3.4.34 \(\int \frac {1}{(d+e x) (b x+c x^2)^{5/2}} \, dx\) [334]

Optimal. Leaf size=230 \[ -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-4 b c d e-3 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {e^4 \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{d^{5/2} (c d-b e)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {754, 836, 12, 738, 212} \begin {gather*} -\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac {2 \left (c x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (c d-b e)^2}+\frac {e^4 \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*c^2*
d^2 - 4*b*c*d*e - 3*b^2*e^2) + c*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2)*x))/(3*b^4*d^2*(c*d - b*e)^
2*Sqrt[b*x + c*x^2]) + (e^4*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(d
^(5/2)*(c*d - b*e)^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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

Rule 754

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

Rule 836

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

Rubi steps

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

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Mathematica [A]
time = 0.77, size = 253, normalized size = 1.10 \begin {gather*} \frac {2 \sqrt {d} \sqrt {-c d+b e} \left (16 c^5 d^3 x^3+24 b c^4 d^2 x^2 (d-e x)+b^5 e^2 (-d+3 e x)+2 b^2 c^3 d x \left (3 d^2-18 d e x+e^2 x^2\right )+2 b^4 c e \left (d^2+3 e^2 x^2\right )-b^3 c^2 \left (d^3+9 d^2 e x-3 d e^2 x^2-3 e^3 x^3\right )\right )-6 b^4 e^4 x^{3/2} (b+c x)^{3/2} \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{3 b^4 d^{5/2} (-c d+b e)^{5/2} (x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(b*x + c*x^2)^(5/2)),x]

[Out]

(2*Sqrt[d]*Sqrt[-(c*d) + b*e]*(16*c^5*d^3*x^3 + 24*b*c^4*d^2*x^2*(d - e*x) + b^5*e^2*(-d + 3*e*x) + 2*b^2*c^3*
d*x*(3*d^2 - 18*d*e*x + e^2*x^2) + 2*b^4*c*e*(d^2 + 3*e^2*x^2) - b^3*c^2*(d^3 + 9*d^2*e*x - 3*d*e^2*x^2 - 3*e^
3*x^3)) - 6*b^4*e^4*x^(3/2)*(b + c*x)^(3/2)*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*S
qrt[-(c*d) + b*e])])/(3*b^4*d^(5/2)*(-(c*d) + b*e)^(5/2)*(x*(b + c*x))^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(659\) vs. \(2(210)=420\).
time = 0.52, size = 660, normalized size = 2.87

method	result	size
risch	\(-\frac {2 \left (c x +b \right ) \left (-3 b e x -8 c d x +b d \right )}{3 b^{4} d^{2} \sqrt {x \left (c x +b \right )}\, x}-\frac {2 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{3} \left (b e -c d \right ) \left (\frac {b}{c}+x \right )^{2}}-\frac {4 c^{2} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{4} \left (b e -c d \right ) \left (\frac {b}{c}+x \right )}-\frac {e^{3} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{2} \left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {6 c^{2} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, e}{b^{3} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )}+\frac {4 d \,c^{3} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{b^{4} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )}\)	\(389\)
default	\(\frac {-\frac {e^{2}}{3 d \left (b e -c d \right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}+\frac {e \left (b e -2 c d \right ) \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}+\frac {2 \left (b e -2 c d \right )}{3 e}}{\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{3 \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right )^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}-\frac {e^{2} \left (-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{d \left (b e -c d \right )}}{e}\)	\(660\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(-1/3/d/(b*e-c*d)*e^2/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/2*e*(b*e-2*c*d)/d/(b*e
-c*d)*(2/3*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*
d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+16/3*c/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)^2*(2*c*(x+d/e)+1/e*(b*e-2*
c*d))/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))-1/d/(b*e-c*d)*e^2*(-1/d/(b*e-c*d)*e^2/(c*(x
+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+e*(b*e-2*c*d)/d/(b*e-c*d)*(2*c*(x+d/e)+1/e*(b*e-2*c*d))
/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/d/(b
*e-c*d)*e^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)
*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))))

