3.4.71 \(\int \frac {(d+e x)^{5/2}}{(b x+c x^2)^2} \, dx\) [371]

Optimal. Leaf size=159 \[ \frac {e (2 c d-b e) \sqrt {d+e x}}{b^2 c}-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {d^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{3/2} (4 c d+b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{3/2}} \]

[Out]

-(e*x+d)^(3/2)*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)+d^(3/2)*(-5*b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3-
(-b*e+c*d)^(3/2)*(b*e+4*c*d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/c^(3/2)+e*(-b*e+2*c*d)*(e*x+d
)^(1/2)/b^2/c

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Rubi [A]
time = 0.20, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {752, 838, 840, 1180, 214} \begin {gather*} -\frac {(c d-b e)^{3/2} (b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{3/2}}+\frac {d^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac {e \sqrt {d+e x} (2 c d-b e)}{b^2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(2*c*d - b*e)*Sqrt[d + e*x])/(b^2*c) - ((d + e*x)^(3/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2)) + (d^(
3/2)*(4*c*d - 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^(3/2)*(4*c*d + b*e)*ArcTanh[(Sqrt[c]*S
qrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(3/2))

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]

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 838

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*
((d + e*x)^m/(c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/
(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} d (4 c d-5 b e)-\frac {1}{2} e (2 c d-b e) x\right )}{b x+c x^2} \, dx}{b^2}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{b^2 c}-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} c d^2 (4 c d-5 b e)+\frac {1}{2} e \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{b^2 c}-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} c d^2 e (4 c d-5 b e)-\frac {1}{2} d e \left (2 c^2 d^2-2 b c d e-b^2 e^2\right )+\frac {1}{2} e \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^2 c}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{b^2 c}-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\left (c d^2 (4 c d-5 b e)\right ) \text {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b^3}+\frac {\left ((c d-b e)^2 (4 c d+b e)\right ) \text {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b^3 c}\\ &=\frac {e (2 c d-b e) \sqrt {d+e x}}{b^2 c}-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {d^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{3/2} (4 c d+b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.54, size = 143, normalized size = 0.90 \begin {gather*} \frac {-\frac {b \sqrt {d+e x} \left (2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)\right )}{c x (b+c x)}+\frac {(-c d+b e)^{3/2} (4 c d+b e) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{3/2}}+d^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b*Sqrt[d + e*x]*(2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x)))/(c*x*(b + c*x))) + ((-(c*d) + b*e)^(3/2)*(4
*c*d + b*e)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(3/2) + d^(3/2)*(4*c*d - 5*b*e)*ArcTanh[Sqrt
[d + e*x]/Sqrt[d]])/b^3

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Maple [A]
time = 0.50, size = 154, normalized size = 0.97

