3.4.46 \(\int \frac {1}{(d+e x)^{7/2} (b x+c x^2)} \, dx\)

Optimal. Leaf size=187 \[ -\frac {2 e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{d^3 \sqrt {d+e x} (c d-b e)^3}+\frac {2 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{7/2}}-\frac {2 e (2 c d-b e)}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 e}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{7/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.38, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, 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 e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )}{d^3 \sqrt {d+e x} (c d-b e)^3}+\frac {2 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{7/2}}-\frac {2 e (2 c d-b e)}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {2 e}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(b*x + c*x^2)),x]

[Out]

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

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)^{7/2} \left (b x+c x^2\right )} \, dx &=-\frac {2 e}{5 d (c d-b e) (d+e x)^{5/2}}+\frac {\int \frac {c d-b e-c e x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{d (c d-b e)}\\ &=-\frac {2 e}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {2 e (2 c d-b e)}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {\int \frac {(c d-b e)^2-c e (2 c d-b e) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{d^2 (c d-b e)^2}\\ &=-\frac {2 e}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {2 e (2 c d-b e)}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )}{d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {\int \frac {(c d-b e)^3-c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{d^3 (c d-b e)^3}\\ &=-\frac {2 e}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {2 e (2 c d-b e)}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )}{d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {2 \operatorname {Subst}\left (\int \frac {e (c d-b e)^3+c d e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )-c e \left (3 c^2 d^2-3 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 )}{d^3 (c d-b e)^3}\\ &=-\frac {2 e}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {2 e (2 c d-b e)}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )}{d^3 (c d-b e)^3 \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^3}-\frac {\left (2 c^4\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)^3}\\ &=-\frac {2 e}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {2 e (2 c d-b e)}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )}{d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{7/2}}+\frac {2 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 83, normalized size = 0.44 \begin {gather*} -\frac {2 \left (c d \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {c (d+e x)}{c d-b e}\right )+(b e-c d) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {e x}{d}+1\right )\right )}{5 b d (d+e x)^{5/2} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*(b*x + c*x^2)),x]

[Out]

(-2*(c*d*Hypergeometric2F1[-5/2, 1, -3/2, (c*(d + e*x))/(c*d - b*e)] + (-(c*d) + b*e)*Hypergeometric2F1[-5/2,
1, -3/2, 1 + (e*x)/d]))/(5*b*d*(c*d - b*e)*(d + e*x)^(5/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.45, size = 225, normalized size = 1.20 \begin {gather*} -\frac {2 e \left (3 b^2 d^2 e^2+5 b^2 d e^2 (d+e x)+15 b^2 e^2 (d+e x)^2-6 b c d^3 e-15 b c d^2 e (d+e x)-45 b c d e (d+e x)^2+3 c^2 d^4+10 c^2 d^3 (d+e x)+45 c^2 d^2 (d+e x)^2\right )}{15 d^3 (d+e x)^{5/2} (c d-b e)^3}-\frac {2 c^{7/2} \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)^{7/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^(7/2)*(b*x + c*x^2)),x]

[Out]

(-2*e*(3*c^2*d^4 - 6*b*c*d^3*e + 3*b^2*d^2*e^2 + 10*c^2*d^3*(d + e*x) - 15*b*c*d^2*e*(d + e*x) + 5*b^2*d*e^2*(
d + e*x) + 45*c^2*d^2*(d + e*x)^2 - 45*b*c*d*e*(d + e*x)^2 + 15*b^2*e^2*(d + e*x)^2))/(15*d^3*(c*d - b*e)^3*(d
+ e*x)^(5/2)) - (2*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(b*(-(c*d) + b*e)^
(7/2)) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(7/2))

