3.8.39 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx\) [739]

Optimal. Leaf size=129 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{3/2}} \]

[Out]

2/5*(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)^(5/2)+4/15*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)^(3/2)

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Rubi [A]
time = 0.09, 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 )^{3/2}}{15 (d+e x)^{3/2} (f+g x)^{3/2} (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}}{5 (d+e x)^{3/2} (f+g x)^{5/2} (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)^(7/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (4*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(15*(c*d*f - a*e*g)^2*(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)^{7/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {(2 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)^{5/2}} \, dx}{5 (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}}{5 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 69, normalized size = 0.53 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{3/2} (-3 a e g+c d (5 f+2 g x))}{15 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{5/2}} \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)^(7/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-3*a*e*g + c*d*(5*f + 2*g*x)))/(15*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f +
g*x)^(5/2))

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Maple [A]
time = 0.15, size = 70, normalized size = 0.54

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right ) \left (-2 c d g x +3 a e g -5 c d f \right )}{15 \left (g x +f \right )^{\frac {5}{2}} \sqrt {e x +d}\, \left (a e g -c d f \right )^{2}}\)	\(70\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +3 a e g -5 c d f \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (g x +f \right )^{\frac {5}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {e x +d}}\)	\(99\)

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)^(7/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/15*((c*d*x+a*e)*(e*x+d))^(1/2)/(g*x+f)^(5/2)/(e*x+d)^(1/2)*(c*d*x+a*e)*(-2*c*d*g*x+3*a*e*g-5*c*d*f)/(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)^(1/2)/(g*x+f)^(7/2)/(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)^(7/2)*sqrt(x*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (119) = 238\).
time = 1.27, size = 424, normalized size = 3.29 \begin {gather*} \frac {2 \, {\left (2 \, c^{2} d^{2} g x^{2} + 5 \, c^{2} d^{2} f x - 3 \, a^{2} g e^{2} - {\left (a c d g x - 5 \, a c d f\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}}{15 \, {\left (c^{2} d^{3} f^{2} g^{3} x^{3} + 3 \, c^{2} d^{3} f^{3} g^{2} x^{2} + 3 \, c^{2} d^{3} f^{4} g x + c^{2} d^{3} f^{5} + {\left (a^{2} g^{5} x^{4} + 3 \, a^{2} f g^{4} x^{3} + 3 \, a^{2} f^{2} g^{3} x^{2} + a^{2} f^{3} g^{2} x\right )} e^{3} - {\left (2 \, a c d f g^{4} x^{4} - a^{2} d f^{3} g^{2} + {\left (6 \, a c d f^{2} g^{3} - a^{2} d g^{5}\right )} x^{3} + 3 \, {\left (2 \, a c d f^{3} g^{2} - a^{2} d f g^{4}\right )} x^{2} + {\left (2 \, a c d f^{4} g - 3 \, a^{2} d f^{2} g^{3}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{3} x^{4} - 2 \, a c d^{2} f^{4} g + {\left (3 \, c^{2} d^{2} f^{3} g^{2} - 2 \, a c d^{2} f g^{4}\right )} x^{3} + 3 \, {\left (c^{2} d^{2} f^{4} g - 2 \, a c d^{2} f^{2} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} f^{5} - 6 \, a c d^{2} f^{3} 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)^(7/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/15*(2*c^2*d^2*g*x^2 + 5*c^2*d^2*f*x - 3*a^2*g*e^2 - (a*c*d*g*x - 5*a*c*d*f)*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^3*x^3 + 3*c^2*d^3*f^3*g^2*x^2 + 3*c^2*d^3*f^4*g*x +
c^2*d^3*f^5 + (a^2*g^5*x^4 + 3*a^2*f*g^4*x^3 + 3*a^2*f^2*g^3*x^2 + a^2*f^3*g^2*x)*e^3 - (2*a*c*d*f*g^4*x^4 - a
^2*d*f^3*g^2 + (6*a*c*d*f^2*g^3 - a^2*d*g^5)*x^3 + 3*(2*a*c*d*f^3*g^2 - a^2*d*f*g^4)*x^2 + (2*a*c*d*f^4*g - 3*
a^2*d*f^2*g^3)*x)*e^2 + (c^2*d^2*f^2*g^3*x^4 - 2*a*c*d^2*f^4*g + (3*c^2*d^2*f^3*g^2 - 2*a*c*d^2*f*g^4)*x^3 + 3
*(c^2*d^2*f^4*g - 2*a*c*d^2*f^2*g^3)*x^2 + (c^2*d^2*f^5 - 6*a*c*d^2*f^3*g^2)*x)*e)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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)**(7/2)/(e*x+d)**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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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)^(7/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 4.08, size = 187, normalized size = 1.45 \begin {gather*} \frac {\left (\frac {x\,\left (10\,c^2\,d^2\,f-2\,a\,c\,d\,e\,g\right )}{15\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {6\,a^2\,e^2\,g-10\,a\,c\,d\,e\,f}{15\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {4\,c^2\,d^2\,x^2}{15\,g\,{\left (a\,e\,g-c\,d\,f\right )}^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 {f^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {2\,f\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{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)^(7/2)*(d + e*x)^(1/2)),x)

[Out]

(((x*(10*c^2*d^2*f - 2*a*c*d*e*g))/(15*g^2*(a*e*g - c*d*f)^2) - (6*a^2*e^2*g - 10*a*c*d*e*f)/(15*g^2*(a*e*g -
c*d*f)^2) + (4*c^2*d^2*x^2)/(15*g*(a*e*g - c*d*f)^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^2*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^2 + (2*f*x*(f + g*x)^(1/2)*(d + e*x)^(1/
2))/g)

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