3.5.91 \(\int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac {4 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

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Rubi [A]  time = 0.15, antiderivative size = 124, 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 = {868, 860} \begin {gather*} -\frac {4 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{\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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/((f + g*x)^(3/2)*(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)*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (4*g*Sqrt[a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/((c*d*f - a*e*g)^2*Sqrt[d + e*x]*Sqrt[f + g*x])

Rule 860

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 868

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)^{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 &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(2 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}{c d f-a e g}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {4 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 64, normalized size = 0.52 \begin {gather*} -\frac {2 \sqrt {d+e x} (a e g+c d (f+2 g x))}{\sqrt {f+g x} \sqrt {(d+e x) (a e+c d x)} (c d f-a e g)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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IntegrateAlgebraic [A]  time = 0.88, size = 132, normalized size = 1.06 \begin {gather*} -\frac {2 (d+e x)^{3/2} (a e g+c d g x)^{3/2} \left (a e g^{3/2}+2 c d \sqrt {g} (f+g x)-c d f \sqrt {g}\right )}{g^{3/2} \sqrt {f+g x} (c d f-a e g)^2 \left (\frac {(d g+e g x) (a e g+c d g x)}{g^2}\right )^{3/2} \sqrt {a e g+c d (f+g x)-c d f}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.44, size = 325, normalized size = 2.62 \begin {gather*} -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {e x + d} \sqrt {g x + f}}{a c^{2} d^{3} e f^{3} - 2 \, a^{2} c d^{2} e^{2} f^{2} g + a^{3} d e^{3} f g^{2} + {\left (c^{3} d^{3} e f^{2} g - 2 \, a c^{2} d^{2} e^{2} f g^{2} + a^{2} c d e^{3} g^{3}\right )} x^{3} + {\left (c^{3} d^{3} e f^{3} + {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{2} g - {\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g^{2} + {\left (a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{3}\right )} x^{2} + {\left (a^{3} d e^{3} g^{3} + {\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} f^{3} - {\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} f^{2} g - {\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f g^{2}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  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)^(3/2)/(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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maple [A]  time = 0.01, size = 97, normalized size = 0.78 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (2 c d g x +a e g +c d f \right ) \left (e x +d \right )^{\frac {3}{2}}}{\sqrt {g x +f}\, \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.98, size = 151, normalized size = 1.22 \begin {gather*} -\frac {\left (\frac {4\,g\,x\,\sqrt {d+e\,x}}{e\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {\left (2\,a\,e\,g+2\,c\,d\,f\right )\,\sqrt {d+e\,x}}{c\,d\,e\,{\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}+\frac {a\,\sqrt {f+g\,x}}{c}+\frac {x\,\sqrt {f+g\,x}\,\left (c\,d^2+a\,e^2\right )}{c\,d\,e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  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)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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