________________________________________________________________________________________

fricas [B]  time = 1.17, size = 2577, normalized size = 13.78

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

[-1/15*(15*(c^3*d^4*e^3*x^3 + 3*c^3*d^5*e^2*x^2 + 3*c^3*d^6*e*x + c^3*d^7)*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)) - 15*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*
c*d^4*e^2 - b^3*d^3*e^3 + (c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*b^2*c*d*e^5 - b^3*e^6)*x^3 + 3*(c^3*d^4*e^2 - 3*b
*c^2*d^3*e^3 + 3*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 3*(c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 - b^3*d^2*e
^4)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(58*b*c^2*d^5*e - 66*b^2*c*d^4*e^2 + 23*b^3*d^
3*e^3 + 15*(3*b*c^2*d^3*e^3 - 3*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 5*(20*b*c^2*d^4*e^2 - 21*b^2*c*d^3*e^3 + 7*b^
3*d^2*e^4)*x)*sqrt(e*x + d))/(b*c^3*d^10 - 3*b^2*c^2*d^9*e + 3*b^3*c*d^8*e^2 - b^4*d^7*e^3 + (b*c^3*d^7*e^3 -
3*b^2*c^2*d^6*e^4 + 3*b^3*c*d^5*e^5 - b^4*d^4*e^6)*x^3 + 3*(b*c^3*d^8*e^2 - 3*b^2*c^2*d^7*e^3 + 3*b^3*c*d^6*e^
4 - b^4*d^5*e^5)*x^2 + 3*(b*c^3*d^9*e - 3*b^2*c^2*d^8*e^2 + 3*b^3*c*d^7*e^3 - b^4*d^6*e^4)*x), 1/15*(30*(c^3*d
^4*e^3*x^3 + 3*c^3*d^5*e^2*x^2 + 3*c^3*d^6*e*x + c^3*d^7)*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)) + 15*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3 + (c^3*d^
3*e^3 - 3*b*c^2*d^2*e^4 + 3*b^2*c*d*e^5 - b^3*e^6)*x^3 + 3*(c^3*d^4*e^2 - 3*b*c^2*d^3*e^3 + 3*b^2*c*d^2*e^4 -
b^3*d*e^5)*x^2 + 3*(c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)*sqrt(d)*log((e*x - 2*sqrt(
e*x + d)*sqrt(d) + 2*d)/x) - 2*(58*b*c^2*d^5*e - 66*b^2*c*d^4*e^2 + 23*b^3*d^3*e^3 + 15*(3*b*c^2*d^3*e^3 - 3*b
^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 5*(20*b*c^2*d^4*e^2 - 21*b^2*c*d^3*e^3 + 7*b^3*d^2*e^4)*x)*sqrt(e*x + d))/(b*c
^3*d^10 - 3*b^2*c^2*d^9*e + 3*b^3*c*d^8*e^2 - b^4*d^7*e^3 + (b*c^3*d^7*e^3 - 3*b^2*c^2*d^6*e^4 + 3*b^3*c*d^5*e
^5 - b^4*d^4*e^6)*x^3 + 3*(b*c^3*d^8*e^2 - 3*b^2*c^2*d^7*e^3 + 3*b^3*c*d^6*e^4 - b^4*d^5*e^5)*x^2 + 3*(b*c^3*d
