3.8.31 \(\int \frac {(d+e x)^{5/2}}{(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [731]

Optimal. Leaf size=194 \[ -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 g \sqrt {d+e x}}{3 (c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

-2/3*(e*x+d)^(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)/(g*x+f)^(1/2)+8/3*g*(e*x+d)^(1/2)/(-
a*e*g+c*d*f)^2/(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+16/3*g^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 = 194, 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 = {882, 874} \begin {gather*} \frac {16 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^3}+\frac {8 g \sqrt {d+e x}}{3 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac {2 (d+e x)^{3/2}}{3 \sqrt {f+g x} \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)/((f + g*x)^(3/2)*(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))/(3*(c*d*f - a*e*g)*Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (8*g*Sq
rt[d + e*x])/(3*(c*d*f - a*e*g)^2*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (16*g^2*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(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 882

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

Rubi steps

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

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Mathematica [A]
time = 0.16, size = 102, normalized size = 0.53 \begin {gather*} -\frac {2 (d+e x)^{3/2} (f+g x)^{3/2} \left (c^2 d^2-\frac {3 g^2 (a e+c d x)^2}{(f+g x)^2}-\frac {6 c d g (a e+c d x)}{f+g x}\right )}{3 (c d f-a e g)^3 ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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Maple [A]
time = 0.14, size = 121, normalized size = 0.62

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (8 g^{2} x^{2} c^{2} d^{2}+12 a c d e \,g^{2} x +4 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}+6 a c d e f g -f^{2} c^{2} d^{2}\right )}{3 \sqrt {e x +d}\, \sqrt {g x +f}\, \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{3}}\)	\(121\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}+12 a c d e \,g^{2} x +4 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}+6 a c d e f g -f^{2} c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \sqrt {g x +f}\, \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(169\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(g*x+f)^(3/2)/(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)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(8*c^2*d^2*g^2*x^2+12*a*c*d*e*g^2*x+4*c^2*d^2*f*g
*x+3*a^2*e^2*g^2+6*a*c*d*e*f*g-c^2*d^2*f^2)/(c*d*x+a*e)^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)^(5/2)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (179) = 358\).
time = 6.90, size = 690, normalized size = 3.56 \begin {gather*} \frac {2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} + 4 \, c^{2} d^{2} f g x - c^{2} d^{2} f^{2} + 3 \, a^{2} g^{2} e^{2} + 6 \, {\left (2 \, a c d g^{2} x + 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}}{3 \, {\left (c^{5} d^{6} f^{3} g x^{3} + c^{5} d^{6} f^{4} x^{2} - {\left (a^{5} g^{4} x^{2} + a^{5} f g^{3} x\right )} e^{6} - {\left (2 \, a^{4} c d g^{4} x^{3} - a^{4} c d f g^{3} x^{2} + a^{5} d f g^{3} - {\left (3 \, a^{4} c d f^{2} g^{2} - a^{5} d g^{4}\right )} x\right )} e^{5} - {\left (a^{3} c^{2} d^{2} g^{4} x^{4} - 5 \, a^{3} c^{2} d^{2} f g^{3} x^{3} - 3 \, a^{4} c d^{2} f^{2} g^{2} - {\left (3 \, a^{3} c^{2} d^{2} f^{2} g^{2} - 2 \, a^{4} c d^{2} g^{4}\right )} x^{2} + {\left (3 \, a^{3} c^{2} d^{2} f^{3} g - a^{4} c d^{2} f g^{3}\right )} x\right )} e^{4} + {\left (3 \, a^{2} c^{3} d^{3} f g^{3} x^{4} - 3 \, a^{3} c^{2} d^{3} f^{3} g - {\left (3 \, a^{2} c^{3} d^{3} f^{2} g^{2} + a^{3} c^{2} d^{3} g^{4}\right )} x^{3} - 5 \, {\left (a^{2} c^{3} d^{3} f^{3} g - a^{3} c^{2} d^{3} f g^{3}\right )} x^{2} + {\left (a^{2} c^{3} d^{3} f^{4} + 3 \, a^{3} c^{2} d^{3} f^{2} g^{2}\right )} x\right )} e^{3} - {\left (3 \, a c^{4} d^{4} f^{2} g^{2} x^{4} + 5 \, a^{2} c^{3} d^{4} f^{3} g x - a^{2} c^{3} d^{4} f^{4} + {\left (a c^{4} d^{4} f^{3} g - 3 \, a^{2} c^{3} d^{4} f g^{3}\right )} x^{3} - {\left (2 \, a c^{4} d^{4} f^{4} - 3 \, a^{2} c^{3} d^{4} f^{2} g^{2}\right )} x^{2}\right )} e^{2} + {\left (c^{5} d^{5} f^{3} g x^{4} - a c^{4} d^{5} f^{3} g x^{2} + 2 \, a c^{4} d^{5} f^{4} x + {\left (c^{5} d^{5} f^{4} - 3 \, a c^{4} d^{5} f^{2} g^{2}\right )} x^{3}\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)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

