3.17.47 \(\int \frac {(d+e x)^{5/2}}{a^2+2 a b x+b^2 x^2} \, dx\) [1647]

Optimal. Leaf size=110 \[ \frac {5 e (b d-a e) \sqrt {d+e x}}{b^3}+\frac {5 e (d+e x)^{3/2}}{3 b^2}-\frac {(d+e x)^{5/2}}{b (a+b x)}-\frac {5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}} \]

[Out]

5/3*e*(e*x+d)^(3/2)/b^2-(e*x+d)^(5/2)/b/(b*x+a)-5*e*(-a*e+b*d)^(3/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^
(1/2))/b^(7/2)+5*e*(-a*e+b*d)*(e*x+d)^(1/2)/b^3

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Rubi [A]
time = 0.04, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}}+\frac {5 e \sqrt {d+e x} (b d-a e)}{b^3}-\frac {(d+e x)^{5/2}}{b (a+b x)}+\frac {5 e (d+e x)^{3/2}}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(5*e*(b*d - a*e)*Sqrt[d + e*x])/b^3 + (5*e*(d + e*x)^(3/2))/(3*b^2) - (d + e*x)^(5/2)/(b*(a + b*x)) - (5*e*(b*
d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(7/2)

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)^{5/2}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^{5/2}}{(a+b x)^2} \, dx\\ &=-\frac {(d+e x)^{5/2}}{b (a+b x)}+\frac {(5 e) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{2 b}\\ &=\frac {5 e (d+e x)^{3/2}}{3 b^2}-\frac {(d+e x)^{5/2}}{b (a+b x)}+\frac {(5 e (b d-a e)) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b^2}\\ &=\frac {5 e (b d-a e) \sqrt {d+e x}}{b^3}+\frac {5 e (d+e x)^{3/2}}{3 b^2}-\frac {(d+e x)^{5/2}}{b (a+b x)}+\frac {\left (5 e (b d-a e)^2\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^3}\\ &=\frac {5 e (b d-a e) \sqrt {d+e x}}{b^3}+\frac {5 e (d+e x)^{3/2}}{3 b^2}-\frac {(d+e x)^{5/2}}{b (a+b x)}+\frac {\left (5 (b d-a e)^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^3}\\ &=\frac {5 e (b d-a e) \sqrt {d+e x}}{b^3}+\frac {5 e (d+e x)^{3/2}}{3 b^2}-\frac {(d+e x)^{5/2}}{b (a+b x)}-\frac {5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 116, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {d+e x} \left (15 a^2 e^2+10 a b e (-2 d+e x)+b^2 \left (3 d^2-14 d e x-2 e^2 x^2\right )\right )}{3 b^3 (a+b x)}+\frac {5 e (-b d+a e)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

-1/3*(Sqrt[d + e*x]*(15*a^2*e^2 + 10*a*b*e*(-2*d + e*x) + b^2*(3*d^2 - 14*d*e*x - 2*e^2*x^2)))/(b^3*(a + b*x))
+ (5*e*(-(b*d) + a*e)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/b^(7/2)

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Maple [A]
time = 0.78, size = 152, normalized size = 1.38

method	result	size
derivativedivides	\(2 e \left (-\frac {-\frac {\left (e x +d \right )^{\frac {3}{2}} b}{3}+2 a e \sqrt {e x +d}-2 d b \sqrt {e x +d}}{b^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} e^{2}+a b d e -\frac {1}{2} b^{2} d^{2}\right ) \sqrt {e x +d}}{\left (e x +d \right ) b +a e -b d}+\frac {5 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}}{b^{3}}\right )\)	\(152\)
default	\(2 e \left (-\frac {-\frac {\left (e x +d \right )^{\frac {3}{2}} b}{3}+2 a e \sqrt {e x +d}-2 d b \sqrt {e x +d}}{b^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} e^{2}+a b d e -\frac {1}{2} b^{2} d^{2}\right ) \sqrt {e x +d}}{\left (e x +d \right ) b +a e -b d}+\frac {5 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{2 \sqrt {b \left (a e -b d \right )}}}{b^{3}}\right )\)	\(152\)
risch	\(-\frac {2 e \left (-b e x +6 a e -7 b d \right ) \sqrt {e x +d}}{3 b^{3}}-\frac {e^{3} \sqrt {e x +d}\, a^{2}}{b^{3} \left (b e x +a e \right )}+\frac {2 e^{2} \sqrt {e x +d}\, a d}{b^{2} \left (b e x +a e \right )}-\frac {e \sqrt {e x +d}\, d^{2}}{b \left (b e x +a e \right )}+\frac {5 e^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2}}{b^{3} \sqrt {b \left (a e -b d \right )}}-\frac {10 e^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a d}{b^{2} \sqrt {b \left (a e -b d \right )}}+\frac {5 e \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d^{2}}{b \sqrt {b \left (a e -b d \right )}}\)	\(242\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