^9*e - 3*b^2*c^2*d^8*e^2 + 3*b^3*c*d^7*e^3 - b^4*d^6*e^4)*x), 1/15*(30*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*
e^2 - b^3*d^3*e^3 + (c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*b^2*c*d*e^5 - b^3*e^6)*x^3 + 3*(c^3*d^4*e^2 - 3*b*c^2*d
^3*e^3 + 3*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 3*(c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)
*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - 15*(c^3*d^4*e^3*x^3 + 3*c^3*d^5*e^2*x^2 + 3*c^3*d^6*e*x + c^3*d^7
)*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*(58*b*c^2*d^5*e - 66*b^2*c*d^4*e^2 + 23*b^3*d^3*e^3 + 15*(3*b*c^2*d^3*e^3 - 3*b^2*c*d^2*e^4 + b^3*d*e^5)*x^
2 + 5*(20*b*c^2*d^4*e^2 - 21*b^2*c*d^3*e^3 + 7*b^3*d^2*e^4)*x)*sqrt(e*x + d))/(b*c^3*d^10 - 3*b^2*c^2*d^9*e +
3*b^3*c*d^8*e^2 - b^4*d^7*e^3 + (b*c^3*d^7*e^3 - 3*b^2*c^2*d^6*e^4 + 3*b^3*c*d^5*e^5 - b^4*d^4*e^6)*x^3 + 3*(b
*c^3*d^8*e^2 - 3*b^2*c^2*d^7*e^3 + 3*b^3*c*d^6*e^4 - b^4*d^5*e^5)*x^2 + 3*(b*c^3*d^9*e - 3*b^2*c^2*d^8*e^2 + 3
*b^3*c*d^7*e^3 - b^4*d^6*e^4)*x), 2/15*(15*(c^3*d^4*e^3*x^3 + 3*c^3*d^5*e^2*x^2 + 3*c^3*d^6*e*x + c^3*d^7)*sqr
t(-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)) + 15*(c^3*d^6 - 3*b*c^
2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3 + (c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*b^2*c*d*e^5 - b^3*e^6)*x^3 + 3*(c
^3*d^4*e^2 - 3*b*c^2*d^3*e^3 + 3*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 3*(c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*b^2*c*d^3
*e^3 - b^3*d^2*e^4)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (58*b*c^2*d^5*e - 66*b^2*c*d^4*e^2 + 23*b^3
*d^3*e^3 + 15*(3*b*c^2*d^3*e^3 - 3*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 5*(20*b*c^2*d^4*e^2 - 21*b^2*c*d^3*e^3 + 7
*b^3*d^2*e^4)*x)*sqrt(e*x + d))/(b*c^3*d^10 - 3*b^2*c^2*d^9*e + 3*b^3*c*d^8*e^2 - b^4*d^7*e^3 + (b*c^3*d^7*e^3
- 3*b^2*c^2*d^6*e^4 + 3*b^3*c*d^5*e^5 - b^4*d^4*e^6)*x^3 + 3*(b*c^3*d^8*e^2 - 3*b^2*c^2*d^7*e^3 + 3*b^3*c*d^6
*e^4 - b^4*d^5*e^5)*x^2 + 3*(b*c^3*d^9*e - 3*b^2*c^2*d^8*e^2 + 3*b^3*c*d^7*e^3 - b^4*d^6*e^4)*x)]