method	result	size
derivativedivides	\(2 e^{3} \left (-\frac {d^{2} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e -4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\left (b e -c d \right )^{2} \left (-\frac {b e \sqrt {e x +d}}{2 c \left (c \left (e x +d \right )+b e -c d \right )}+\frac {\left (b e +4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 c \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}\right )\)	\(154\)
default	\(2 e^{3} \left (-\frac {d^{2} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e -4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\left (b e -c d \right )^{2} \left (-\frac {b e \sqrt {e x +d}}{2 c \left (c \left (e x +d \right )+b e -c d \right )}+\frac {\left (b e +4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 c \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}\right )\)	\(154\)
risch	\(-\frac {d^{2} \sqrt {e x +d}}{b^{2} x}-\frac {e^{3} \sqrt {e x +d}}{c \left (c e x +b e \right )}+\frac {2 e^{2} \sqrt {e x +d}\, d}{b \left (c e x +b e \right )}-\frac {e c \sqrt {e x +d}\, d^{2}}{b^{2} \left (c e x +b e \right )}+\frac {e^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{c \sqrt {\left (b e -c d \right ) c}}+\frac {2 e^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d}{b \sqrt {\left (b e -c d \right ) c}}-\frac {7 e c \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{2}}{b^{2} \sqrt {\left (b e -c d \right ) c}}+\frac {4 c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{3}}{b^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {5 e \,d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {4 d^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{3}}\)	\(313\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^3*(-d^2/b^3/e^3*(1/2*b*(e*x+d)^(1/2)/x+1/2*(5*b*e-4*c*d)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))+(b*e-c*d)
^2/b^3/e^3*(-1/2*b*e/c*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d)+1/2*(b*e+4*c*d)/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d
)^(1/2)/((b*e-c*d)*c)^(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)/(c*x^2+b*x)^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(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 2.00, size = 1042, normalized size = 6.55 \begin {gather*} \left [-\frac {{\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - {\left (b^{2} c x^{2} + b^{3} x\right )} e^{2} - 3 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {2 \, c d + 2 \, \sqrt {x e + d} c \sqrt {\frac {c d - b e}{c}} + {\left (c x - b\right )} e}{c x + b}\right ) + {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - 5 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (2 \, b c^{2} d^{2} x - 2 \, b^{2} c d x e + b^{2} c d^{2} + b^{3} x e^{2}\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c^{2} x^{2} + b^{4} c x\right )}}, -\frac {2 \, {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - {\left (b^{2} c x^{2} + b^{3} x\right )} e^{2} - 3 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {x e + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - 5 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (2 \, b c^{2} d^{2} x - 2 \, b^{2} c d x e + b^{2} c d^{2} + b^{3} x e^{2}\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c^{2} x^{2} + b^{4} c x\right )}}, -\frac {2 \, {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - 5 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - {\left (b^{2} c x^{2} + b^{3} x\right )} e^{2} - 3 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {2 \, c d + 2 \, \sqrt {x e + d} c \sqrt {\frac {c d - b e}{c}} + {\left (c x - b\right )} e}{c x + b}\right ) + 2 \, {\left (2 \, b c^{2} d^{2} x - 2 \, b^{2} c d x e + b^{2} c d^{2} + b^{3} x e^{2}\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c^{2} x^{2} + b^{4} c x\right )}}, -\frac {{\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - {\left (b^{2} c x^{2} + b^{3} x\right )} e^{2} - 3 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {x e + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + {\left (4 \, c^{3} d^{2} x^{2} + 4 \, b c^{2} d^{2} x - 5 \, {\left (b c^{2} d x^{2} + b^{2} c d x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (2 \, b c^{2} d^{2} x - 2 \, b^{2} c d x e + b^{2} c d^{2} + b^{3} x e^{2}\right )} \sqrt {x e + d}}{b^{3} c^{2} x^{2} + b^{4} c x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*((4*c^3*d^2*x^2 + 4*b*c^2*d^2*x - (b^2*c*x^2 + b^3*x)*e^2 - 3*(b*c^2*d*x^2 + b^2*c*d*x)*e)*sqrt((c*d - b
*e)/c)*log((2*c*d + 2*sqrt(x*e + d)*c*sqrt((c*d - b*e)/c) + (c*x - b)*e)/(c*x + b)) + (4*c^3*d^2*x^2 + 4*b*c^2
*d^2*x - 5*(b*c^2*d*x^2 + b^2*c*d*x)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(2*b*c^2*d^2*
x - 2*b^2*c*d*x*e + b^2*c*d^2 + b^3*x*e^2)*sqrt(x*e + d))/(b^3*c^2*x^2 + b^4*c*x), -1/2*(2*(4*c^3*d^2*x^2 + 4*
b*c^2*d^2*x - (b^2*c*x^2 + b^3*x)*e^2 - 3*(b*c^2*d*x^2 + b^2*c*d*x)*e)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(x*e +
d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + (4*c^3*d^2*x^2 + 4*b*c^2*d^2*x - 5*(b*c^2*d*x^2 + b^2*c*d*x)*e)*sqrt
(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(2*b*c^2*d^2*x - 2*b^2*c*d*x*e + b^2*c*d^2 + b^3*x*e^2)*s
qrt(x*e + d))/(b^3*c^2*x^2 + b^4*c*x), -1/2*(2*(4*c^3*d^2*x^2 + 4*b*c^2*d^2*x - 5*(b*c^2*d*x^2 + b^2*c*d*x)*e)
*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (4*c^3*d^2*x^2 + 4*b*c^2*d^2*x - (b^2*c*x^2 + b^3*x)*e^2 - 3*(b*c
^2*d*x^2 + b^2*c*d*x)*e)*sqrt((c*d - b*e)/c)*log((2*c*d + 2*sqrt(x*e + d)*c*sqrt((c*d - b*e)/c) + (c*x - b)*e)
/(c*x + b)) + 2*(2*b*c^2*d^2*x - 2*b^2*c*d*x*e + b^2*c*d^2 + b^3*x*e^2)*sqrt(x*e + d))/(b^3*c^2*x^2 + b^4*c*x)
, -((4*c^3*d^2*x^2 + 4*b*c^2*d^2*x - (b^2*c*x^2 + b^3*x)*e^2 - 3*(b*c^2*d*x^2 + b^2*c*d*x)*e)*sqrt(-(c*d - b*e
)/c)*arctan(-sqrt(x*e + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + (4*c^3*d^2*x^2 + 4*b*c^2*d^2*x - 5*(b*c^2*d*x
^2 + b^2*c*d*x)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (2*b*c^2*d^2*x - 2*b^2*c*d*x*e + b^2*c*d^2 + b^
3*x*e^2)*sqrt(x*e + d))/(b^3*c^2*x^2 + b^4*c*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1411 vs. \(2 (143) = 286\).
time = 132.30, size = 1411, normalized size = 8.87 \begin {gather*} - \frac {2 b e^{4} \sqrt {d + e x}}{2 b^{2} c e^{2} - 2 b c^{2} d e + 2 b c^{2} e^{2} x - 2 c^{3} d e x} + \frac {b e^{4} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 c} - \frac {b e^{4} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 c} + \frac {2 c^{2} d^{3} e \sqrt {d + e x}}{2 b^{4} e^{2} - 2 b^{3} c d e + 2 b^{3} c e^{2} x - 2 b^{2} c^{2} d e x} - \frac {6 c d^{2} e^{2} \sqrt {d + e x}}{2 b^{3} e^{2} - 2 b^{2} c d e + 2 b^{2} c e^{2} x - 2 b c^{2} d e x} - \frac {3 d e^{3} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {3 d e^{3} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {6 d e^{3} \sqrt {d + e x}}{2 b^{2} e^{2} - 2 