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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(1/(e*x+d)/(c*x^2+b*x)^(5/2),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 [B] Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (218) = 436\).
time = 1.71, size = 1025, normalized size = 4.46 \begin {gather*} \left [\frac {3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )} \sqrt {c d^{2} - b d e} e^{4} \log \left (\frac {2 \, c d x - b x e + b d + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{x e + d}\right ) + 2 \, {\left (16 \, c^{6} d^{5} x^{3} + 24 \, b c^{5} d^{5} x^{2} + 6 \, b^{2} c^{4} d^{5} x - b^{3} c^{3} d^{5} - 3 \, {\left (b^{4} c^{2} d x^{3} + 2 \, b^{5} c d x^{2} + b^{6} d x\right )} e^{4} + {\left (b^{3} c^{3} d^{2} x^{3} + 3 \, b^{4} c^{2} d^{2} x^{2} + 3 \, b^{5} c d^{2} x + b^{6} d^{2}\right )} e^{3} + {\left (26 \, b^{2} c^{4} d^{3} x^{3} + 39 \, b^{3} c^{3} d^{3} x^{2} + 9 \, b^{4} c^{2} d^{3} x - 3 \, b^{5} c d^{3}\right )} e^{2} - {\left (40 \, b c^{5} d^{4} x^{3} + 60 \, b^{2} c^{4} d^{4} x^{2} + 15 \, b^{3} c^{3} d^{4} x - 3 \, b^{4} c^{2} d^{4}\right )} e\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{5} d^{6} x^{4} + 2 \, b^{5} c^{4} d^{6} x^{3} + b^{6} c^{3} d^{6} x^{2} - {\left (b^{7} c^{2} d^{3} x^{4} + 2 \, b^{8} c d^{3} x^{3} + b^{9} d^{3} x^{2}\right )} e^{3} + 3 \, {\left (b^{6} c^{3} d^{4} x^{4} + 2 \, b^{7} c^{2} d^{4} x^{3} + b^{8} c d^{4} x^{2}\right )} e^{2} - 3 \, {\left (b^{5} c^{4} d^{5} x^{4} + 2 \, b^{6} c^{3} d^{5} x^{3} + b^{7} c^{2} d^{5} x^{2}\right )} e\right )}}, \frac {2 \, {\left (3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) e^{4} + {\left (16 \, c^{6} d^{5} x^{3} + 24 \, b c^{5} d^{5} x^{2} + 6 \, b^{2} c^{4} d^{5} x - b^{3} c^{3} d^{5} - 3 \, {\left (b^{4} c^{2} d x^{3} + 2 \, b^{5} c d x^{2} + b^{6} d x\right )} e^{4} + {\left (b^{3} c^{3} d^{2} x^{3} + 3 \, b^{4} c^{2} d^{2} x^{2} + 3 \, b^{5} c d^{2} x + b^{6} d^{2}\right )} e^{3} + {\left (26 \, b^{2} c^{4} d^{3} x^{3} + 39 \, b^{3} c^{3} d^{3} x^{2} + 9 \, b^{4} c^{2} d^{3} x - 3 \, b^{5} c d^{3}\right )} e^{2} - {\left (40 \, b c^{5} d^{4} x^{3} + 60 \, b^{2} c^{4} d^{4} x^{2} + 15 \, b^{3} c^{3} d^{4} x - 3 \, b^{4} c^{2} d^{4}\right )} e\right )} \sqrt {c x^{2} + b x}\right )}}{3 \, {\left (b^{4} c^{5} d^{6} x^{4} + 2 \, b^{5} c^{4} d^{6} x^{3} + b^{6} c^{3} d^{6} x^{2} - {\left (b^{7} c^{2} d^{3} x^{4} + 2 \, b^{8} c d^{3} x^{3} + b^{9} d^{3} x^{2}\right )} e^{3} + 3 \, {\left (b^{6} c^{3} d^{4} x^{4} + 2 \, b^{7} c^{2} d^{4} x^{3} + b^{8} c d^{4} x^{2}\right )} e^{2} - 3 \, {\left (b^{5} c^{4} d^{5} x^{4} + 2 \, b^{6} c^{3} d^{5} x^{3} + b^{7} c^{2} d^{5} x^{2}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)*sqrt(c*d^2 - b*d*e)*e^4*log((2*c*d*x - b*x*e + b*d + 2*sqrt(c*d^
2 - b*d*e)*sqrt(c*x^2 + b*x))/(x*e + d)) + 2*(16*c^6*d^5*x^3 + 24*b*c^5*d^5*x^2 + 6*b^2*c^4*d^5*x - b^3*c^3*d^
5 - 3*(b^4*c^2*d*x^3 + 2*b^5*c*d*x^2 + b^6*d*x)*e^4 + (b^3*c^3*d^2*x^3 + 3*b^4*c^2*d^2*x^2 + 3*b^5*c*d^2*x + b
^6*d^2)*e^3 + (26*b^2*c^4*d^3*x^3 + 39*b^3*c^3*d^3*x^2 + 9*b^4*c^2*d^3*x - 3*b^5*c*d^3)*e^2 - (40*b*c^5*d^4*x^
3 + 60*b^2*c^4*d^4*x^2 + 15*b^3*c^3*d^4*x - 3*b^4*c^2*d^4)*e)*sqrt(c*x^2 + b*x))/(b^4*c^5*d^6*x^4 + 2*b^5*c^4*
d^6*x^3 + b^6*c^3*d^6*x^2 - (b^7*c^2*d^3*x^4 + 2*b^8*c*d^3*x^3 + b^9*d^3*x^2)*e^3 + 3*(b^6*c^3*d^4*x^4 + 2*b^7
*c^2*d^4*x^3 + b^8*c*d^4*x^2)*e^2 - 3*(b^5*c^4*d^5*x^4 + 2*b^6*c^3*d^5*x^3 + b^7*c^2*d^5*x^2)*e), 2/3*(3*(b^4*
c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x -
b*x*e))*e^4 + (16*c^6*d^5*x^3 + 24*b*c^5*d^5*x^2 + 6*b^2*c^4*d^5*x - b^3*c^3*d^5 - 3*(b^4*c^2*d*x^3 + 2*b^5*c*
d*x^2 + b^6*d*x)*e^4 + (b^3*c^3*d^2*x^3 + 3*b^4*c^2*d^2*x^2 + 3*b^5*c*d^2*x + b^6*d^2)*e^3 + (26*b^2*c^4*d^3*x
^3 + 39*b^3*c^3*d^3*x^2 + 9*b^4*c^2*d^3*x - 3*b^5*c*d^3)*e^2 - (40*b*c^5*d^4*x^3 + 60*b^2*c^4*d^4*x^2 + 15*b^3
*c^3*d^4*x - 3*b^4*c^2*d^4)*e)*sqrt(c*x^2 + b*x))/(b^4*c^5*d^6*x^4 + 2*b^5*c^4*d^6*x^3 + b^6*c^3*d^6*x^2 - (b^
7*c^2*d^3*x^4 + 2*b^8*c*d^3*x^3 + b^9*d^3*x^2)*e^3 + 3*(b^6*c^3*d^4*x^4 + 2*b^7*c^2*d^4*x^3 + b^8*c*d^4*x^2)*e
^2 - 3*(b^5*c^4*d^5*x^4 + 2*b^6*c^3*d^5*x^3 + b^7*c^2*d^5*x^2)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral(1/((x*(b + c*x))**(5/2)*(d + e*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 628 vs. \(2 (218) = 436\).
time = 1.40, size = 628, normalized size = 2.73 \begin {gather*} -\frac {2 \, \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right ) e^{4}}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} + \frac {2 \, {\left ({\left ({\left (\frac {{\left (16 \, c^{7} d^{10} - 56 \, b c^{6} d^{9} e + 66 \, b^{2} c^{5} d^{8} e^{2} - 25 \, b^{3} c^{4} d^{7} e^{3} - 4 \, b^{4} c^{3} d^{6} e^{4} + 3 \, b^{5} c^{2} d^{5} e^{5}\right )} x}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}} + \frac {3 \, {\left (8 \, b c^{6} d^{10} - 28 \, b^{2} c^{5} d^{9} e + 33 \, b^{3} c^{4} d^{8} e^{2} - 12 \, b^{4} c^{3} d^{7} e^{3} - 3 \, b^{5} c^{2} d^{6} e^{4} + 2 \, b^{6} c d^{5} e^{5}\right )}}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c^{5} d^{10} - 7 \, b^{3} c^{4} d^{9} e + 8 \, b^{4} c^{3} d^{8} e^{2} - 2 \, b^{5} c^{2} d^{7} e^{3} - 2 \, b^{6} c d^{6} e^{4} + b^{7} d^{5} e^{5}\right )}}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}}\right )} x - \frac {b^{3} c^{4} d^{10} - 4 \, b^{4} c^{3} d^{9} e + 6 \, b^{5} c^{2} d^{8} e^{2} - 4 \, b^{6} c d^{7} e^{3} + b^{7} d^{6} e^{4}}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

