3.4.80 \(\int \frac {(d+e x)^{5/2}}{(b x+c x^2)^3} \, dx\) [380]

Optimal. Leaf size=232 \[ -\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}-\frac {3 \sqrt {d} \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 \sqrt {c d-b e} \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c}} \]

[Out]

-1/2*(e*x+d)^(3/2)*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)^2-3/4*(5*b^2*e^2-20*b*c*d*e+16*c^2*d^2)*arctanh((e*x+d
)^(1/2)/d^(1/2))*d^(1/2)/b^5+3/4*(b^2*e^2-12*b*c*d*e+16*c^2*d^2)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2
))*(-b*e+c*d)^(1/2)/b^5/c^(1/2)+3/4*(b*d*(-3*b*e+4*c*d)+(b^2*e^2-8*b*c*d*e+8*c^2*d^2)*x)*(e*x+d)^(1/2)/b^4/(c*
x^2+b*x)

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Rubi [A]
time = 0.21, antiderivative size = 232, 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, 834, 840, 1180, 214} \begin {gather*} -\frac {(d+e x)^{3/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {3 \sqrt {d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 \sqrt {c d-b e} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c}}+\frac {3 \sqrt {d+e x} \left (x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+b d (4 c d-3 b e)\right )}{4 b^4 \left (b x+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(b*x + c*x^2)^3,x]

[Out]

-1/2*((d + e*x)^(3/2)*(b*d + (2*c*d - b*e)*x))/(b^2*(b*x + c*x^2)^2) + (3*Sqrt[d + e*x]*(b*d*(4*c*d - 3*b*e) +
(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*x))/(4*b^4*(b*x + c*x^2)) - (3*Sqrt[d]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2
)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (3*Sqrt[c*d - b*e]*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*ArcTanh[(Sq
rt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*Sqrt[c])

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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], 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] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

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 )^3} \, dx &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {3}{2} d (4 c d-3 b e)+\frac {3}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )+\frac {3}{4} e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4}\\ &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} d e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )+\frac {3}{4} d e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )+\frac {3}{4} e \left (8 c^2 d^2-8 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^4}\\ &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}-\frac {\left (3 (c d-b e) \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )\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 )}{4 b^5}+\frac {\left (3 c d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )\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 )}{4 b^5}\\ &=-\frac {(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {3 \sqrt {d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}-\frac {3 \sqrt {d} \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {3 \sqrt {c d-b e} \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c}}\\ \end {align*}

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Mathematica [A]
time = 0.90, size = 222, normalized size = 0.96 \begin {gather*} \frac {\frac {b \sqrt {d+e x} \left (24 c^3 d^2 x^3+12 b c^2 d x^2 (3 d-2 e x)+b^2 c x \left (8 d^2-37 d e x+3 e^2 x^2\right )+b^3 \left (-2 d^2-9 d e x+5 e^2 x^2\right )\right )}{x^2 (b+c x)^2}+\frac {3 \sqrt {-c d+b e} \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{\sqrt {c}}-3 \sqrt {d} \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(b*x + c*x^2)^3,x]

[Out]

((b*Sqrt[d + e*x]*(24*c^3*d^2*x^3 + 12*b*c^2*d*x^2*(3*d - 2*e*x) + b^2*c*x*(8*d^2 - 37*d*e*x + 3*e^2*x^2) + b^
3*(-2*d^2 - 9*d*e*x + 5*e^2*x^2)))/(x^2*(b + c*x)^2) + (3*Sqrt[-(c*d) + b*e]*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^
2)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/Sqrt[c] - 3*Sqrt[d]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^
2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5)

