3.17.48 \(\int \frac {(d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx\) [1648]

Optimal. Leaf size=85 \[ \frac {3 e \sqrt {d+e x}}{b^2}-\frac {(d+e x)^{3/2}}{b (a+b x)}-\frac {3 e \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}} \]

[Out]

-(e*x+d)^(3/2)/b/(b*x+a)-3*e*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(5/2)+3*e*(e*x
+d)^(1/2)/b^2

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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 52, 65, 214} \begin {gather*} -\frac {3 e \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}-\frac {(d+e x)^{3/2}}{b (a+b x)}+\frac {3 e \sqrt {d+e x}}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(3*e*Sqrt[d + e*x])/b^2 - (d + e*x)^(3/2)/(b*(a + b*x)) - (3*e*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])
/Sqrt[b*d - a*e]])/b^(5/2)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx\\ &=-\frac {(d+e x)^{3/2}}{b (a+b x)}+\frac {(3 e) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b}\\ &=\frac {3 e \sqrt {d+e x}}{b^2}-\frac {(d+e x)^{3/2}}{b (a+b x)}+\frac {(3 e (b d-a e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^2}\\ &=\frac {3 e \sqrt {d+e x}}{b^2}-\frac {(d+e x)^{3/2}}{b (a+b x)}+\frac {(3 (b d-a e)) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^2}\\ &=\frac {3 e \sqrt {d+e x}}{b^2}-\frac {(d+e x)^{3/2}}{b (a+b x)}-\frac {3 e \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 83, normalized size = 0.98 \begin {gather*} \frac {\sqrt {d+e x} (-b d+3 a e+2 b e x)}{b^2 (a+b x)}-\frac {3 e \sqrt {-b d+a e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(Sqrt[d + e*x]*(-(b*d) + 3*a*e + 2*b*e*x))/(b^2*(a + b*x)) - (3*e*Sqrt[-(b*d) + a*e]*ArcTan[(Sqrt[b]*Sqrt[d +
e*x])/Sqrt[-(b*d) + a*e]])/b^(5/2)

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Maple [A]
time = 0.70, size = 100, normalized size = 1.18

method	result	size
derivativedivides	\(2 e \left (\frac {\sqrt {e x +d}}{b^{2}}-\frac {\frac {\left (-\frac {a e}{2}+\frac {b d}{2}\right ) \sqrt {e x +d}}{\left (e x +d \right ) b +a e -b d}+\frac {3 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}}{b^{2}}\right )\)	\(100\)
default	\(2 e \left (\frac {\sqrt {e x +d}}{b^{2}}-\frac {\frac {\left (-\frac {a e}{2}+\frac {b d}{2}\right ) \sqrt {e x +d}}{\left (e x +d \right ) b +a e -b d}+\frac {3 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}}{b^{2}}\right )\)	\(100\)
risch	\(\frac {2 e \sqrt {e x +d}}{b^{2}}+\frac {e^{2} \sqrt {e x +d}\, a}{b^{2} \left (b e x +a e \right )}-\frac {e \sqrt {e x +d}\, d}{b \left (b e x +a e \right )}-\frac {3 e^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a}{b^{2} \sqrt {b \left (a e -b d \right )}}+\frac {3 e \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d}{b \sqrt {b \left (a e -b d \right )}}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