________________________________________________________________________________________

giac [A]  time = 0.20, size = 288, normalized size = 1.54 \begin {gather*} -\frac {2 \, c^{4} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} c^{2} d^{2} e + 10 \, {\left (x e + d\right )} c^{2} d^{3} e + 3 \, c^{2} d^{4} e - 45 \, {\left (x e + d\right )}^{2} b c d e^{2} - 15 \, {\left (x e + d\right )} b c d^{2} e^{2} - 6 \, b c d^{3} e^{2} + 15 \, {\left (x e + d\right )}^{2} b^{2} e^{3} + 5 \, {\left (x e + d\right )} b^{2} d e^{3} + 3 \, b^{2} d^{2} e^{3}\right )}}{15 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {5}{2}}} + \frac {2 \, \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b \sqrt {-d} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x),x, algorithm="giac")

[Out]

-2*c^4*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b*c^3*d^3 - 3*b^2*c^2*d^2*e + 3*b^3*c*d*e^2 - b^4*e^3)*s
qrt(-c^2*d + b*c*e)) - 2/15*(45*(x*e + d)^2*c^2*d^2*e + 10*(x*e + d)*c^2*d^3*e + 3*c^2*d^4*e - 45*(x*e + d)^2*
b*c*d*e^2 - 15*(x*e + d)*b*c*d^2*e^2 - 6*b*c*d^3*e^2 + 15*(x*e + d)^2*b^2*e^3 + 5*(x*e + d)*b^2*d*e^3 + 3*b^2*
d^2*e^3)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*(x*e + d)^(5/2)) + 2*arctan(sqrt(x*e + d)/
sqrt(-d))/(b*sqrt(-d)*d^3)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 228, normalized size = 1.22 \begin {gather*} \frac {2 c^{4} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}\, b}+\frac {2 b^{2} e^{3}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{3}}-\frac {6 b c \,e^{2}}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d^{2}}+\frac {6 c^{2} e}{\left (b e -c d \right )^{3} \sqrt {e x +d}\, d}+\frac {2 b \,e^{2}}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} d^{2}}-\frac {4 c e}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} d}+\frac {2 e}{5 \left (b e -c d \right ) \left (e x +d \right )^{\frac {5}{2}} d}-\frac {2 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b \,d^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(c*x^2+b*x),x)

[Out]

2/(b*e-c*d)^3*c^4/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^(7/2)+2/3/(b*e-c*d)^2/d^2/(e*x+d)^(3/2)*b*e^2-4/3*e/(b*e-c*d)^2/d/(e*x+d)^(3/2)*c+2/(b*e-c*d)^3/d^3
/(e*x+d)^(1/2)*b^2*e^3-6/(b*e-c*d)^3/d^2/(e*x+d)^(1/2)*b*c*e^2+6*e/(b*e-c*d)^3/d/(e*x+d)^(1/2)*c^2+2/5*e/(b*e-
c*d)/d/(e*x+d)^(5/2)

________________________________________________________________________________________

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)^(7/2)/(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?

________________________________________________________________________________________

mupad [B]  time = 2.25, size = 4068, normalized size = 21.75

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)^(7/2)),x)

[Out]