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Maple [A]
time = 0.52, size = 267, normalized size = 1.15

method	result	size
derivativedivides	\(2 e^{5} \left (\frac {\left (b e -c d \right ) \left (\frac {\left (\frac {3}{8} b^{2} e^{2} c -\frac {3}{2} d b e \,c^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (5 b^{2} e^{2}-17 b c d e +12 d^{2} c^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (b^{2} e^{2}-12 b c d e +16 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}-\frac {d \left (\frac {\left (\frac {9}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {7}{8} b^{2} d \,e^{2}+\frac {3}{2} b c \,d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (5 b^{2} e^{2}-20 b c d e +16 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}\right )\)	\(267\)
default	\(2 e^{5} \left (\frac {\left (b e -c d \right ) \left (\frac {\left (\frac {3}{8} b^{2} e^{2} c -\frac {3}{2} d b e \,c^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (5 b^{2} e^{2}-17 b c d e +12 d^{2} c^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (b^{2} e^{2}-12 b c d e +16 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}-\frac {d \left (\frac {\left (\frac {9}{8} b^{2} e^{2}-\frac {3}{2} b c d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {7}{8} b^{2} d \,e^{2}+\frac {3}{2} b c \,d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (5 b^{2} e^{2}-20 b c d e +16 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{5}}\right )\)	\(267\)
risch	\(-\frac {d \sqrt {e x +d}\, \left (9 b e x -12 c d x +2 b d \right )}{4 b^{4} x^{2}}+\frac {3 e^{3} \left (e x +d \right )^{\frac {3}{2}} c}{4 b^{2} \left (c e x +b e \right )^{2}}-\frac {15 e^{2} \left (e x +d \right )^{\frac {3}{2}} c^{2} d}{4 b^{3} \left (c e x +b e \right )^{2}}+\frac {3 e \left (e x +d \right )^{\frac {3}{2}} c^{3} d^{2}}{b^{4} \left (c e x +b e \right )^{2}}+\frac {5 e^{4} \sqrt {e x +d}}{4 b \left (c e x +b e \right )^{2}}-\frac {11 e^{3} \sqrt {e x +d}\, c d}{2 b^{2} \left (c e x +b e \right )^{2}}+\frac {29 e^{2} \sqrt {e x +d}\, c^{2} d^{2}}{4 b^{3} \left (c e x +b e \right )^{2}}-\frac {3 e \sqrt {e x +d}\, c^{3} d^{3}}{b^{4} \left (c e x +b e \right )^{2}}+\frac {3 e^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 b^{2} \sqrt {\left (b e -c d \right ) c}}-\frac {39 e^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d c}{4 b^{3} \sqrt {\left (b e -c d \right ) c}}+\frac {21 e \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) c^{2} d^{2}}{b^{4} \sqrt {\left (b e -c d \right ) c}}-\frac {12 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) c^{3} d^{3}}{b^{5} \sqrt {\left (b e -c d \right ) c}}-\frac {15 e^{2} \sqrt {d}\, \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3}}+\frac {15 e \,d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{4}}-\frac {12 d^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c^{2}}{b^{5}}\)	\(474\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*((b*e-c*d)/b^5/e^5*(((3/8*b^2*e^2*c-3/2*d*b*e*c^2)*(e*x+d)^(3/2)+1/8*b*e*(5*b^2*e^2-17*b*c*d*e+12*c^2*d^
2)*(e*x+d)^(1/2))/(c*(e*x+d)+b*e-c*d)^2+3/8*(b^2*e^2-12*b*c*d*e+16*c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+
d)^(1/2)/((b*e-c*d)*c)^(1/2)))-d/b^5/e^5*(((9/8*b^2*e^2-3/2*b*c*d*e)*(e*x+d)^(3/2)+(-7/8*b^2*d*e^2+3/2*b*c*d^2
*e)*(e*x+d)^(1/2))/e^2/x^2+3/8*(5*b^2*e^2-20*b*c*d*e+16*c^2*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/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)/(c*x^2+b*x)^3,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 [B] Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (212) = 424\).
time = 3.02, size = 1756, normalized size = 7.57 \begin {gather*} \left [\frac {3 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 12 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\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 ) + 3 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 5 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 20 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (24 \, b c^{3} d^{2} x^{3} + 36 \, b^{2} c^{2} d^{2} x^{2} + 8 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2} + {\left (3 \, b^{3} c x^{3} + 5 \, b^{4} x^{2}\right )} e^{2} - {\left (24 \, b^{2} c^{2} d x^{3} + 37 \, b^{3} c d x^{2} + 9 \, b^{4} d x\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}}, \frac {6 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 12 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\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 ) + 3 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 5 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 20 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (24 \, b c^{3} d^{2} x^{3} + 36 \, b^{2} c^{2} d^{2} x^{2} + 8 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2} + {\left (3 \, b^{3} c x^{3} + 5 \, b^{4} x^{2}\right )} e^{2} - {\left (24 \, b^{2} c^{2} d x^{3} + 37 \, b^{3} c d x^{2} + 9 \, b^{4} d x\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}}, \frac {6 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 5 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 20 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + 3 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 12 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\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 (24 \, b c^{3} d^{2} x^{3} + 36 \, b^{2} c^{2} d^{2} x^{2} + 8 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2} + {\left (3 \, b^{3} c x^{3} + 5 \, b^{4} x^{2}\right )} e^{2} - {\left (24 \, b^{2} c^{2} d x^{3} + 37 \, b^{3} c d x^{2} + 9 \, b^{4} d x\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}}, \frac {3 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 12 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\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 ) + 3 \, {\left (16 \, c^{4} d^{2} x^{4} + 32 \, b c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 5 \, {\left (b^{2} c^{2} x^{4} + 2 \, b^{3} c x^{3} + b^{4} x^{2}\right )} e^{2} - 20 \, {\left (b c^{3} d x^{4} + 2 \, b^{2} c^{2} d x^{3} + b^{3} c d x^{2}\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (24 \, b c^{3} d^{2} x^{3} + 36 \, b^{2} c^{2} d^{2} x^{2} + 8 \, b^{3} c d^{2} x - 2 \, b^{4} d^{2} + {\left (3 \, b^{3} c x^{3} + 5 \, b^{4} x^{2}\right )} e^{2} - {\left (24 \, b^{2} c^{2} d x^{3} + 37 \, b^{3} c d x^{2} + 9 \, b^{4} d x\right )} e\right )} \sqrt {x e + d}}{4 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}}\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)^3,x, algorithm="fricas")