b c d e + 2 b c e^{2} x - 2 c^{2} d e x} + \frac {2 e^{3} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e}{c} - d}} \right )}}{c^{2} \sqrt {\frac {b e}{c} - d}} + \frac {3 c d^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {3 c d^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {c^{2} d^{3} e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} + \frac {c^{2} d^{3} e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} - \frac {d^{3} e \sqrt {\frac {1}{d^{3}}} \log {\left (- d^{2} \sqrt {\frac {1}{d^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} + \frac {d^{3} e \sqrt {\frac {1}{d^{3}}} \log {\left (d^{2} \sqrt {\frac {1}{d^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} - \frac {6 d^{2} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e}{c} - d}} \right )}}{b^{2} \sqrt {\frac {b e}{c} - d}} + \frac {6 d^{2} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b^{2} \sqrt {- d}} - \frac {d^{2} \sqrt {d + e x}}{b^{2} x} + \frac {4 c d^{3} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e}{c} - d}} \right )}}{b^{3} \sqrt {\frac {b e}{c} - d}} - \frac {4 c d^{3} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b^{3} \sqrt {- d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b*e**4*sqrt(d + e*x)/(2*b**2*c*e**2 - 2*b*c**2*d*e + 2*b*c**2*e**2*x - 2*c**3*d*e*x) + b*e**4*sqrt(-1/(c*(b
*e - c*d)**3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) - c**2*d**2*
sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*c) - b*e**4*sqrt(-1/(c*(b*e - c*d)**3))*log(b**2*e**2*sqrt(-1/
(c*(b*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d +
e*x))/(2*c) + 2*c**2*d**3*e*sqrt(d + e*x)/(2*b**4*e**2 - 2*b**3*c*d*e + 2*b**3*c*e**2*x - 2*b**2*c**2*d*e*x)
- 6*c*d**2*e**2*sqrt(d + e*x)/(2*b**3*e**2 - 2*b**2*c*d*e + 2*b**2*c*e**2*x - 2*b*c**2*d*e*x) - 3*d*e**3*sqrt(
-1/(c*(b*e - c*d)**3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) - c*
*2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/2 + 3*d*e**3*sqrt(-1/(c*(b*e - c*d)**3))*log(b**2*e**2*sq
rt(-1/(c*(b*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sq
rt(d + e*x))/2 + 6*d*e**3*sqrt(d + e*x)/(2*b**2*e**2 - 2*b*c*d*e + 2*b*c*e**2*x - 2*c**2*d*e*x) + 2*e**3*atan(
sqrt(d + e*x)/sqrt(b*e/c - d))/(c**2*sqrt(b*e/c - d)) + 3*c*d**2*e**2*sqrt(-1/(c*(b*e - c*d)**3))*log(-b**2*e*
*2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) - c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3))
+ sqrt(d + e*x))/(2*b) - 3*c*d**2*e**2*sqrt(-1/(c*(b*e - c*d)**3))*log(b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3))
- 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b) - c**2*
d**3*e*sqrt(-1/(c*(b*e - c*d)**3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*
d)**3)) - c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b**2) + c**2*d**3*e*sqrt(-1/(c*(b*e - c*d)
**3))*log(b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c
*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b**2) - d**3*e*sqrt(d**(-3))*log(-d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2
*b**2) + d**3*e*sqrt(d**(-3))*log(d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2*b**2) - 6*d**2*e*atan(sqrt(d + e*x)/s
qrt(b*e/c - d))/(b**2*sqrt(b*e/c - d)) + 6*d**2*e*atan(sqrt(d + e*x)/sqrt(-d))/(b**2*sqrt(-d)) - d**2*sqrt(d +
e*x)/(b**2*x) + 4*c*d**3*atan(sqrt(d + e*x)/sqrt(b*e/c - d))/(b**3*sqrt(b*e/c - d)) - 4*c*d**3*atan(sqrt(d +
e*x)/sqrt(-d))/(b**3*sqrt(-d))

