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

Optimal. Leaf size=138 \[ \frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{5/2}}-\frac {2 e (2 c d-b e)}{d^2 \sqrt {d+e x} (c d-b e)^2}-\frac {2 e}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}} \]

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Rubi [A]  time = 0.26, antiderivative size = 138, 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 = {709, 828, 826, 1166, 208} \begin {gather*} \frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{5/2}}-\frac {2 e (2 c d-b e)}{d^2 \sqrt {d+e x} (c d-b e)^2}-\frac {2 e}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 208

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

Rule 709

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 826

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 828

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

Rule 1166

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

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Mathematica [C]  time = 0.02, size = 83, normalized size = 0.60 \begin {gather*} -\frac {2 \left (c d \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c (d+e x)}{c d-b e}\right )+(b e-c d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {e x}{d}+1\right )\right )}{3 b d (d+e x)^{3/2} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(c*d*Hypergeometric2F1[-3/2, 1, -1/2, (c*(d + e*x))/(c*d - b*e)] + (-(c*d) + b*e)*Hypergeometric2F1[-3/2,
1, -1/2, 1 + (e*x)/d]))/(3*b*d*(c*d - b*e)*(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.63, size = 170, normalized size = 1.23 \begin {gather*} \frac {2 \left (b^2 c^{5/2} e^2-2 b c^{7/2} d e+c^{9/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b (b e-c d)^{9/2}}-\frac {2 e \left (-3 b e (d+e x)-b d e+c d^2+6 c d (d+e x)\right )}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e*(c*d^2 - b*d*e + 6*c*d*(d + e*x) - 3*b*e*(d + e*x)))/(3*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) + (2*(c^(9/2)
*d^2 - 2*b*c^(7/2)*d*e + b^2*c^(5/2)*e^2)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(b*(
-(c*d) + b*e)^(9/2)) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(5/2))

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fricas [B]  time = 0.59, size = 1469, normalized size = 10.64

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3*(3*(c^2*d^3*e^2*x^2 + 2*c^2*d^4*e*x + c^2*d^5)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*
e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 3*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (c^2*d^2*e^2 - 2*b
*c*d*e^3 + b^2*e^4)*x^2 + 2*(c^2*d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt
(d) + 2*d)/x) - 2*(7*b*c*d^3*e - 4*b^2*d^2*e^2 + 3*(2*b*c*d^2*e^2 - b^2*d*e^3)*x)*sqrt(e*x + d))/(b*c^2*d^7 -
2*b^2*c*d^6*e + b^3*d^5*e^2 + (b*c^2*d^5*e^2 - 2*b^2*c*d^4*e^3 + b^3*d^3*e^4)*x^2 + 2*(b*c^2*d^6*e - 2*b^2*c*d
^5*e^2 + b^3*d^4*e^3)*x), 1/3*(6*(c^2*d^3*e^2*x^2 + 2*c^2*d^4*e*x + c^2*d^5)*sqrt(-c/(c*d - b*e))*arctan(-(c*d
- b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) + 3*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (c^2*d^2*
e^2 - 2*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(c^2*d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x
+ d)*sqrt(d) + 2*d)/x) - 2*(7*b*c*d^3*e - 4*b^2*d^2*e^2 + 3*(2*b*c*d^2*e^2 - b^2*d*e^3)*x)*sqrt(e*x + d))/(b*c
^2*d^7 - 2*b^2*c*d^6*e + b^3*d^5*e^2 + (b*c^2*d^5*e^2 - 2*b^2*c*d^4*e^3 + b^3*d^3*e^4)*x^2 + 2*(b*c^2*d^6*e -
2*b^2*c*d^5*e^2 + b^3*d^4*e^3)*x), 1/3*(6*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (c^2*d^2*e^2 - 2*b*c*d*e^3 +
b^2*e^4)*x^2 + 2*(c^2*d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 3*(c^2
*d^3*e^2*x^2 + 2*c^2*d^4*e*x + c^2*d^5)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x
+ d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 2*(7*b*c*d^3*e - 4*b^2*d^2*e^2 + 3*(2*b*c*d^2*e^2 - b^2*d*e^3)*x)*sqrt(
e*x + d))/(b*c^2*d^7 - 2*b^2*c*d^6*e + b^3*d^5*e^2 + (b*c^2*d^5*e^2 - 2*b^2*c*d^4*e^3 + b^3*d^3*e^4)*x^2 + 2*(
b*c^2*d^6*e - 2*b^2*c*d^5*e^2 + b^3*d^4*e^3)*x), 2/3*(3*(c^2*d^3*e^2*x^2 + 2*c^2*d^4*e*x + c^2*d^5)*sqrt(-c/(c
*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) + 3*(c^2*d^4 - 2*b*c*d^3*e +
b^2*d^2*e^2 + (c^2*d^2*e^2 - 2*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(c^2*d^3*e - 2*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(-d
)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (7*b*c*d^3*e - 4*b^2*d^2*e^2 + 3*(2*b*c*d^2*e^2 - b^2*d*e^3)*x)*sqrt(e*x
+ d))/(b*c^2*d^7 - 2*b^2*c*d^6*e + b^3*d^5*e^2 + (b*c^2*d^5*e^2 - 2*b^2*c*d^4*e^3 + b^3*d^3*e^4)*x^2 + 2*(b*c^
2*d^6*e - 2*b^2*c*d^5*e^2 + b^3*d^4*e^3)*x)]