[Out]

[1/8*(3*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + (b^2*c^2*x^4 + 2*b^3*c*x^3 + b^4*x^2)*e^2 -
12*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*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)) + 3*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + 5*(b^2*c^
2*x^4 + 2*b^3*c*x^3 + b^4*x^2)*e^2 - 20*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e)*sqrt(d)*log((x*e - 2*
sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(24*b*c^3*d^2*x^3 + 36*b^2*c^2*d^2*x^2 + 8*b^3*c*d^2*x - 2*b^4*d^2 + (3*b^
3*c*x^3 + 5*b^4*x^2)*e^2 - (24*b^2*c^2*d*x^3 + 37*b^3*c*d*x^2 + 9*b^4*d*x)*e)*sqrt(x*e + d))/(b^5*c^2*x^4 + 2*
b^6*c*x^3 + b^7*x^2), 1/8*(6*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + (b^2*c^2*x^4 + 2*b^3*c*
x^3 + b^4*x^2)*e^2 - 12*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(x*e
+ d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + 5*(b^2
*c^2*x^4 + 2*b^3*c*x^3 + b^4*x^2)*e^2 - 20*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e)*sqrt(d)*log((x*e -
2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(24*b*c^3*d^2*x^3 + 36*b^2*c^2*d^2*x^2 + 8*b^3*c*d^2*x - 2*b^4*d^2 + (3
*b^3*c*x^3 + 5*b^4*x^2)*e^2 - (24*b^2*c^2*d*x^3 + 37*b^3*c*d*x^2 + 9*b^4*d*x)*e)*sqrt(x*e + d))/(b^5*c^2*x^4 +
2*b^6*c*x^3 + b^7*x^2), 1/8*(6*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + 5*(b^2*c^2*x^4 + 2*b
^3*c*x^3 + b^4*x^2)*e^2 - 20*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sq
rt(-d)/d) + 3*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + (b^2*c^2*x^4 + 2*b^3*c*x^3 + b^4*x^2)*
e^2 - 12*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e)*sqrt((c*d - b*e)/c)*log((2*c*d + 2*sqrt(x*e + d)*c*s
qrt((c*d - b*e)/c) + (c*x - b)*e)/(c*x + b)) + 2*(24*b*c^3*d^2*x^3 + 36*b^2*c^2*d^2*x^2 + 8*b^3*c*d^2*x - 2*b^
4*d^2 + (3*b^3*c*x^3 + 5*b^4*x^2)*e^2 - (24*b^2*c^2*d*x^3 + 37*b^3*c*d*x^2 + 9*b^4*d*x)*e)*sqrt(x*e + d))/(b^5
*c^2*x^4 + 2*b^6*c*x^3 + b^7*x^2), 1/4*(3*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2*c^2*d^2*x^2 + (b^2*c^2*x
^4 + 2*b^3*c*x^3 + b^4*x^2)*e^2 - 12*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e)*sqrt(-(c*d - b*e)/c)*arc
tan(-sqrt(x*e + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*(16*c^4*d^2*x^4 + 32*b*c^3*d^2*x^3 + 16*b^2*c^2*d^2
*x^2 + 5*(b^2*c^2*x^4 + 2*b^3*c*x^3 + b^4*x^2)*e^2 - 20*(b*c^3*d*x^4 + 2*b^2*c^2*d*x^3 + b^3*c*d*x^2)*e)*sqrt(
-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (24*b*c^3*d^2*x^3 + 36*b^2*c^2*d^2*x^2 + 8*b^3*c*d^2*x - 2*b^4*d^2 + (3
*b^3*c*x^3 + 5*b^4*x^2)*e^2 - (24*b^2*c^2*d*x^3 + 37*b^3*c*d*x^2 + 9*b^4*d*x)*e)*sqrt(x*e + d))/(b^5*c^2*x^4 +
2*b^6*c*x^3 + b^7*x^2)]

