3.8.29 \(\int \frac {(d+e x)^{5/2} \sqrt {f+g x}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [729]

Optimal. Leaf size=63 \[ -\frac {2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(5/2)*Sqrt[f + g*x])/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 52, normalized size = 0.83 \begin {gather*} -\frac {2 (d+e x)^{3/2} (f+g x)^{3/2}}{3 (c d f-a e g) ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(5/2)*Sqrt[f + g*x])/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

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

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Maple [A]
time = 0.15, size = 55, normalized size = 0.87

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

[Out]

integrate(sqrt(g*x + f)*(x*e + d)^(5/2)/(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (58) = 116\).
time = 4.63, size = 188, normalized size = 2.98 \begin {gather*} -\frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (g x + f\right )}^{\frac {3}{2}} \sqrt {x e + d}}{3 \, {\left (c^{3} d^{4} f x^{2} - a^{3} g x e^{4} - {\left (2 \, a^{2} c d g x^{2} - a^{2} c d f x + a^{3} d g\right )} e^{3} - {\left (a c^{2} d^{2} g x^{3} - 2 \, a c^{2} d^{2} f x^{2} + 2 \, a^{2} c d^{2} g x - a^{2} c d^{2} f\right )} e^{2} + {\left (c^{3} d^{3} f x^{3} - a c^{2} d^{3} g x^{2} + 2 \, a c^{2} d^{3} f x\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4369 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(g*x + f)*(x*e + d)^(5/2)/(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(5/2), x)

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Mupad [B]
time = 4.32, size = 169, normalized size = 2.68 \begin {gather*} \frac {\left (\frac {2\,f\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,\left (a\,e\,g-c\,d\,f\right )}+\frac {2\,g\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{3\,c^2\,d^2\,e\,\left (a\,e\,g-c\,d\,f\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (c\,d^2+2\,a\,e^2\right )}{c\,d\,e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^(1/2)*(d + e*x)^(5/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

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

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