2/3*(8*c^2*d^2*g^2*x^2 + 4*c^2*d^2*f*g*x - c^2*d^2*f^2 + 3*a^2*g^2*e^2 + 6*(2*a*c*d*g^2*x + 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^5*d^6*f^3*g*x^3 + c^5*d^6*f^4*x^2 - (a^
5*g^4*x^2 + a^5*f*g^3*x)*e^6 - (2*a^4*c*d*g^4*x^3 - a^4*c*d*f*g^3*x^2 + a^5*d*f*g^3 - (3*a^4*c*d*f^2*g^2 - a^5
*d*g^4)*x)*e^5 - (a^3*c^2*d^2*g^4*x^4 - 5*a^3*c^2*d^2*f*g^3*x^3 - 3*a^4*c*d^2*f^2*g^2 - (3*a^3*c^2*d^2*f^2*g^2
- 2*a^4*c*d^2*g^4)*x^2 + (3*a^3*c^2*d^2*f^3*g - a^4*c*d^2*f*g^3)*x)*e^4 + (3*a^2*c^3*d^3*f*g^3*x^4 - 3*a^3*c^
2*d^3*f^3*g - (3*a^2*c^3*d^3*f^2*g^2 + a^3*c^2*d^3*g^4)*x^3 - 5*(a^2*c^3*d^3*f^3*g - a^3*c^2*d^3*f*g^3)*x^2 +
(a^2*c^3*d^3*f^4 + 3*a^3*c^2*d^3*f^2*g^2)*x)*e^3 - (3*a*c^4*d^4*f^2*g^2*x^4 + 5*a^2*c^3*d^4*f^3*g*x - a^2*c^3*
d^4*f^4 + (a*c^4*d^4*f^3*g - 3*a^2*c^3*d^4*f*g^3)*x^3 - (2*a*c^4*d^4*f^4 - 3*a^2*c^3*d^4*f^2*g^2)*x^2)*e^2 + (
c^5*d^5*f^3*g*x^4 - a*c^4*d^5*f^3*g*x^2 + 2*a*c^4*d^5*f^4*x + (c^5*d^5*f^4 - 3*a*c^4*d^5*f^2*g^2)*x^3)*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)**(3/2)/(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 8570 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)^(5/2)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 5.28, size = 255, normalized size = 1.31 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,g^2\,x^2\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {\sqrt {d+e\,x}\,\left (6\,a^2\,e^2\,g^2+12\,a\,c\,d\,e\,f\,g-2\,c^2\,d^2\,f^2\right )}{3\,c^2\,d^2\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {8\,g\,x\,\left (3\,a\,e\,g+c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^3\,\sqrt {f+g\,x}+\frac {a^2\,e\,\sqrt {f+g\,x}}{c^2\,d}+\frac {x^2\,\sqrt {f+g\,x}\,\left (c\,d^2+2\,a\,e^2\right )}{c\,d\,e}+\frac {a\,x\,\sqrt {f+g\,x}\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((16*g^2*x^2*(d + e*x)^(1/2))/(3*e*(a*e*g - c*d*f)^3) + ((d +
e*x)^(1/2)*(6*a^2*e^2*g^2 - 2*c^2*d^2*f^2 + 12*a*c*d*e*f*g))/(3*c^2*d^2*e*(a*e*g - c*d*f)^3) + (8*g*x*(3*a*e*g
+ c*d*f)*(d + e*x)^(1/2))/(3*c*d*e*(a*e*g - c*d*f)^3)))/(x^3*(f + g*x)^(1/2) + (a^2*e*(f + g*x)^(1/2))/(c^2*d
) + (x^2*(f + g*x)^(1/2)*(2*a*e^2 + c*d^2))/(c*d*e) + (a*x*(f + g*x)^(1/2)*(a*e^2 + 2*c*d^2))/(c^2*d^2))

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