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Sympy [F(-1)] Timed out
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)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (212) = 424\).
time = 1.87, size = 448, normalized size = 1.93 \begin {gather*} -\frac {3 \, {\left (16 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 13 \, 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 )}{4 \, \sqrt {-c^{2} d + b c e} b^{5}} + \frac {3 \, {\left (16 \, c^{2} d^{3} - 20 \, b c d^{2} e + 5 \, b^{2} d e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{2} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{3} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{4} e - 24 \, \sqrt {x e + d} c^{3} d^{5} e - 24 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} d e^{2} + 108 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d^{2} e^{2} - 144 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{3} e^{2} + 60 \, \sqrt {x e + d} b c^{2} d^{4} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c e^{3} - 46 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c d e^{3} + 91 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d^{2} e^{3} - 48 \, \sqrt {x e + d} b^{2} c d^{3} e^{3} + 5 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} e^{4} - 19 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d e^{4} + 12 \, \sqrt {x e + d} b^{3} d^{2} e^{4}}{4 \, {\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 )}^{2} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-3/4*(16*c^3*d^3 - 28*b*c^2*d^2*e + 13*b^2*c*d*e^2 - b^3*e^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sq
rt(-c^2*d + b*c*e)*b^5) + 3/4*(16*c^2*d^3 - 20*b*c*d^2*e + 5*b^2*d*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sq
rt(-d)) + 1/4*(24*(x*e + d)^(7/2)*c^3*d^2*e - 72*(x*e + d)^(5/2)*c^3*d^3*e + 72*(x*e + d)^(3/2)*c^3*d^4*e - 24
*sqrt(x*e + d)*c^3*d^5*e - 24*(x*e + d)^(7/2)*b*c^2*d*e^2 + 108*(x*e + d)^(5/2)*b*c^2*d^2*e^2 - 144*(x*e + d)^
(3/2)*b*c^2*d^3*e^2 + 60*sqrt(x*e + d)*b*c^2*d^4*e^2 + 3*(x*e + d)^(7/2)*b^2*c*e^3 - 46*(x*e + d)^(5/2)*b^2*c*
d*e^3 + 91*(x*e + d)^(3/2)*b^2*c*d^2*e^3 - 48*sqrt(x*e + d)*b^2*c*d^3*e^3 + 5*(x*e + d)^(5/2)*b^3*e^4 - 19*(x*
e + d)^(3/2)*b^3*d*e^4 + 12*sqrt(x*e + d)*b^3*d^2*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b
*e - b*d*e)^2*b^4)