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giac [A]  time = 0.19, size = 174, normalized size = 1.26 \begin {gather*} -\frac {2 \, c^{3} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (6 \, {\left (x e + d\right )} c d e + c d^{2} e - 3 \, {\left (x e + d\right )} b e^{2} - b d e^{2}\right )}}{3 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {2 \, \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b \sqrt {-d} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.07, size = 147, normalized size = 1.07 \begin {gather*} -\frac {2 c^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{2} \sqrt {\left (b e -c d \right ) c}\, b}+\frac {2 b \,e^{2}}{\left (b e -c d \right )^{2} \sqrt {e x +d}\, d^{2}}-\frac {4 c e}{\left (b e -c d \right )^{2} \sqrt {e x +d}\, d}+\frac {2 e}{3 \left (b e -c d \right ) \left (e x +d \right )^{\frac {3}{2}} d}-\frac {2 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^(5/2)/(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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 1.17, size = 4509, normalized size = 32.67

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((d + e*x)^(1/2)*(16*c^13*d^16*e^2 - 128*b*c^12*d^15*e^3 + 480*b^2*c^11*d^14*e^4 - 1120*b^3*c^10*d^13*
e^5 + 1800*b^4*c^9*d^12*e^6 - 2064*b^5*c^8*d^11*e^7 + 1688*b^6*c^7*d^10*e^8 - 960*b^7*c^6*d^9*e^9 + 360*b^8*c^
5*d^8*e^10 - 80*b^9*c^4*d^7*e^11 + 8*b^10*c^3*d^6*e^12) + ((-c^5*(b*e - c*d)^5)^(1/2)*(24*b^2*c^12*d^18*e^3 -
216*b^3*c^11*d^17*e^4 + 872*b^4*c^10*d^16*e^5 - 2080*b^5*c^9*d^15*e^6 + 3248*b^6*c^8*d^14*e^7 - 3472*b^7*c^7*d
^13*e^8 + 2576*b^8*c^6*d^12*e^9 - 1312*b^9*c^5*d^11*e^10 + 440*b^10*c^4*d^10*e^11 - 88*b^11*c^3*d^9*e^12 + 8*b
^12*c^2*d^8*e^13 - ((-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 +
800*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 + 4320*b^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7
*d^15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*d^13*e^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 -
8*b^13*c^2*d^10*e^13))/(b*(b*e - c*d)^5)))/(b*(b*e - c*d)^5))*(-c^5*(b*e - c*d)^5)^(1/2)*1i)/(b*(b*e - c*d)^5
) + (((d + e*x)^(1/2)*(16*c^13*d^16*e^2 - 128*b*c^12*d^15*e^3 + 480*b^2*c^11*d^14*e^4 - 1120*b^3*c^10*d^13*e^5
+ 1800*b^4*c^9*d^12*e^6 - 2064*b^5*c^8*d^11*e^7 + 1688*b^6*c^7*d^10*e^8 - 960*b^7*c^6*d^9*e^9 + 360*b^8*c^5*d
^8*e^10 - 80*b^9*c^4*d^7*e^11 + 8*b^10*c^3*d^6*e^12) - ((-c^5*(b*e - c*d)^5)^(1/2)*(24*b^2*c^12*d^18*e^3 - 216
*b^3*c^11*d^17*e^4 + 872*b^4*c^10*d^16*e^5 - 2080*b^5*c^9*d^15*e^6 + 3248*b^6*c^8*d^14*e^7 - 3472*b^7*c^7*d^13
*e^8 + 2576*b^8*c^6*d^12*e^9 - 1312*b^9*c^5*d^11*e^10 + 440*b^10*c^4*d^10*e^11 - 88*b^11*c^3*d^9*e^12 + 8*b^12
*c^2*d^8*e^13 + ((-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 + 80
0*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 + 4320*b^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7*d^
15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*d^13*e^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 - 8*
