3.8.25 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [725]

Optimal. Leaf size=192 \[ -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-8/3*g*(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)-16/3*c*d*g*(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 = 192, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {882, 886, 874} \begin {gather*} -\frac {16 c d g \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 {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d + e*x])/((c*d*f - a*e*g)*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8*g*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(3/2)) - (16*c*d*g*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]

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 {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(4 g) \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}{c d f-a e g}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 g \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)^2 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {(8 c d g) \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 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 g \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)^2 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {16 c d g \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 = 0.15, size = 105, normalized size = 0.55 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (-a^2 e^2 g^2+2 a c d e g (3 f+2 g x)+c^2 d^2 \left (3 f^2+12 f g x+8 g^2 x^2\right )\right )}{3 (c d f-a e g)^3 \sqrt {(a e+c d x) (d+e x)} (f+g x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(-(a^2*e^2*g^2) + 2*a*c*d*e*g*(3*f + 2*g*x) + c^2*d^2*(3*f^2 + 12*f*g*x + 8*g^2*x^2)))/(3*(c
*d*f - a*e*g)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^(3/2))

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Maple [A]
time = 0.14, size = 120, normalized size = 0.62

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-8 g^{2} x^{2} c^{2} d^{2}-4 a c d e \,g^{2} x -12 c^{2} d^{2} f g x +a^{2} e^{2} g^{2}-6 a c d e f g -3 f^{2} c^{2} d^{2}\right )}{3 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {3}{2}} \left (c d x +a e \right ) \left (a e g -c d f \right )^{3}}\)	\(120\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-8 g^{2} x^{2} c^{2} d^{2}-4 a c d e \,g^{2} x -12 c^{2} d^{2} f g x +a^{2} e^{2} g^{2}-6 a c d e f g -3 f^{2} c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (179) = 358\).
time = 3.23, size = 679, normalized size = 3.54 \begin {gather*} -\frac {2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} + 12 \, c^{2} d^{2} f g x + 3 \, c^{2} d^{2} f^{2} - a^{2} g^{2} e^{2} + 2 \, {\left (2 \, a c d g^{2} x + 3 \, 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^{4} d^{5} f^{3} g^{2} x^{3} + 2 \, c^{4} d^{5} f^{4} g x^{2} + c^{4} d^{5} f^{5} x - {\left (a^{4} g^{5} x^{3} + 2 \, a^{4} f g^{4} x^{2} + a^{4} f^{2} g^{3} x\right )} e^{5} - {\left (a^{3} c d g^{5} x^{4} - a^{3} c d f g^{4} x^{3} + a^{4} d f^{2} g^{3} - {\left (5 \, a^{3} c d f^{2} g^{3} - a^{4} d g^{5}\right )} x^{2} - {\left (3 \, a^{3} c d f^{3} g^{2} - 2 \, a^{4} d f g^{4}\right )} x\right )} e^{4} + {\left (3 \, a^{2} c^{2} d^{2} f g^{4} x^{4} + 3 \, a^{3} c d^{2} f^{3} g^{2} + {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{3} - a^{3} c d^{2} g^{5}\right )} x^{3} - {\left (3 \, a^{2} c^{2} d^{2} f^{3} g^{2} - a^{3} c d^{2} f g^{4}\right )} x^{2} - {\left (3 \, a^{2} c^{2} d^{2} f^{4} g - 5 \, a^{3} c d^{2} f^{2} g^{3}\right )} x\right )} e^{3} - {\left (3 \, a c^{3} d^{3} f^{2} g^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} f^{4} g + {\left (5 \, a c^{3} d^{3} f^{3} g^{2} - 3 \, a^{2} c^{2} d^{3} f g^{4}\right )} x^{3} + {\left (a c^{3} d^{3} f^{4} g - 3 \, a^{2} c^{2} d^{3} f^{2} g^{3}\right )} x^{2} - {\left (a c^{3} d^{3} f^{5} - 3 \, a^{2} c^{2} d^{3} f^{3} g^{2}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{3} g^{2} x^{4} - a c^{3} d^{4} f^{4} g x + a c^{3} d^{4} f^{5} + {\left (2 \, c^{4} d^{4} f^{4} g - 3 \, a c^{3} d^{4} f^{2} g^{3}\right )} x^{3} + {\left (c^{4} d^{4} f^{5} - 5 \, a c^{3} d^{4} f^{3} g^{2}\right )} x^{2}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(8*c^2*d^2*g^2*x^2 + 12*c^2*d^2*f*g*x + 3*c^2*d^2*f^2 - a^2*g^2*e^2 + 2*(2*a*c*d*g^2*x + 3*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^4*d^5*f^3*g^2*x^3 + 2*c^4*d^5*f^4*g
*x^2 + c^4*d^5*f^5*x - (a^4*g^5*x^3 + 2*a^4*f*g^4*x^2 + a^4*f^2*g^3*x)*e^5 - (a^3*c*d*g^5*x^4 - a^3*c*d*f*g^4*
x^3 + a^4*d*f^2*g^3 - (5*a^3*c*d*f^2*g^3 - a^4*d*g^5)*x^2 - (3*a^3*c*d*f^3*g^2 - 2*a^4*d*f*g^4)*x)*e^4 + (3*a^
2*c^2*d^2*f*g^4*x^4 + 3*a^3*c*d^2*f^3*g^2 + (3*a^2*c^2*d^2*f^2*g^3 - a^3*c*d^2*g^5)*x^3 - (3*a^2*c^2*d^2*f^3*g
^2 - a^3*c*d^2*f*g^4)*x^2 - (3*a^2*c^2*d^2*f^4*g - 5*a^3*c*d^2*f^2*g^3)*x)*e^3 - (3*a*c^3*d^3*f^2*g^3*x^4 + 3*
a^2*c^2*d^3*f^4*g + (5*a*c^3*d^3*f^3*g^2 - 3*a^2*c^2*d^3*f*g^4)*x^3 + (a*c^3*d^3*f^4*g - 3*a^2*c^2*d^3*f^2*g^3
)*x^2 - (a*c^3*d^3*f^5 - 3*a^2*c^2*d^3*f^3*g^2)*x)*e^2 + (c^4*d^4*f^3*g^2*x^4 - a*c^3*d^4*f^4*g*x + a*c^3*d^4*
f^5 + (2*c^4*d^4*f^4*g - 3*a*c^3*d^4*f^2*g^3)*x^3 + (c^4*d^4*f^5 - 5*a*c^3*d^4*f^3*g^2)*x^2)*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)**(3/2)/(g*x+f)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4370 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)^(3/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 5.33, size = 268, normalized size = 1.40 \begin {gather*} \frac {\left (\frac {8\,x\,\left (a\,e\,g+3\,c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {\sqrt {d+e\,x}\,\left (-2\,a^2\,e^2\,g^2+12\,a\,c\,d\,e\,f\,g+6\,c^2\,d^2\,f^2\right )}{3\,c\,d\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c\,d\,g\,x^2\,\sqrt {d+e\,x}}{3\,e\,{\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 {a\,f\,\sqrt {f+g\,x}}{c\,g}+\frac {x\,\sqrt {f+g\,x}\,\left (c\,f\,d^2+a\,g\,d\,e+a\,f\,e^2\right )}{c\,d\,e\,g}+\frac {x^2\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+c\,f\,d\,e+a\,g\,e^2\right )}{c\,d\,e\,g}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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