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

Optimal. Leaf size=202 \[ -\frac {3 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac {3 c d \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2}} \]

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Rubi [A]  time = 0.26, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {868, 872, 874, 205} \begin {gather*} -\frac {3 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(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)}-\frac {3 c d \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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]

Rule 872

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]

Rule 874

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{a, b, c, d, e, f, g}, 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]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{(f+g x)^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) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(3 g) \int \frac {\sqrt {d+e x}}{(f+g x)^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) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 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} (f+g x)}-\frac {(3 c d g) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 (c d f-a e g)^2}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 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} (f+g x)}-\frac {\left (3 c d e^2 g\right ) \operatorname {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{(c d f-a e g)^2}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 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} (f+g x)}-\frac {3 c d \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(c d f-a e g)^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 73, normalized size = 0.36 \begin {gather*} -\frac {2 c d \sqrt {d+e x} \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{\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)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*c*d*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 2, 1/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/((c*d*f - a*e*g)^
2*Sqrt[(a*e + c*d*x)*(d + e*x)])

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IntegrateAlgebraic [F]  time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [B]  time = 0.45, size = 1067, normalized size = 5.28 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{2} e g x^{3} + a c d^{2} e f + {\left (c^{2} d^{2} e f + {\left (c^{2} d^{3} + a c d e^{2}\right )} g\right )} x^{2} + {\left (a c d^{2} e g + {\left (c^{2} d^{3} + a c d e^{2}\right )} f\right )} x\right )} \sqrt {-\frac {g}{c d f - a e g}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d f - a e g\right )} \sqrt {e x + d} \sqrt {-\frac {g}{c d f - a e g}} - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d g x + 2 \, c d f + a e g\right )} \sqrt {e x + d}}{2 \, {\left (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\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e g x^{3} + a c d^{2} e f + {\left (c^{2} d^{2} e f + {\left (c^{2} d^{3} + a c d e^{2}\right )} g\right )} x^{2} + {\left (a c d^{2} e g + {\left (c^{2} d^{3} + a c d e^{2}\right )} f\right )} x\right )} \sqrt {\frac {g}{c d f - a e g}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d f - a e g\right )} \sqrt {e x + d} \sqrt {\frac {g}{c d f - a e g}}}{c d e g x^{2} + a d e g + {\left (c d^{2} + a e^{2}\right )} g x}\right ) + \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d g x + 2 \, c d f + a e g\right )} \sqrt {e x + d}}{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}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(3*(c^2*d^2*e*g*x^3 + a*c*d^2*e*f + (c^2*d^2*e*f + (c^2*d^3 + a*c*d*e^2)*g)*x^2 + (a*c*d^2*e*g + (c^2*d^3
+ a*c*d*e^2)*f)*x)*sqrt(-g/(c*d*f - a*e*g))*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - 2*sqrt(c*d*e*x^2 + a*d*
e + (c*d^2 + a*e^2)*x)*(c*d*f - a*e*g)*sqrt(e*x + d)*sqrt(-g/(c*d*f - a*e*g)) - (c*d*e*f - (c*d^2 + 2*a*e^2)*g
)*x)/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*g*x + 2*c*d*f + a
*e*g)*sqrt(e*x + d))/(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), -(3*(c^2*d^2*e*g*x^3 + a*c*d^2*e*f + (c^2*
d^2*e*f + (c^2*d^3 + a*c*d*e^2)*g)*x^2 + (a*c*d^2*e*g + (c^2*d^3 + a*c*d*e^2)*f)*x)*sqrt(g/(c*d*f - a*e*g))*ar
ctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*f - a*e*g)*sqrt(e*x + d)*sqrt(g/(c*d*f - a*e*g))/(c*d*e
*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*g*x)) + sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*g*x + 2*c*d*f +
a*e*g)*sqrt(e*x + d))/(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(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 2.58Unable to transpose Error: Bad Argument Value

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maple [A]  time = 0.03, size = 225, normalized size = 1.11 \begin {gather*} \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (3 \sqrt {c d x +a e}\, c d \,g^{2} x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+3 \sqrt {c d x +a e}\, c d f g \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-3 \sqrt {\left (a e g -c d f \right ) g}\, c d g x -\sqrt {\left (a e g -c d f \right ) g}\, a e g -2 \sqrt {\left (a e g -c d f \right ) g}\, c d f \right )}{\sqrt {e x +d}\, \left (c d x +a e \right ) \left (a e g -c d f \right )^{2} \left (g x +f \right ) \sqrt {\left (a e g -c d f \right ) g}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}^{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)^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)^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)**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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