b^13*c^2*d^10*e^13))/(b*(b*e - c*d)^5)))/(b*(b*e - c*d)^5))*(-c^5*(b*e - c*d)^5)^(1/2)*1i)/(b*(b*e - c*d)^5))/
(32*c^12*d^13*e^3 - 208*b*c^11*d^12*e^4 + 576*b^2*c^10*d^11*e^5 - 880*b^3*c^9*d^10*e^6 + 800*b^4*c^8*d^9*e^7 -
432*b^5*c^7*d^8*e^8 + 128*b^6*c^6*d^7*e^9 - 16*b^7*c^5*d^6*e^10 + (((d + e*x)^(1/2)*(16*c^13*d^16*e^2 - 128*b
*c^12*d^15*e^3 + 480*b^2*c^11*d^14*e^4 - 1120*b^3*c^10*d^13*e^5 + 1800*b^4*c^9*d^12*e^6 - 2064*b^5*c^8*d^11*e^
7 + 1688*b^6*c^7*d^10*e^8 - 960*b^7*c^6*d^9*e^9 + 360*b^8*c^5*d^8*e^10 - 80*b^9*c^4*d^7*e^11 + 8*b^10*c^3*d^6*
e^12) + ((-c^5*(b*e - c*d)^5)^(1/2)*(24*b^2*c^12*d^18*e^3 - 216*b^3*c^11*d^17*e^4 + 872*b^4*c^10*d^16*e^5 - 20
80*b^5*c^9*d^15*e^6 + 3248*b^6*c^8*d^14*e^7 - 3472*b^7*c^7*d^13*e^8 + 2576*b^8*c^6*d^12*e^9 - 1312*b^9*c^5*d^1
1*e^10 + 440*b^10*c^4*d^10*e^11 - 88*b^11*c^3*d^9*e^12 + 8*b^12*c^2*d^8*e^13 - ((-c^5*(b*e - c*d)^5)^(1/2)*(d
+ e*x)^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 + 800*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 +
4320*b^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7*d^15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*
d^13*e^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 - 8*b^13*c^2*d^10*e^13))/(b*(b*e - c*d)^5)))/(b*(b*
e - c*d)^5))*(-c^5*(b*e - c*d)^5)^(1/2))/(b*(b*e - c*d)^5) - (((d + e*x)^(1/2)*(16*c^13*d^16*e^2 - 128*b*c^12*
d^15*e^3 + 480*b^2*c^11*d^14*e^4 - 1120*b^3*c^10*d^13*e^5 + 1800*b^4*c^9*d^12*e^6 - 2064*b^5*c^8*d^11*e^7 + 16
88*b^6*c^7*d^10*e^8 - 960*b^7*c^6*d^9*e^9 + 360*b^8*c^5*d^8*e^10 - 80*b^9*c^4*d^7*e^11 + 8*b^10*c^3*d^6*e^12)
- ((-c^5*(b*e - c*d)^5)^(1/2)*(24*b^2*c^12*d^18*e^3 - 216*b^3*c^11*d^17*e^4 + 872*b^4*c^10*d^16*e^5 - 2080*b^5
*c^9*d^15*e^6 + 3248*b^6*c^8*d^14*e^7 - 3472*b^7*c^7*d^13*e^8 + 2576*b^8*c^6*d^12*e^9 - 1312*b^9*c^5*d^11*e^10
+ 440*b^10*c^4*d^10*e^11 - 88*b^11*c^3*d^9*e^12 + 8*b^12*c^2*d^8*e^13 + ((-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)
^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 + 800*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 + 4320*b
^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7*d^15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*d^13*e
^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 - 8*b^13*c^2*d^10*e^13))/(b*(b*e - c*d)^5)))/(b*(b*e - c*
d)^5))*(-c^5*(b*e - c*d)^5)^(1/2))/(b*(b*e - c*d)^5)))*(-c^5*(b*e - c*d)^5)^(1/2)*2i)/(b*(b*e - c*d)^5) - (2*a
tanh((80*c^12*d^15*e^3*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d
^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^
7*c^5*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) + (2320*b^2*c^10*d^13*e^
5*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^
9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^7*c^5*d^6*e^10 + 880*
b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) - (5040*b^3*c^9*d^12*e^6*(d + e*x)^(1/2))/((d
^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*
