3.8.18 \(\int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [718]

Optimal. Leaf size=198 \[ \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g) \sqrt {d+e x} (f+g x)^{5/2}}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^3 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

2/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^(5/2)/(e*x+d)^(1/2)+8/15*c*d*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^(3/2)/(e*x+d)^(1/2)+16/15*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^3/(e*x+d)^(1/2)/(g*x+f)^(1/2)

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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 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^3}+\frac {8 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^(5/2)) + (8*c*d*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (16*c^2*d^2*S
qrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*Sqrt[f + g*x])

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 {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g) \sqrt {d+e x} (f+g x)^{5/2}}+\frac {(4 c d) \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{5 (c d f-a e g)}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g) \sqrt {d+e x} (f+g x)^{5/2}}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {\left (8 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{15 (c d f-a e g)^2}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g) \sqrt {d+e x} (f+g x)^{5/2}}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^3 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 105, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (3 a^2 e^2 g^2-2 a c d e g (5 f+2 g x)+c^2 d^2 \left (15 f^2+20 f g x+8 g^2 x^2\right )\right )}{15 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(3*a^2*e^2*g^2 - 2*a*c*d*e*g*(5*f + 2*g*x) + c^2*d^2*(15*f^2 + 20*f*g*x + 8*g
^2*x^2)))/(15*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(5/2))

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Maple [A]
time = 0.14, size = 111, normalized size = 0.56

