3.4.66 \(\int \frac {1}{(d+e x)^{3/2} (b x+c x^2)} \, dx\) [366]

Optimal. Leaf size=102 \[ -\frac {2 e}{d (c d-b e) \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{3/2}}+\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {723, 840, 1180, 214} \begin {gather*} \frac {2 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{3/2}}-\frac {2 e}{d \sqrt {d+e x} (c d-b e)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)),x]

[Out]

(-2*e)/(d*(c*d - b*e)*Sqrt[d + e*x]) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(3/2)) + (2*c^(3/2)*ArcTanh[(Sq
rt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(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 723

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m
+ 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c
*e*x, x]/(a + b*x + c*x^2)), 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[m, -1]

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

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Mathematica [A]
time = 0.30, size = 102, normalized size = 1.00 \begin {gather*} \frac {2 e}{d (-c d+b e) \sqrt {d+e x}}+\frac {2 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{b (-c d+b e)^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)),x]

[Out]

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

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Maple [A]
time = 0.43, size = 103, normalized size = 1.01

method	result	size
derivativedivides	\(2 e \left (-\frac {\arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{d^{\frac {3}{2}} b e}+\frac {c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) b e \sqrt {\left (b e -c d \right ) c}}+\frac {1}{d \left (b e -c d \right ) \sqrt {e x +d}}\right )\)	\(103\)
default	\(2 e \left (-\frac {\arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{d^{\frac {3}{2}} b e}+\frac {c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) b e \sqrt {\left (b e -c d \right ) c}}+\frac {1}{d \left (b e -c d \right ) \sqrt {e x +d}}\right )\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+b*x),x,method=_RETURNVERBOSE)

[Out]

2*e*(-1/d^(3/2)/b/e*arctanh((e*x+d)^(1/2)/d^(1/2))+1/(b*e-c*d)*c^2/b/e/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1
/2)/((b*e-c*d)*c)^(1/2))+1/d/(b*e-c*d)/(e*x+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(1/(e*x+d)^(3/2)/(c*x^2+b*x),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. 183 vs. \(2 (91) = 182\).
time = 1.79, size = 766, normalized size = 7.51 \begin {gather*} \left [-\frac {2 \, \sqrt {x e + d} b d e + {\left (c d^{2} x e + c d^{3}\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {2 \, c d - 2 \, {\left (c d - b e\right )} \sqrt {x e + d} \sqrt {\frac {c}{c d - b e}} + {\left (c x - b\right )} e}{c x + b}\right ) - {\left (c d^{2} - b x e^{2} + {\left (c d x - b d\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right )}{b c d^{4} - b^{2} d^{2} x e^{2} + {\left (b c d^{3} x - b^{2} d^{3}\right )} e}, -\frac {2 \, \sqrt {x e + d} b d e - 2 \, {\left (c d^{2} x e + c d^{3}\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {x e + d} \sqrt {-\frac {c}{c d - b e}}}{c x e + c d}\right ) - {\left (c d^{2} - b x e^{2} + {\left (c d x - b d\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right )}{b c d^{4} - b^{2} d^{2} x e^{2} + {\left (b c d^{3} x - b^{2} d^{3}\right )} e}, -\frac {2 \, \sqrt {x e + d} b d e - 2 \, {\left (c d^{2} - b x e^{2} + {\left (c d x - b d\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (c d^{2} x e + c d^{3}\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {2 \, c d - 2 \, {\left (c d - b e\right )} \sqrt {x e + d} \sqrt {\frac {c}{c d - b e}} + {\left (c x - b\right )} e}{c x + b}\right )}{b c d^{4} - b^{2} d^{2} x e^{2} + {\left (b c d^{3} x - b^{2} d^{3}\right )} e}, -\frac {2 \, {\left (\sqrt {x e + d} b d e - {\left (c d^{2} x e + c d^{3}\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {x e + d} \sqrt {-\frac {c}{c d - b e}}}{c x e + c d}\right ) - {\left (c d^{2} - b x e^{2} + {\left (c d x - b d\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right )\right )}}{b c d^{4} - b^{2} d^{2} x e^{2} + {\left (b c d^{3} x - b^{2} d^{3}\right )} e}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