c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^7*c^5*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176
*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) + (7296*b^4*c^8*d^11*e^7*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^1
3*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5
*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^7*c^5*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16
*b^10*c^2*d^3*e^13)) - (7376*b^5*c^7*d^10*e^8*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^1
2*e^4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^
6*c^6*d^7*e^9 - 2640*b^7*c^5*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) +
(5280*b^6*c^6*d^9*e^9*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d
^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^
7*c^5*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) - (2640*b^7*c^5*d^8*e^10
*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^9
*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^7*c^5*d^6*e^10 + 880*b
^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) + (880*b^8*c^4*d^7*e^11*(d + e*x)^(1/2))/((d^5
)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*c^
8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^7*c^5*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176*b
^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) - (176*b^9*c^3*d^6*e^12*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e
^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^
7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^7*c^5*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^
10*c^2*d^3*e^13)) + (16*b^10*c^2*d^5*e^13*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^
4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^
6*d^7*e^9 - 2640*b^7*c^5*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) - (64
0*b*c^11*d^14*e^4*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d^11*e
^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^7*c^5
*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13))))/(b*(d^5)^(1/2)) - ((2*e)/(3
*(c*d^2 - b*d*e)) - (2*e*(b*e - 2*c*d)*(d + e*x))/(c*d^2 - b*d*e)^2)/(d + e*x)^(3/2)

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sympy [A]  time = 21.14, size = 133, normalized size = 0.96 \begin {gather*} \frac {2 e}{3 d \left (d + e x\right )^{\frac {3}{2}} \left (b e - c d\right )} + \frac {2 e \left (b e - 2 c d\right )}{d^{2} \sqrt {d + e x} \left (b e - c d\right )^{2}} - \frac {2 c^{2} \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 )^{2}} + \frac {2 \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b d^{2} \sqrt {- d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e/(3*d*(d + e*x)**(3/2)*(b*e - c*d)) + 2*e*(b*e - 2*c*d)/(d**2*sqrt(d + e*x)*(b*e - c*d)**2) - 2*c**2*atan(s
qrt(d + e*x)/sqrt((b*e - c*d)/c))/(b*sqrt((b*e - c*d)/c)*(b*e - c*d)**2) + 2*atan(sqrt(d + e*x)/sqrt(-d))/(b*d
**2*sqrt(-d))

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