^{2} + a^{3} f^{4} g^{3} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{6} x^{5} - a^{3} d f^{4} g^{3} + {\left (12 \, a^{2} c d f^{2} g^{5} - a^{3} d g^{7}\right )} x^{4} + 2 \, {\left (9 \, a^{2} c d f^{3} g^{4} - 2 \, a^{3} d f g^{6}\right )} x^{3} + 6 \, {\left (2 \, a^{2} c d f^{4} g^{3} - a^{3} d f^{2} g^{5}\right )} x^{2} + {\left (3 \, a^{2} c d f^{5} g^{2} - 4 \, a^{3} d f^{3} g^{4}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{5} x^{5} - a^{2} c d^{2} f^{5} g^{2} + {\left (4 \, a c^{2} d^{2} f^{3} g^{4} - a^{2} c d^{2} f g^{6}\right )} x^{4} + 2 \, {\left (3 \, a c^{2} d^{2} f^{4} g^{3} - 2 \, a^{2} c d^{2} f^{2} g^{5}\right )} x^{3} + 2 \, {\left (2 \, a c^{2} d^{2} f^{5} g^{2} - 3 \, a^{2} c d^{2} f^{3} g^{4}\right )} x^{2} + {\left (a c^{2} d^{2} f^{6} g - 4 \, a^{2} c d^{2} f^{4} g^{3}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{4} x^{5} - 3 \, a c^{2} d^{3} f^{6} g + {\left (4 \, c^{3} d^{3} f^{4} g^{3} - 3 \, a c^{2} d^{3} f^{2} g^{5}\right )} x^{4} + 6 \, {\left (c^{3} d^{3} f^{5} g^{2} - 2 \, a c^{2} d^{3} f^{3} g^{4}\right )} x^{3} + 2 \, {\left (2 \, c^{3} d^{3} f^{6} g - 9 \, a c^{2} d^{3} f^{4} g^{3}\right )} x^{2} + {\left (c^{3} d^{3} f^{7} - 12 \, a c^{2} d^{3} f^{5} 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)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/105*(8*c^3*d^3*g^2*x^3 + 28*c^3*d^3*f*g*x^2 + 35*c^3*d^3*f^2*x + 15*a^3*g^2*e^3 + 3*(a^2*c*d*g^2*x - 14*a^2*
c*d*f*g)*e^2 - (4*a*c^2*d^2*g^2*x^2 + 14*a*c^2*d^2*f*g*x - 35*a*c^2*d^2*f^2)*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^3*d^4*f^3*g^4*x^4 + 4*c^3*d^4*f^4*g^3*x^3 + 6*c^3*d^4*f^5*g^2*x^2
+ 4*c^3*d^4*f^6*g*x + c^3*d^4*f^7 - (a^3*g^7*x^5 + 4*a^3*f*g^6*x^4 + 6*a^3*f^2*g^5*x^3 + 4*a^3*f^3*g^4*x^2 +
a^3*f^4*g^3*x)*e^4 + (3*a^2*c*d*f*g^6*x^5 - a^3*d*f^4*g^3 + (12*a^2*c*d*f^2*g^5 - a^3*d*g^7)*x^4 + 2*(9*a^2*c*
d*f^3*g^4 - 2*a^3*d*f*g^6)*x^3 + 6*(2*a^2*c*d*f^4*g^3 - a^3*d*f^2*g^5)*x^2 + (3*a^2*c*d*f^5*g^2 - 4*a^3*d*f^3*
g^4)*x)*e^3 - 3*(a*c^2*d^2*f^2*g^5*x^5 - a^2*c*d^2*f^5*g^2 + (4*a*c^2*d^2*f^3*g^4 - a^2*c*d^2*f*g^6)*x^4 + 2*(
3*a*c^2*d^2*f^4*g^3 - 2*a^2*c*d^2*f^2*g^5)*x^3 + 2*(2*a*c^2*d^2*f^5*g^2 - 3*a^2*c*d^2*f^3*g^4)*x^2 + (a*c^2*d^
2*f^6*g - 4*a^2*c*d^2*f^4*g^3)*x)*e^2 + (c^3*d^3*f^3*g^4*x^5 - 3*a*c^2*d^3*f^6*g + (4*c^3*d^3*f^4*g^3 - 3*a*c^
2*d^3*f^2*g^5)*x^4 + 6*(c^3*d^3*f^5*g^2 - 2*a*c^2*d^3*f^3*g^4)*x^3 + 2*(2*c^3*d^3*f^6*g - 9*a*c^2*d^3*f^4*g^3)
*x^2 + (c^3*d^3*f^7 - 12*a*c^2*d^3*f^5*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)**(1/2)/(g*x+f)**(9/2)/(e*x+d)**(1/2),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)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 4.29, size = 289, normalized size = 1.46 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {30\,a^3\,e^3\,g^2-84\,a^2\,c\,d\,e^2\,f\,g+70\,a\,c^2\,d^2\,e\,f^2}{105\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {x\,\left (6\,a^2\,c\,d\,e^2\,g^2-28\,a\,c^2\,d^2\,e\,f\,g+70\,c^3\,d^3\,f^2\right )}{105\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^3\,d^3\,x^3}{105\,g\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c^2\,d^2\,x^2\,\left (a\,e\,g-7\,c\,d\,f\right )}{105\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^3}\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)^(1/2)/((f + g*x)^(9/2)*(d + e*x)^(1/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((30*a^3*e^3*g^2 + 70*a*c^2*d^2*e*f^2 - 84*a^2*c*d*e^2*f*g)/(1
05*g^3*(a*e*g - c*d*f)^3) + (x*(70*c^3*d^3*f^2 + 6*a^2*c*d*e^2*g^2 - 28*a*c^2*d^2*e*f*g))/(105*g^3*(a*e*g - c*
d*f)^3) + (16*c^3*d^3*x^3)/(105*g*(a*e*g - c*d*f)^3) - (8*c^2*d^2*x^2*(a*e*g - 7*c*d*f))/(105*g^2*(a*e*g - c*d
*f)^3)))/(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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