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (8 g^{2} x^{2} c^{2} d^{2}-4 a c d e \,g^{2} x +20 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}-10 a c d e f g +15 f^{2} c^{2} d^{2}\right )}{15 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {5}{2}} \left (a e g -c d f \right )^{3}}\)	\(111\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-4 a c d e \,g^{2} x +20 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}-10 a c d e f g +15 f^{2} c^{2} d^{2}\right ) \sqrt {e x +d}}{15 \left (g x +f \right )^{\frac {5}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\)	\(169\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/15/(e*x+d)^(1/2)/(g*x+f)^(5/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(8*c^2*d^2*g^2*x^2-4*a*c*d*e*g^2*x+20*c^2*d^2*f*
g*x+3*a^2*e^2*g^2-10*a*c*d*e*f*g+15*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((e*x+d)^(1/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (183) = 366\).
time = 5.08, size = 599, normalized size = 3.03 \begin {gather*} \frac {2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} + 20 \, c^{2} d^{2} f g x + 15 \, c^{2} d^{2} f^{2} + 3 \, a^{2} g^{2} e^{2} - 2 \, {\left (2 \, a c d g^{2} x + 5 \, a c d f g\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^{3} d^{4} f^{3} g^{3} x^{3} + 3 \, c^{3} d^{4} f^{4} g^{2} x^{2} + 3 \, c^{3} d^{4} f^{5} g x + c^{3} d^{4} f^{6} - {\left (a^{3} g^{6} x^{4} + 3 \, a^{3} f g^{5} x^{3} + 3 \, a^{3} f^{2} g^{4} x^{2} + a^{3} f^{3} g^{3} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{5} x^{4} - a^{3} d f^{3} g^{3} + {\left (9 \, a^{2} c d f^{2} g^{4} - a^{3} d g^{6}\right )} x^{3} + 3 \, {\left (3 \, a^{2} c d f^{3} g^{3} - a^{3} d f g^{5}\right )} x^{2} + 3 \, {\left (a^{2} c d f^{4} g^{2} - a^{3} d f^{2} g^{4}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{4} x^{4} - a^{2} c d^{2} f^{4} g^{2} + {\left (3 \, a c^{2} d^{2} f^{3} g^{3} - a^{2} c d^{2} f g^{5}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} f^{4} g^{2} - a^{2} c d^{2} f^{2} g^{4}\right )} x^{2} + {\left (a c^{2} d^{2} f^{5} g - 3 \, a^{2} c d^{2} f^{3} g^{3}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{3} x^{4} - 3 \, a c^{2} d^{3} f^{5} g + 3 \, {\left (c^{3} d^{3} f^{4} g^{2} - a c^{2} d^{3} f^{2} g^{4}\right )} x^{3} + 3 \, {\left (c^{3} d^{3} f^{5} g - 3 \, a c^{2} d^{3} f^{3} g^{3}\right )} x^{2} + {\left (c^{3} d^{3} f^{6} - 9 \, a c^{2} d^{3} f^{4} g^{2}\right )} x\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/15*(8*c^2*d^2*g^2*x^2 + 20*c^2*d^2*f*g*x + 15*c^2*d^2*f^2 + 3*a^2*g^2*e^2 - 2*(2*a*c*d*g^2*x + 5*a*c*d*f*g)*
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^3*x^3 + 3*c^3*d^4*f^
4*g^2*x^2 + 3*c^3*d^4*f^5*g*x + c^3*d^4*f^6 - (a^3*g^6*x^4 + 3*a^3*f*g^5*x^3 + 3*a^3*f^2*g^4*x^2 + a^3*f^3*g^3
*x)*e^4 + (3*a^2*c*d*f*g^5*x^4 - a^3*d*f^3*g^3 + (9*a^2*c*d*f^2*g^4 - a^3*d*g^6)*x^3 + 3*(3*a^2*c*d*f^3*g^3 -
a^3*d*f*g^5)*x^2 + 3*(a^2*c*d*f^4*g^2 - a^3*d*f^2*g^4)*x)*e^3 - 3*(a*c^2*d^2*f^2*g^4*x^4 - a^2*c*d^2*f^4*g^2 +
(3*a*c^2*d^2*f^3*g^3 - a^2*c*d^2*f*g^5)*x^3 + 3*(a*c^2*d^2*f^4*g^2 - a^2*c*d^2*f^2*g^4)*x^2 + (a*c^2*d^2*f^5*
g - 3*a^2*c*d^2*f^3*g^3)*x)*e^2 + (c^3*d^3*f^3*g^3*x^4 - 3*a*c^2*d^3*f^5*g + 3*(c^3*d^3*f^4*g^2 - a*c^2*d^3*f^
2*g^4)*x^3 + 3*(c^3*d^3*f^5*g - 3*a*c^2*d^3*f^3*g^3)*x^2 + (c^3*d^3*f^6 - 9*a*c^2*d^3*f^4*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((e*x+d)**(1/2)/(g*x+f)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(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((e*x+d)^(1/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 5.17, size = 242, normalized size = 1.22 \begin {gather*} -\frac {\left (\frac {\sqrt {d+e\,x}\,\left (6\,a^2\,e^2\,g^2-20\,a\,c\,d\,e\,f\,g+30\,c^2\,d^2\,f^2\right )}{15\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^2\,d^2\,x^2\,\sqrt {d+e\,x}}{15\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c\,d\,x\,\left (a\,e\,g-5\,c\,d\,f\right )\,\sqrt {d+e\,x}}{15\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3\,\sqrt {f+g\,x}+\frac {d\,f^2\,\sqrt {f+g\,x}}{e\,g^2}+\frac {x^2\,\sqrt {f+g\,x}\,\left (d\,g+2\,e\,f\right )}{e\,g}+\frac {f\,x\,\sqrt {f+g\,x}\,\left (2\,d\,g+e\,f\right )}{e\,g^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/((f + g*x)^(7/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)

[Out]

-((((d + e*x)^(1/2)*(6*a^2*e^2*g^2 + 30*c^2*d^2*f^2 - 20*a*c*d*e*f*g))/(15*e*g^2*(a*e*g - c*d*f)^3) + (16*c^2*
d^2*x^2*(d + e*x)^(1/2))/(15*e*(a*e*g - c*d*f)^3) - (8*c*d*x*(a*e*g - 5*c*d*f)*(d + e*x)^(1/2))/(15*e*g*(a*e*g
- c*d*f)^3))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(x^3*(f + g*x)^(1/2) + (d*f^2*(f + g*x)^(1/2))/(e
*g^2) + (x^2*(f + g*x)^(1/2)*(d*g + 2*e*f))/(e*g) + (f*x*(f + g*x)^(1/2)*(2*d*g + e*f))/(e*g^2))

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