[-(2*sqrt(x*e + d)*b*d*e + (c*d^2*x*e + c*d^3)*sqrt(c/(c*d - b*e))*log((2*c*d - 2*(c*d - b*e)*sqrt(x*e + d)*sq
rt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) - (c*d^2 - b*x*e^2 + (c*d*x - b*d)*e)*sqrt(d)*log((x*e - 2*sqrt(x*
e + d)*sqrt(d) + 2*d)/x))/(b*c*d^4 - b^2*d^2*x*e^2 + (b*c*d^3*x - b^2*d^3)*e), -(2*sqrt(x*e + d)*b*d*e - 2*(c*
d^2*x*e + c*d^3)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) -
(c*d^2 - b*x*e^2 + (c*d*x - b*d)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x))/(b*c*d^4 - b^2*d^2*x
*e^2 + (b*c*d^3*x - b^2*d^3)*e), -(2*sqrt(x*e + d)*b*d*e - 2*(c*d^2 - b*x*e^2 + (c*d*x - b*d)*e)*sqrt(-d)*arct
an(sqrt(x*e + d)*sqrt(-d)/d) + (c*d^2*x*e + c*d^3)*sqrt(c/(c*d - b*e))*log((2*c*d - 2*(c*d - b*e)*sqrt(x*e + d
)*sqrt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)))/(b*c*d^4 - b^2*d^2*x*e^2 + (b*c*d^3*x - b^2*d^3)*e), -2*(sqrt
(x*e + d)*b*d*e - (c*d^2*x*e + c*d^3)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*
e))/(c*x*e + c*d)) - (c*d^2 - b*x*e^2 + (c*d*x - b*d)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d))/(b*c*d^4 -
b^2*d^2*x*e^2 + (b*c*d^3*x - b^2*d^3)*e)]

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Sympy [A]
time = 6.24, size = 94, normalized size = 0.92 \begin {gather*} \frac {2 e}{d \sqrt {d + e x} \left (b e - c d\right )} + \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b \sqrt {\frac {b e - c d}{c}} \left (b e - c d\right )} + \frac {2 \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b d \sqrt {- d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x),x)

[Out]

2*e/(d*sqrt(d + e*x)*(b*e - c*d)) + 2*c*atan(sqrt(d + e*x)/sqrt((b*e - c*d)/c))/(b*sqrt((b*e - c*d)/c)*(b*e -
c*d)) + 2*atan(sqrt(d + e*x)/sqrt(-d))/(b*d*sqrt(-d))