2*e*(-1/b^3*(-1/3*(e*x+d)^(3/2)*b+2*a*e*(e*x+d)^(1/2)-2*d*b*(e*x+d)^(1/2))+1/b^3*((-1/2*a^2*e^2+a*b*d*e-1/2*b^
2*d^2)*(e*x+d)^(1/2)/((e*x+d)*b+a*e-b*d)+5/2*(a^2*e^2-2*a*b*d*e+b^2*d^2)/(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)^(5/2)/(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 = 2.94, size = 329, normalized size = 2.99 \begin {gather*} \left [\frac {15 \, {\left ({\left (a b x + a^{2}\right )} e^{2} - {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {\frac {b d - a e}{b}} \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 (3 \, b^{2} d^{2} - {\left (2 \, b^{2} x^{2} - 10 \, a b x - 15 \, a^{2}\right )} e^{2} - 2 \, {\left (7 \, b^{2} d x + 10 \, a b d\right )} e\right )} \sqrt {x e + d}}{6 \, {\left (b^{4} x + a b^{3}\right )}}, \frac {15 \, {\left ({\left (a b x + a^{2}\right )} e^{2} - {\left (b^{2} d x + a b d\right )} e\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 ) - {\left (3 \, b^{2} d^{2} - {\left (2 \, b^{2} x^{2} - 10 \, a b x - 15 \, a^{2}\right )} e^{2} - 2 \, {\left (7 \, b^{2} d x + 10 \, a b d\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (b^{4} x + a b^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

[1/6*(15*((a*b*x + a^2)*e^2 - (b^2*d*x + a*b*d)*e)*sqrt((b*d - a*e)/b)*log((2*b*d + 2*sqrt(x*e + d)*b*sqrt((b*
d - a*e)/b) + (b*x - a)*e)/(b*x + a)) - 2*(3*b^2*d^2 - (2*b^2*x^2 - 10*a*b*x - 15*a^2)*e^2 - 2*(7*b^2*d*x + 10
*a*b*d)*e)*sqrt(x*e + d))/(b^4*x + a*b^3), 1/3*(15*((a*b*x + a^2)*e^2 - (b^2*d*x + a*b*d)*e)*sqrt(-(b*d - a*e)
/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (3*b^2*d^2 - (2*b^2*x^2 - 10*a*b*x - 15*a^2)*e
^2 - 2*(7*b^2*d*x + 10*a*b*d)*e)*sqrt(x*e + d))/(b^4*x + a*b^3)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1312 vs. \(2 (97) = 194\).
time = 124.36, size = 1312, normalized size = 11.93 \begin {gather*} - \frac {2 a^{3} e^{4} \sqrt {d + e x}}{2 a^{2} b^{3} e^{2} - 2 a b^{4} d e + 2 a b^{4} e^{2} x - 2 b^{5} d e x} + \frac {a^{3} e^{4} \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^{3}} - \frac {a^{3} e^{4} \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^{3}} + \frac {6 a^{2} d 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 {3 a^{2} d 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 {3 a^{2} d 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 {6 a^{2} e^{3} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e}{b} - d}} \right )}}{b^{4} \sqrt {\frac {a e}{b} - d}} - \frac {6 a d^{2} 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 {3 a d^{2} 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 )}}{2 b} - \frac {3 a d^{2} 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 )}}{2 b} - \frac {12 a d 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 {4 a e^{2} \sqrt {d + e x}}{b^{3}} - \frac {d^{3} 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^{3} 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^{3} 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 {6 d^{2} 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 {4 d e \sqrt {d + e x}}{b^{2}} + \frac {2 e \left (d + e x\right )^{\frac {3}{2}}}{3 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