2*e*(1/b^2*(e*x+d)^(1/2)-1/b^2*((-1/2*a*e+1/2*b*d)*(e*x+d)^(1/2)/((e*x+d)*b+a*e-b*d)+3/2*(a*e-b*d)/(b*(a*e-b*d
))^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 2.67, size = 222, normalized size = 2.61 \begin {gather*} \left [\frac {3 \, {\left (b x + a\right )} \sqrt {\frac {b d - a e}{b}} e \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (b d - {\left (2 \, b x + 3 \, a\right )} e\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} x + a b^{2}\right )}}, -\frac {3 \, {\left (b x + a\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) e + {\left (b d - {\left (2 \, b x + 3 \, a\right )} e\right )} \sqrt {x e + d}}{b^{3} x + a b^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(3*(b*x + a)*sqrt((b*d - a*e)/b)*e*log((2*b*d - 2*sqrt(x*e + d)*b*sqrt((b*d - a*e)/b) + (b*x - a)*e)/(b*x
+ a)) - 2*(b*d - (2*b*x + 3*a)*e)*sqrt(x*e + d))/(b^3*x + a*b^2), -(3*(b*x + a)*sqrt(-(b*d - a*e)/b)*arctan(-
sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e))*e + (b*d - (2*b*x + 3*a)*e)*sqrt(x*e + d))/(b^3*x + a*b^2)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 923 vs. \(2 (73) = 146\).
time = 68.43, size = 923, normalized size = 10.86 \begin {gather*} \frac {2 a^{2} e^{3} \sqrt {d + e x}}{2 a^{2} b^{2} e^{2} - 2 a b^{3} d e + 2 a b^{3} e^{2} x - 2 b^{4} d e x} - \frac {a^{2} e^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} + \frac {a^{2} e^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} - \frac {4 a d e^{2} \sqrt {d + e x}}{2 a^{2} b e^{2} - 2 a b^{2} d e + 2 a b^{2} e^{2} x - 2 b^{3} d e x} + \frac {a d e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{b} - \frac {a d e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{b} - \frac {4 a e^{2} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e}{b} - d}} \right )}}{b^{3} \sqrt {\frac {a e}{b} - d}} - \frac {d^{2} e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {d^{2} e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {2 d^{2} e \sqrt {d + e x}}{2 a^{2} e^{2} - 2 a b d e + 2 a b e^{2} x - 2 b^{2} d e x} + \frac {4 d e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e}{b} - d}} \right )}}{b^{2} \sqrt {\frac {a e}{b} - d}} + \frac {2 e \sqrt {d + e x}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

2*a**2*e**3*sqrt(d + e*x)/(2*a**2*b**2*e**2 - 2*a*b**3*d*e + 2*a*b**3*e**2*x - 2*b**4*d*e*x) - a**2*e**3*sqrt(
-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b*
*2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**2) + a**2*e**3*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2
*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**
3)) + sqrt(d + e*x))/(2*b**2) - 4*a*d*e**2*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b
**3*d*e*x) + a*d*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(
-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/b - a*d*e**2*sqrt(-1/(b*(a*e -
b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(
-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/b - 4*a*e**2*atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b**3*sqrt(a*e/b - d)
) - d**2*e*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e
- b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + d**2*e*sqrt(-1/(b*(a*e - b*d)**3))*lo
g(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e -
b*d)**3)) + sqrt(d + e*x))/2 + 2*d**2*e*sqrt(d + e*x)/(2*a**2*e**2 - 2*a*b*d*e + 2*a*b*e**2*x - 2*b**2*d*e*x)
+ 4*d*e*atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b**2*sqrt(a*e/b - d)) + 2*e*sqrt(d + e*x)/b**2

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Giac [A]
time = 0.76, size = 122, normalized size = 1.44 \begin {gather*} \frac {3 \, {\left (b d e - a e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{2}} + \frac {2 \, \sqrt {x e + d} e}{b^{2}} - \frac {\sqrt {x e + d} b d e - \sqrt {x e + d} a e^{2}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b*d*e - a*e^2)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^2) + 2*sqrt(x*e + d)*e/
b^2 - (sqrt(x*e + d)*b*d*e - sqrt(x*e + d)*a*e^2)/(((x*e + d)*b - b*d + a*e)*b^2)

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Mupad [B]
time = 0.11, size = 109, normalized size = 1.28 \begin {gather*} \frac {\left (a\,e^2-b\,d\,e\right )\,\sqrt {d+e\,x}}{b^3\,\left (d+e\,x\right )-b^3\,d+a\,b^2\,e}+\frac {2\,e\,\sqrt {d+e\,x}}{b^2}-\frac {3\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,\sqrt {a\,e-b\,d}\,\sqrt {d+e\,x}}{a\,e^2-b\,d\,e}\right )\,\sqrt {a\,e-b\,d}}{b^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

((a*e^2 - b*d*e)*(d + e*x)^(1/2))/(b^3*(d + e*x) - b^3*d + a*b^2*e) + (2*e*(d + e*x)^(1/2))/b^2 - (3*e*atan((b
^(1/2)*e*(a*e - b*d)^(1/2)*(d + e*x)^(1/2))/(a*e^2 - b*d*e))*(a*e - b*d)^(1/2))/b^(5/2)

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