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Giac [A]
time = 1.27, size = 113, normalized size = 1.11 \begin {gather*} -\frac {2 \, c^{2} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b c d - b^{2} e\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, e}{{\left (c d^{2} - b d e\right )} \sqrt {x e + d}} + \frac {2 \, \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b \sqrt {-d} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.63, size = 2258, normalized size = 22.14 \begin {gather*} -\frac {2\,e}{\left (c\,d^2-b\,d\,e\right )\,\sqrt {d+e\,x}}-\frac {2\,\mathrm {atanh}\left (\frac {48\,c^7\,d^7\,e^3\,\sqrt {d+e\,x}}{\sqrt {d^3}\,\left (-16\,b^5\,c^2\,d\,e^8+96\,b^4\,c^3\,d^2\,e^7-240\,b^3\,c^4\,d^3\,e^6+304\,b^2\,c^5\,d^4\,e^5-192\,b\,c^6\,d^5\,e^4+48\,c^7\,d^6\,e^3\right )}-\frac {192\,b\,c^6\,d^6\,e^4\,\sqrt {d+e\,x}}{\sqrt {d^3}\,\left (-16\,b^5\,c^2\,d\,e^8+96\,b^4\,c^3\,d^2\,e^7-240\,b^3\,c^4\,d^3\,e^6+304\,b^2\,c^5\,d^4\,e^5-192\,b\,c^6\,d^5\,e^4+48\,c^7\,d^6\,e^3\right )}+\frac {304\,b^2\,c^5\,d^5\,e^5\,\sqrt {d+e\,x}}{\sqrt {d^3}\,\left (-16\,b^5\,c^2\,d\,e^8+96\,b^4\,c^3\,d^2\,e^7-240\,b^3\,c^4\,d^3\,e^6+304\,b^2\,c^5\,d^4\,e^5-192\,b\,c^6\,d^5\,e^4+48\,c^7\,d^6\,e^3\right )}-\frac {240\,b^3\,c^4\,d^4\,e^6\,\sqrt {d+e\,x}}{\sqrt {d^3}\,\left (-16\,b^5\,c^2\,d\,e^8+96\,b^4\,c^3\,d^2\,e^7-240\,b^3\,c^4\,d^3\,e^6+304\,b^2\,c^5\,d^4\,e^5-192\,b\,c^6\,d^5\,e^4+48\,c^7\,d^6\,e^3\right )}+\frac {96\,b^4\,c^3\,d^3\,e^7\,\sqrt {d+e\,x}}{\sqrt {d^3}\,\left (-16\,b^5\,c^2\,d\,e^8+96\,b^4\,c^3\,d^2\,e^7-240\,b^3\,c^4\,d^3\,e^6+304\,b^2\,c^5\,d^4\,e^5-192\,b\,c^6\,d^5\,e^4+48\,c^7\,d^6\,e^3\right )}-\frac {16\,b^5\,c^2\,d^2\,e^8\,\sqrt {d+e\,x}}{\sqrt {d^3}\,\left (-16\,b^5\,c^2\,d\,e^8+96\,b^4\,c^3\,d^2\,e^7-240\,b^3\,c^4\,d^3\,e^6+304\,b^2\,c^5\,d^4\,e^5-192\,b\,c^6\,d^5\,e^4+48\,c^7\,d^6\,e^3\right )}\right )}{b\,\sqrt {d^3}}+\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (\sqrt {d+e\,x}\,\left (-8\,b^5\,c^3\,d^3\,e^7+40\,b^4\,c^4\,d^4\,e^6-88\,b^3\,c^5\,d^5\,e^5+104\,b^2\,c^6\,d^6\,e^4-64\,b\,c^7\,d^7\,e^3+16\,c^8\,d^8\,e^2\right )-\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (72\,b^3\,c^6\,d^8\,e^4-16\,b^2\,c^7\,d^9\,e^3-128\,b^4\,c^5\,d^7\,e^5+112\,b^5\,c^4\,d^6\,e^6-48\,b^6\,c^3\,d^5\,e^7+8\,b^7\,c^2\,d^4\,e^8+\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\sqrt {d+e\,x}\,\left (8\,b^8\,c^2\,d^5\,e^8-56\,b^7\,c^3\,d^6\,e^7+160\,b^6\,c^4\,d^7\,e^6-240\,b^5\,c^5\,d^8\,e^5+200\,b^4\,c^6\,d^9\,e^4-88\,b^3\,c^7\,d^{10}\,e^3+16\,b^2\,c^8\,d^{11}\,e^2\right )}{b\,{\left (b\,e-c\,d\right )}^3}\right )}{b\,{\left (b\,e-c\,d\right )}^3}\right )\,1{}\mathrm {i}}{b\,{\left (b\,e-c\,d\right )}^3}+\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (\sqrt {d+e\,x}\,\left (-8\,b^5\,c^3\,d^3\,e^7+40\,b^4\,c^4\,d^4\,e^6-88\,b^3\,c^5\,d^5\,e^5+104\,b^2\,c^6\,d^6\,e^4-64\,b\,c^7\,d^7\,e^3+16\,c^8\,d^8\,e^2\right )-\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (16\,b^2\,c^7\,d^9\,e^3-72\,b^3\,c^6\,d^8\,e^4+128\,b^4\,c^5\,d^7\,e^5-112\,b^5\,c^4\,d^6\,e^6+48\,b^6\,c^3\,d^5\,e^7-8\,b^7\,c^2\,d^4\,e^8+\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\sqrt {d+e\,x}\,\left (8\,b^8\,c^2\,d^5\,e^8-56\,b^7\,c^3\,d^6\,e^7+160\,b^6\,c^4\,d^7\,e^6-240\,b^5\,c^5\,d^8\,e^5+200\,b^4\,c^6\,d^9\,e^4-88\,b^3\,c^7\,d^{10}\,e^3+16\,b^2\,c^8\,d^{11}\,e^2\right )}{b\,{\left (b\,e-c\,d\right )}^3}\right )}{b\,{\left (b\,e-c\,d\right )}^3}\right )\,1{}\mathrm {i}}{b\,{\left (b\,e-c\,d\right )}^3}}{16\,c^7\,d^6\,e^3-48\,b\,c^6\,d^5\,e^4+48\,b^2\,c^5\,d^4\,e^5-16\,b^3\,c^4\,d^3\,e^6+\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (\sqrt {d+e\,x}\,\left (-8\,b^5\,c^3\,d^3\,e^7+40\,b^4\,c^4\,d^4\,e^6-88\,b^3\,c^5\,d^5\,e^5+104\,b^2\,c^6\,d^6\,e^4-64\,b\,c^7\,d^7\,e^3+16\,c^8\,d^8\,e^2\right )-\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (72\,b^3\,c^6\,d^8\,e^4-16\,b^2\,c^7\,d^9\,e^3-128\,b^4\,c^5\,d^7\,e^5+112\,b^5\,c^4\,d^6\,e^6-48\,b^6\,c^3\,d^5\,e^7+8\,b^7\,c^2\,d^4\,e^8+\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\sqrt {d+e\,x}\,\left (8\,b^8\,c^2\,d^5\,e^8-56\,b^7\,c^3\,d^6\,e^7+160\,b^6\,c^4\,d^7\,e^6-240\,b^5\,c^5\,d^8\,e^5+200\,b^4\,c^6\,d^9\,e^4-88\,b^3\,c^7\,d^{10}\,e^3+16\,b^2\,c^8\,d^{11}\,e^2\right )}{b\,{\left (b\,e-c\,d\right )}^3}\right )}{b\,{\left (b\,e-c\,d\right )}^3}\right )}{b\,{\left (b\,e-c\,d\right )}^3}-\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (\sqrt {d+e\,x}\,\left (-8\,b^5\,c^3\,d^3\,e^7+40\,b^4\,c^4\,d^4\,e^6-88\,b^3\,c^5\,d^5\,e^5+104\,b^2\,c^6\,d^6\,e^4-64\,b\,c^7\,d^7\,e^3+16\,c^8\,d^8\,e^2\right )-\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\left (16\,b^2\,c^7\,d^9\,e^3-72\,b^3\,c^6\,d^8\,e^4+128\,b^4\,c^5\,d^7\,e^5-112\,b^5\,c^4\,d^6\,e^6+48\,b^6\,c^3\,d^5\,e^7-8\,b^7\,c^2\,d^4\,e^8+\frac {\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,\sqrt {d+e\,x}\,\left (8\,b^8\,c^2\,d^5\,e^8-56\,b^7\,c^3\,d^6\,e^7+160\,b^6\,c^4\,d^7\,e^6-240\,b^5\,c^5\,d^8\,e^5+200\,b^4\,c^6\,d^9\,e^4-88\,b^3\,c^7\,d^{10}\,e^3+16\,b^2\,c^8\,d^{11}\,e^2\right )}{b\,{\left (b\,e-c\,d\right )}^3}\right )}{b\,{\left (b\,e-c\,d\right )}^3}\right )}{b\,{\left (b\,e-c\,d\right )}^3}}\right )\,\sqrt {-c^3\,{\left (b\,e-c\,d\right )}^3}\,2{}\mathrm {i}}{b\,{\left (b\,e-c\,d\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)*(d + e*x)^(3/2)),x)