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Giac [A]
time = 1.36, size = 273, normalized size = 1.72 \begin {gather*} -\frac {{\left (4 \, c d^{3} - 5 \, b d^{2} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} + \frac {{\left (4 \, c^{3} d^{3} - 7 \, b c^{2} d^{2} e + 2 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} e - 2 \, \sqrt {x e + d} c^{2} d^{3} e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} b c d e^{2} + 3 \, \sqrt {x e + d} b c d^{2} e^{2} + {\left (x e + d\right )}^{\frac {3}{2}} b^{2} e^{3} - \sqrt {x e + d} b^{2} d e^{3}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )} b^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*c*d^3 - 5*b*d^2*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) + (4*c^3*d^3 - 7*b*c^2*d^2*e + 2*b^2*c*d*
e^2 + b^3*e^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3*c) - (2*(x*e + d)^(3/2)*
c^2*d^2*e - 2*sqrt(x*e + d)*c^2*d^3*e - 2*(x*e + d)^(3/2)*b*c*d*e^2 + 3*sqrt(x*e + d)*b*c*d^2*e^2 + (x*e + d)^
(3/2)*b^2*e^3 - sqrt(x*e + d)*b^2*d*e^3)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^
2*c)

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Mupad [B]
time = 0.50, size = 1127, normalized size = 7.09 \begin {gather*} \frac {\frac {\sqrt {d+e\,x}\,\left (b^2\,d\,e^3-3\,b\,c\,d^2\,e^2+2\,c^2\,d^3\,e\right )}{b^2\,c}-\frac {e\,{\left (d+e\,x\right )}^{3/2}\,\left (b^2\,e^2-2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^2\,c}}{\left (b\,e-2\,c\,d\right )\,\left (d+e\,x\right )+c\,{\left (d+e\,x\right )}^2+c\,d^2-b\,d\,e}-\frac {\mathrm {atanh}\left (\frac {10\,e^9\,\sqrt {d^3}\,\sqrt {d+e\,x}}{10\,d^2\,e^9+\frac {32\,c\,d^3\,e^8}{b}-\frac {132\,c^2\,d^4\,e^7}{b^2}+\frac {130\,c^3\,d^5\,e^6}{b^3}-\frac {40\,c^4\,d^6\,e^5}{b^4}}+\frac {32\,d\,e^8\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,d^3\,e^8+\frac {10\,b\,d^2\,e^9}{c}-\frac {132\,c\,d^4\,e^7}{b}+\frac {130\,c^2\,d^5\,e^6}{b^2}-\frac {40\,c^3\,d^6\,e^5}{b^3}}-\frac {132\,c\,d^2\,e^7\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,b\,d^3\,e^8-132\,c\,d^4\,e^7+\frac {130\,c^2\,d^5\,e^6}{b}+\frac {10\,b^2\,d^2\,e^9}{c}-\frac {40\,c^3\,d^6\,e^5}{b^2}}+\frac {130\,c^2\,d^3\,e^6\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,b^2\,d^3\,e^8+130\,c^2\,d^5\,e^6-\frac {40\,c^3\,d^6\,e^5}{b}+\frac {10\,b^3\,d^2\,e^9}{c}-132\,b\,c\,d^4\,e^7}-\frac {40\,c^3\,d^4\,e^5\,\sqrt {d^3}\,\sqrt {d+e\,x}}{32\,b^3\,d^3\,e^8-40\,c^3\,d^6\,e^5+130\,b\,c^2\,d^5\,e^6-132\,b^2\,c\,d^4\,e^7+\frac {10\,b^4\,d^2\,e^9}{c}}\right )\,\left (5\,b\,e-4\,c\,d\right )\,\sqrt {d^3}}{b^3}-\frac {\mathrm {atanh}\left (\frac {30\,d^3\,e^6\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{14\,b^3\,d^2\,e^9+110\,c^3\,d^5\,e^6-82\,b\,c^2\,d^4\,e^7-4\,b^2\,c\,d^3\,e^8+\frac {2\,b^4\,d\,e^{10}}{c}-\frac {40\,c^4\,d^6\,e^5}{b}}-\frac {2\,d\,e^8\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{4\,c^3\,d^3\,e^8-14\,b\,c^2\,d^2\,e^9+\frac {82\,c^4\,d^4\,e^7}{b}-\frac {110\,c^5\,d^5\,e^6}{b^2}+\frac {40\,c^6\,d^6\,e^5}{b^3}-2\,b^2\,c\,d\,e^{10}}+\frac {18\,d^2\,e^7\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{2\,b^3\,d\,e^{10}-82\,c^3\,d^4\,e^7-4\,b\,c^2\,d^3\,e^8+14\,b^2\,c\,d^2\,e^9+\frac {110\,c^4\,d^5\,e^6}{b}-\frac {40\,c^5\,d^6\,e^5}{b^2}}+\frac {40\,d^4\,e^5\,\sqrt {d+e\,x}\,\sqrt {-b^3\,c^3\,e^3+3\,b^2\,c^4\,d\,e^2-3\,b\,c^5\,d^2\,e+c^6\,d^3}}{4\,b^3\,d^3\,e^8+40\,c^3\,d^6\,e^5-110\,b\,c^2\,d^5\,e^6+82\,b^2\,c\,d^4\,e^7-\frac {2\,b^5\,d\,e^{10}}{c^2}-\frac {14\,b^4\,d^2\,e^9}{c}}\right )\,\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (b\,e+4\,c\,d\right )}{b^3\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d + e*x)^(1/2)*(b^2*d*e^3 + 2*c^2*d^3*e - 3*b*c*d^2*e^2))/(b^2*c) - (e*(d + e*x)^(3/2)*(b^2*e^2 + 2*c^2*d^2
- 2*b*c*d*e))/(b^2*c))/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + c*d^2 - b*d*e) - (atanh((10*e^9*(d^3)^(1/2)
*(d + e*x)^(1/2))/(10*d^2*e^9 + (32*c*d^3*e^8)/b - (132*c^2*d^4*e^7)/b^2 + (130*c^3*d^5*e^6)/b^3 - (40*c^4*d^6
*e^5)/b^4) + (32*d*e^8*(d^3)^(1/2)*(d + e*x)^(1/2))/(32*d^3*e^8 + (10*b*d^2*e^9)/c - (132*c*d^4*e^7)/b + (130*
c^2*d^5*e^6)/b^2 - (40*c^3*d^6*e^5)/b^3) - (132*c*d^2*e^7*(d^3)^(1/2)*(d + e*x)^(1/2))/(32*b*d^3*e^8 - 132*c*d
^4*e^7 + (130*c^2*d^5*e^6)/b + (10*b^2*d^2*e^9)/c - (40*c^3*d^6*e^5)/b^2) + (130*c^2*d^3*e^6*(d^3)^(1/2)*(d +
e*x)^(1/2))/(32*b^2*d^3*e^8 + 130*c^2*d^5*e^6 - (40*c^3*d^6*e^5)/b + (10*b^3*d^2*e^9)/c - 132*b*c*d^4*e^7) - (
40*c^3*d^4*e^5*(d^3)^(1/2)*(d + e*x)^(1/2))/(32*b^3*d^3*e^8 - 40*c^3*d^6*e^5 + 130*b*c^2*d^5*e^6 - 132*b^2*c*d
^4*e^7 + (10*b^4*d^2*e^9)/c))*(5*b*e - 4*c*d)*(d^3)^(1/2))/b^3 - (atanh((30*d^3*e^6*(d + e*x)^(1/2)*(c^6*d^3 -
b^3*c^3*e^3 + 3*b^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(14*b^3*d^2*e^9 + 110*c^3*d^5*e^6 - 82*b*c^2*d^4*e^7 -
4*b^2*c*d^3*e^8 + (2*b^4*d*e^10)/c - (40*c^4*d^6*e^5)/b) - (2*d*e^8*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3
*b^2*c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(4*c^3*d^3*e^8 - 14*b*c^2*d^2*e^9 + (82*c^4*d^4*e^7)/b - (110*c^5*d^5*e
^6)/b^2 + (40*c^6*d^6*e^5)/b^3 - 2*b^2*c*d*e^10) + (18*d^2*e^7*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*
c^4*d*e^2 - 3*b*c^5*d^2*e)^(1/2))/(2*b^3*d*e^10 - 82*c^3*d^4*e^7 - 4*b*c^2*d^3*e^8 + 14*b^2*c*d^2*e^9 + (110*c
^4*d^5*e^6)/b - (40*c^5*d^6*e^5)/b^2) + (40*d^4*e^5*(d + e*x)^(1/2)*(c^6*d^3 - b^3*c^3*e^3 + 3*b^2*c^4*d*e^2 -
3*b*c^5*d^2*e)^(1/2))/(4*b^3*d^3*e^8 + 40*c^3*d^6*e^5 - 110*b*c^2*d^5*e^6 + 82*b^2*c*d^4*e^7 - (2*b^5*d*e^10)
/c^2 - (14*b^4*d^2*e^9)/c))*(-c^3*(b*e - c*d)^3)^(1/2)*(b*e + 4*c*d))/(b^3*c^3)

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