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Mupad [B]
time = 0.58, size = 910, normalized size = 3.92 \begin {gather*} \frac {3\,\mathrm {atanh}\left (\frac {81\,c^2\,d^2\,e^8\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{8\,\left (\frac {189\,c^3\,d^3\,e^8}{8}-\frac {351\,b\,c^2\,d^2\,e^9}{32}-\frac {27\,c^4\,d^4\,e^7}{2\,b}+\frac {27\,b^2\,c\,d\,e^{10}}{32}\right )}+\frac {27\,c^3\,d^3\,e^7\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{2\,\left (-\frac {27\,b^3\,c\,d\,e^{10}}{32}+\frac {351\,b^2\,c^2\,d^2\,e^9}{32}-\frac {189\,b\,c^3\,d^3\,e^8}{8}+\frac {27\,c^4\,d^4\,e^7}{2}\right )}+\frac {27\,c\,d\,e^9\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{32\,\left (\frac {351\,c^2\,d^2\,e^9}{32}-\frac {27\,b\,c\,d\,e^{10}}{32}-\frac {189\,c^3\,d^3\,e^8}{8\,b}+\frac {27\,c^4\,d^4\,e^7}{2\,b^2}\right )}\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (b^2\,e^2-12\,b\,c\,d\,e+16\,c^2\,d^2\right )}{4\,b^5\,c}-\frac {3\,\sqrt {d}\,\mathrm {atanh}\left (\frac {135\,c\,\sqrt {d}\,e^{10}\,\sqrt {d+e\,x}}{32\,\left (\frac {135\,c\,d\,e^{10}}{32}-\frac {675\,c^2\,d^2\,e^9}{32\,b}+\frac {243\,c^3\,d^3\,e^8}{8\,b^2}-\frac {27\,c^4\,d^4\,e^7}{2\,b^3}\right )}+\frac {675\,c^2\,d^{3/2}\,e^9\,\sqrt {d+e\,x}}{32\,\left (\frac {675\,c^2\,d^2\,e^9}{32}-\frac {135\,b\,c\,d\,e^{10}}{32}-\frac {243\,c^3\,d^3\,e^8}{8\,b}+\frac {27\,c^4\,d^4\,e^7}{2\,b^2}\right )}+\frac {243\,c^3\,d^{5/2}\,e^8\,\sqrt {d+e\,x}}{8\,\left (\frac {243\,c^3\,d^3\,e^8}{8}-\frac {675\,b\,c^2\,d^2\,e^9}{32}-\frac {27\,c^4\,d^4\,e^7}{2\,b}+\frac {135\,b^2\,c\,d\,e^{10}}{32}\right )}+\frac {27\,c^4\,d^{7/2}\,e^7\,\sqrt {d+e\,x}}{2\,\left (-\frac {135\,b^3\,c\,d\,e^{10}}{32}+\frac {675\,b^2\,c^2\,d^2\,e^9}{32}-\frac {243\,b\,c^3\,d^3\,e^8}{8}+\frac {27\,c^4\,d^4\,e^7}{2}\right )}\right )\,\left (5\,b^2\,e^2-20\,b\,c\,d\,e+16\,c^2\,d^2\right )}{4\,b^5}-\frac {\frac {{\left (d+e\,x\right )}^{3/2}\,\left (19\,b^3\,d\,e^4-91\,b^2\,c\,d^2\,e^3+144\,b\,c^2\,d^3\,e^2-72\,c^3\,d^4\,e\right )}{4\,b^4}+\frac {3\,\sqrt {d+e\,x}\,\left (-b^3\,d^2\,e^4+4\,b^2\,c\,d^3\,e^3-5\,b\,c^2\,d^4\,e^2+2\,c^3\,d^5\,e\right )}{b^4}-\frac {\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (5\,b^2\,e^3-36\,b\,c\,d\,e^2+36\,c^2\,d^2\,e\right )}{4\,b^4}-\frac {3\,c\,e\,{\left (d+e\,x\right )}^{7/2}\,\left (b^2\,e^2-8\,b\,c\,d\,e+8\,c^2\,d^2\right )}{4\,b^4}}{c^2\,{\left (d+e\,x\right )}^4-\left (d+e\,x\right )\,\left (2\,b^2\,d\,e^2-6\,b\,c\,d^2\,e+4\,c^2\,d^3\right )-\left (4\,c^2\,d-2\,b\,c\,e\right )\,{\left (d+e\,x\right )}^3+{\left (d+e\,x\right )}^2\,\left (b^2\,e^2-6\,b\,c\,d\,e+6\,c^2\,d^2\right )+c^2\,d^4+b^2\,d^2\,e^2-2\,b\,c\,d^3\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/(b*x + c*x^2)^3,x)