-2*a**3*e**4*sqrt(d + e*x)/(2*a**2*b**3*e**2 - 2*a*b**4*d*e + 2*a*b**4*e**2*x - 2*b**5*d*e*x) + a**3*e**4*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**3) - a**3*e**4*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**3) + 6*a**2*d*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) - 3*a**2*d*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) + 3*a**2*
d*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) + 6*a**2*e**3*atan(sqrt(d + e*x)/sqrt
(a*e/b - d))/(b**4*sqrt(a*e/b - d)) - 6*a*d**2*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) + 3*a*d**2*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))/(2*b) - 3*a*d**2
*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))/(2*b) - 12*a*d*e**2*atan(sqrt(d + e*x)/sqrt(a*e
/b - d))/(b**3*sqrt(a*e/b - d)) - 4*a*e**2*sqrt(d + e*x)/b**3 - d**3*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**3*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 + 2*d**3*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) + 6*d**2*e*atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b*
*2*sqrt(a*e/b - d)) + 4*d*e*sqrt(d + e*x)/b**2 + 2*e*(d + e*x)**(3/2)/(3*b**2)

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Giac [A]
time = 0.76, size = 191, normalized size = 1.74 \begin {gather*} \frac {5 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\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^{3}} - \frac {\sqrt {x e + d} b^{2} d^{2} e - 2 \, \sqrt {x e + d} a b d e^{2} + \sqrt {x e + d} a^{2} e^{3}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{3}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{4} e + 6 \, \sqrt {x e + d} b^{4} d e - 6 \, \sqrt {x e + d} a b^{3} e^{2}\right )}}{3 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

5*(b^2*d^2*e - 2*a*b*d*e^2 + a^2*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^3)
- (sqrt(x*e + d)*b^2*d^2*e - 2*sqrt(x*e + d)*a*b*d*e^2 + sqrt(x*e + d)*a^2*e^3)/(((x*e + d)*b - b*d + a*e)*b^3
) + 2/3*((x*e + d)^(3/2)*b^4*e + 6*sqrt(x*e + d)*b^4*d*e - 6*sqrt(x*e + d)*a*b^3*e^2)/b^6

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Mupad [B]
time = 0.59, size = 161, normalized size = 1.46 \begin {gather*} \frac {2\,e\,{\left (d+e\,x\right )}^{3/2}}{3\,b^2}-\frac {\sqrt {d+e\,x}\,\left (a^2\,e^3-2\,a\,b\,d\,e^2+b^2\,d^2\,e\right )}{b^4\,\left (d+e\,x\right )-b^4\,d+a\,b^3\,e}+\frac {5\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^3-2\,a\,b\,d\,e^2+b^2\,d^2\,e}\right )\,{\left (a\,e-b\,d\right )}^{3/2}}{b^{7/2}}+\frac {2\,e\,\left (2\,b^2\,d-2\,a\,b\,e\right )\,\sqrt {d+e\,x}}{b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

(2*e*(d + e*x)^(3/2))/(3*b^2) - ((d + e*x)^(1/2)*(a^2*e^3 + b^2*d^2*e - 2*a*b*d*e^2))/(b^4*(d + e*x) - b^4*d +
a*b^3*e) + (5*e*atan((b^(1/2)*e*(a*e - b*d)^(3/2)*(d + e*x)^(1/2))/(a^2*e^3 + b^2*d^2*e - 2*a*b*d*e^2))*(a*e
- b*d)^(3/2))/b^(7/2) + (2*e*(2*b^2*d - 2*a*b*e)*(d + e*x)^(1/2))/b^4

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