-2*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))*e^4/((c^2*d^4 - 2*b*c*d^3*e +
b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) + 2/3*((((16*c^7*d^10 - 56*b*c^6*d^9*e + 66*b^2*c^5*d^8*e^2 - 25*b^3*c^4*d^
7*e^3 - 4*b^4*c^3*d^6*e^4 + 3*b^5*c^2*d^5*e^5)*x/(b^4*c^4*d^11 - 4*b^5*c^3*d^10*e + 6*b^6*c^2*d^9*e^2 - 4*b^7*
c*d^8*e^3 + b^8*d^7*e^4) + 3*(8*b*c^6*d^10 - 28*b^2*c^5*d^9*e + 33*b^3*c^4*d^8*e^2 - 12*b^4*c^3*d^7*e^3 - 3*b^
5*c^2*d^6*e^4 + 2*b^6*c*d^5*e^5)/(b^4*c^4*d^11 - 4*b^5*c^3*d^10*e + 6*b^6*c^2*d^9*e^2 - 4*b^7*c*d^8*e^3 + b^8*
d^7*e^4))*x + 3*(2*b^2*c^5*d^10 - 7*b^3*c^4*d^9*e + 8*b^4*c^3*d^8*e^2 - 2*b^5*c^2*d^7*e^3 - 2*b^6*c*d^6*e^4 +
b^7*d^5*e^5)/(b^4*c^4*d^11 - 4*b^5*c^3*d^10*e + 6*b^6*c^2*d^9*e^2 - 4*b^7*c*d^8*e^3 + b^8*d^7*e^4))*x - (b^3*c
^4*d^10 - 4*b^4*c^3*d^9*e + 6*b^5*c^2*d^8*e^2 - 4*b^6*c*d^7*e^3 + b^7*d^6*e^4)/(b^4*c^4*d^11 - 4*b^5*c^3*d^10*
e + 6*b^6*c^2*d^9*e^2 - 4*b^7*c*d^8*e^3 + b^8*d^7*e^4))/(c*x^2 + b*x)^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^(5/2)*(d + e*x)),x)

[Out]

int(1/((b*x + c*x^2)^(5/2)*(d + e*x)), x)

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