[Out]

(3*atanh((81*c^2*d^2*e^8*(c^2*d - b*c*e)^(1/2)*(d + e*x)^(1/2))/(8*((189*c^3*d^3*e^8)/8 - (351*b*c^2*d^2*e^9)/
32 - (27*c^4*d^4*e^7)/(2*b) + (27*b^2*c*d*e^10)/32)) + (27*c^3*d^3*e^7*(c^2*d - b*c*e)^(1/2)*(d + e*x)^(1/2))/
(2*((27*c^4*d^4*e^7)/2 - (189*b*c^3*d^3*e^8)/8 + (351*b^2*c^2*d^2*e^9)/32 - (27*b^3*c*d*e^10)/32)) + (27*c*d*e
^9*(c^2*d - b*c*e)^(1/2)*(d + e*x)^(1/2))/(32*((351*c^2*d^2*e^9)/32 - (27*b*c*d*e^10)/32 - (189*c^3*d^3*e^8)/(
8*b) + (27*c^4*d^4*e^7)/(2*b^2))))*(-c*(b*e - c*d))^(1/2)*(b^2*e^2 + 16*c^2*d^2 - 12*b*c*d*e))/(4*b^5*c) - (3*
d^(1/2)*atanh((135*c*d^(1/2)*e^10*(d + e*x)^(1/2))/(32*((135*c*d*e^10)/32 - (675*c^2*d^2*e^9)/(32*b) + (243*c^
3*d^3*e^8)/(8*b^2) - (27*c^4*d^4*e^7)/(2*b^3))) + (675*c^2*d^(3/2)*e^9*(d + e*x)^(1/2))/(32*((675*c^2*d^2*e^9)
/32 - (135*b*c*d*e^10)/32 - (243*c^3*d^3*e^8)/(8*b) + (27*c^4*d^4*e^7)/(2*b^2))) + (243*c^3*d^(5/2)*e^8*(d + e
*x)^(1/2))/(8*((243*c^3*d^3*e^8)/8 - (675*b*c^2*d^2*e^9)/32 - (27*c^4*d^4*e^7)/(2*b) + (135*b^2*c*d*e^10)/32))
+ (27*c^4*d^(7/2)*e^7*(d + e*x)^(1/2))/(2*((27*c^4*d^4*e^7)/2 - (243*b*c^3*d^3*e^8)/8 + (675*b^2*c^2*d^2*e^9)
/32 - (135*b^3*c*d*e^10)/32)))*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(4*b^5) - (((d + e*x)^(3/2)*(19*b^3*d*e^
4 - 72*c^3*d^4*e + 144*b*c^2*d^3*e^2 - 91*b^2*c*d^2*e^3))/(4*b^4) + (3*(d + e*x)^(1/2)*(2*c^3*d^5*e - b^3*d^2*
e^4 - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3))/b^4 - ((b*e - 2*c*d)*(d + e*x)^(5/2)*(5*b^2*e^3 + 36*c^2*d^2*e - 36*
b*c*d*e^2))/(4*b^4) - (3*c*e*(d + e*x)^(7/2)*(b^2*e^2 + 8*c^2*d^2 - 8*b*c*d*e))/(4*b^4))/(c^2*(d + e*x)^4 - (d
+ e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c
^2*d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e)

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