[Out]

(atan((((-c^3*(b*e - c*d)^3)^(1/2)*((d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 64*b*c^7*d^7*e^3 + 104*b^2*c^6*d^6*e^4 -
88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7) - ((-c^3*(b*e - c*d)^3)^(1/2)*(72*b^3*c^6*d^8*e^
4 - 16*b^2*c^7*d^9*e^3 - 128*b^4*c^5*d^7*e^5 + 112*b^5*c^4*d^6*e^6 - 48*b^6*c^3*d^5*e^7 + 8*b^7*c^2*d^4*e^8 +
((-c^3*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(16*b^2*c^8*d^11*e^2 - 88*b^3*c^7*d^10*e^3 + 200*b^4*c^6*d^9*e^4 -
240*b^5*c^5*d^8*e^5 + 160*b^6*c^4*d^7*e^6 - 56*b^7*c^3*d^6*e^7 + 8*b^8*c^2*d^5*e^8))/(b*(b*e - c*d)^3)))/(b*(
b*e - c*d)^3))*1i)/(b*(b*e - c*d)^3) + ((-c^3*(b*e - c*d)^3)^(1/2)*((d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 64*b*c^7
*d^7*e^3 + 104*b^2*c^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7) - ((-c^3*(b*e -
c*d)^3)^(1/2)*(16*b^2*c^7*d^9*e^3 - 72*b^3*c^6*d^8*e^4 + 128*b^4*c^5*d^7*e^5 - 112*b^5*c^4*d^6*e^6 + 48*b^6*c^
3*d^5*e^7 - 8*b^7*c^2*d^4*e^8 + ((-c^3*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(16*b^2*c^8*d^11*e^2 - 88*b^3*c^7*
d^10*e^3 + 200*b^4*c^6*d^9*e^4 - 240*b^5*c^5*d^8*e^5 + 160*b^6*c^4*d^7*e^6 - 56*b^7*c^3*d^6*e^7 + 8*b^8*c^2*d^
5*e^8))/(b*(b*e - c*d)^3)))/(b*(b*e - c*d)^3))*1i)/(b*(b*e - c*d)^3))/(16*c^7*d^6*e^3 - 48*b*c^6*d^5*e^4 + 48*
b^2*c^5*d^4*e^5 - 16*b^3*c^4*d^3*e^6 + ((-c^3*(b*e - c*d)^3)^(1/2)*((d + e*x)^(1/2)*(16*c^8*d^8*e^2 - 64*b*c^7
*d^7*e^3 + 104*b^2*c^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*c^3*d^3*e^7) - ((-c^3*(b*e -
c*d)^3)^(1/2)*(72*b^3*c^6*d^8*e^4 - 16*b^2*c^7*d^9*e^3 - 128*b^4*c^5*d^7*e^5 + 112*b^5*c^4*d^6*e^6 - 48*b^6*c^
3*d^5*e^7 + 8*b^7*c^2*d^4*e^8 + ((-c^3*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(16*b^2*c^8*d^11*e^2 - 88*b^3*c^7*
d^10*e^3 + 200*b^4*c^6*d^9*e^4 - 240*b^5*c^5*d^8*e^5 + 160*b^6*c^4*d^7*e^6 - 56*b^7*c^3*d^6*e^7 + 8*b^8*c^2*d^
5*e^8))/(b*(b*e - c*d)^3)))/(b*(b*e - c*d)^3)))/(b*(b*e - c*d)^3) - ((-c^3*(b*e - c*d)^3)^(1/2)*((d + e*x)^(1/
2)*(16*c^8*d^8*e^2 - 64*b*c^7*d^7*e^3 + 104*b^2*c^6*d^6*e^4 - 88*b^3*c^5*d^5*e^5 + 40*b^4*c^4*d^4*e^6 - 8*b^5*
c^3*d^3*e^7) - ((-c^3*(b*e - c*d)^3)^(1/2)*(16*b^2*c^7*d^9*e^3 - 72*b^3*c^6*d^8*e^4 + 128*b^4*c^5*d^7*e^5 - 11
2*b^5*c^4*d^6*e^6 + 48*b^6*c^3*d^5*e^7 - 8*b^7*c^2*d^4*e^8 + ((-c^3*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(16*b
^2*c^8*d^11*e^2 - 88*b^3*c^7*d^10*e^3 + 200*b^4*c^6*d^9*e^4 - 240*b^5*c^5*d^8*e^5 + 160*b^6*c^4*d^7*e^6 - 56*b
^7*c^3*d^6*e^7 + 8*b^8*c^2*d^5*e^8))/(b*(b*e - c*d)^3)))/(b*(b*e - c*d)^3)))/(b*(b*e - c*d)^3)))*(-c^3*(b*e -
c*d)^3)^(1/2)*2i)/(b*(b*e - c*d)^3) - (2*atanh((48*c^7*d^7*e^3*(d + e*x)^(1/2))/((d^3)^(1/2)*(48*c^7*d^6*e^3 -
192*b*c^6*d^5*e^4 - 16*b^5*c^2*d*e^8 + 304*b^2*c^5*d^4*e^5 - 240*b^3*c^4*d^3*e^6 + 96*b^4*c^3*d^2*e^7)) - (19
2*b*c^6*d^6*e^4*(d + e*x)^(1/2))/((d^3)^(1/2)*(48*c^7*d^6*e^3 - 192*b*c^6*d^5*e^4 - 16*b^5*c^2*d*e^8 + 304*b^2
*c^5*d^4*e^5 - 240*b^3*c^4*d^3*e^6 + 96*b^4*c^3*d^2*e^7)) + (304*b^2*c^5*d^5*e^5*(d + e*x)^(1/2))/((d^3)^(1/2)
*(48*c^7*d^6*e^3 - 192*b*c^6*d^5*e^4 - 16*b^5*c^2*d*e^8 + 304*b^2*c^5*d^4*e^5 - 240*b^3*c^4*d^3*e^6 + 96*b^4*c
^3*d^2*e^7)) - (240*b^3*c^4*d^4*e^6*(d + e*x)^(1/2))/((d^3)^(1/2)*(48*c^7*d^6*e^3 - 192*b*c^6*d^5*e^4 - 16*b^5
*c^2*d*e^8 + 304*b^2*c^5*d^4*e^5 - 240*b^3*c^4*d^3*e^6 + 96*b^4*c^3*d^2*e^7)) + (96*b^4*c^3*d^3*e^7*(d + e*x)^
(1/2))/((d^3)^(1/2)*(48*c^7*d^6*e^3 - 192*b*c^6*d^5*e^4 - 16*b^5*c^2*d*e^8 + 304*b^2*c^5*d^4*e^5 - 240*b^3*c^4
*d^3*e^6 + 96*b^4*c^3*d^2*e^7)) - (16*b^5*c^2*d^2*e^8*(d + e*x)^(1/2))/((d^3)^(1/2)*(48*c^7*d^6*e^3 - 192*b*c^
6*d^5*e^4 - 16*b^5*c^2*d*e^8 + 304*b^2*c^5*d^4*e^5 - 240*b^3*c^4*d^3*e^6 + 96*b^4*c^3*d^2*e^7))))/(b*(d^3)^(1/
2)) - (2*e)/((c*d^2 - b*d*e)*(d + e*x)^(1/2))

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