(atan((((-c^7*(b*e - c*d)^7)^(1/2)*((d + e*x)^(1/2)*(16*c^18*d^24*e^2 - 192*b*c^17*d^23*e^3 + 1128*b^2*c^16*d^
22*e^4 - 4312*b^3*c^15*d^21*e^5 + 11928*b^4*c^14*d^20*e^6 - 25032*b^5*c^13*d^19*e^7 + 40712*b^6*c^12*d^18*e^8
- 51768*b^7*c^11*d^17*e^9 + 51552*b^8*c^10*d^16*e^10 - 40048*b^9*c^9*d^15*e^11 + 24024*b^10*c^8*d^14*e^12 - 10
920*b^11*c^7*d^13*e^13 + 3640*b^12*c^6*d^12*e^14 - 840*b^13*c^5*d^11*e^15 + 120*b^14*c^4*d^10*e^16 - 8*b^15*c^
3*d^9*e^17) - ((-c^7*(b*e - c*d)^7)^(1/2)*(432*b^3*c^16*d^26*e^4 - 32*b^2*c^17*d^27*e^3 - 2720*b^4*c^15*d^25*e
^5 + 10600*b^5*c^14*d^24*e^6 - 28608*b^6*c^13*d^23*e^7 + 56672*b^7*c^12*d^22*e^8 - 85184*b^8*c^11*d^21*e^9 + 9
9000*b^9*c^10*d^20*e^10 - 89760*b^10*c^9*d^19*e^11 + 63536*b^11*c^8*d^18*e^12 - 34848*b^12*c^7*d^17*e^13 + 145
52*b^13*c^6*d^16*e^14 - 4480*b^14*c^5*d^15*e^15 + 960*b^15*c^4*d^14*e^16 - 128*b^16*c^3*d^13*e^17 + 8*b^17*c^2
*d^12*e^18 + ((-c^7*(b*e - c*d)^7)^(1/2)*(d + e*x)^(1/2)*(16*b^2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 + 1800*
b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480*b^6*c^14*d^27*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b^8*c^
12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^10*c^10*d^23*e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b^12*c
^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^14*c^6*d^19*e^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*c^4*d
^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^15*e^18))/(b*(b*e - c*d)^7)))/(b*(b*e - c*d)^7))*1i)/(b*(b*e
- c*d)^7) + ((-c^7*(b*e - c*d)^7)^(1/2)*((d + e*x)^(1/2)*(16*c^18*d^24*e^2 - 192*b*c^17*d^23*e^3 + 1128*b^2*c^
16*d^22*e^4 - 4312*b^3*c^15*d^21*e^5 + 11928*b^4*c^14*d^20*e^6 - 25032*b^5*c^13*d^19*e^7 + 40712*b^6*c^12*d^18
*e^8 - 51768*b^7*c^11*d^17*e^9 + 51552*b^8*c^10*d^16*e^10 - 40048*b^9*c^9*d^15*e^11 + 24024*b^10*c^8*d^14*e^12
- 10920*b^11*c^7*d^13*e^13 + 3640*b^12*c^6*d^12*e^14 - 840*b^13*c^5*d^11*e^15 + 120*b^14*c^4*d^10*e^16 - 8*b^
15*c^3*d^9*e^17) - ((-c^7*(b*e - c*d)^7)^(1/2)*(32*b^2*c^17*d^27*e^3 - 432*b^3*c^16*d^26*e^4 + 2720*b^4*c^15*d
^25*e^5 - 10600*b^5*c^14*d^24*e^6 + 28608*b^6*c^13*d^23*e^7 - 56672*b^7*c^12*d^22*e^8 + 85184*b^8*c^11*d^21*e^
9 - 99000*b^9*c^10*d^20*e^10 + 89760*b^10*c^9*d^19*e^11 - 63536*b^11*c^8*d^18*e^12 + 34848*b^12*c^7*d^17*e^13
- 14552*b^13*c^6*d^16*e^14 + 4480*b^14*c^5*d^15*e^15 - 960*b^15*c^4*d^14*e^16 + 128*b^16*c^3*d^13*e^17 - 8*b^1
7*c^2*d^12*e^18 + ((-c^7*(b*e - c*d)^7)^(1/2)*(d + e*x)^(1/2)*(16*b^2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 +
1800*b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480*b^6*c^14*d^27*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b
^8*c^12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^10*c^10*d^23*e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b
^12*c^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^14*c^6*d^19*e^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*
c^4*d^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^15*e^18))/(b*(b*e - c*d)^7)))/(b*(b*e - c*d)^7))*1i)/(b*
(b*e - c*d)^7))/(48*c^17*d^20*e^3 - 480*b*c^16*d^19*e^4 + ((-c^7*(b*e - c*d)^7)^(1/2)*((d + e*x)^(1/2)*(16*c^1
8*d^24*e^2 - 192*b*c^17*d^23*e^3 + 1128*b^2*c^16*d^22*e^4 - 4312*b^3*c^15*d^21*e^5 + 11928*b^4*c^14*d^20*e^6 -
25032*b^5*c^13*d^19*e^7 + 40712*b^6*c^12*d^18*e^8 - 51768*b^7*c^11*d^17*e^9 + 51552*b^8*c^10*d^16*e^10 - 4004
8*b^9*c^9*d^15*e^11 + 24024*b^10*c^8*d^14*e^12 - 10920*b^11*c^7*d^13*e^13 + 3640*b^12*c^6*d^12*e^14 - 840*b^13
*c^5*d^11*e^15 + 120*b^14*c^4*d^10*e^16 - 8*b^15*c^3*d^9*e^17) - ((-c^7*(b*e - c*d)^7)^(1/2)*(432*b^3*c^16*d^2
6*e^4 - 32*b^2*c^17*d^27*e^3 - 2720*b^4*c^15*d^25*e^5 + 10600*b^5*c^14*d^24*e^6 - 28608*b^6*c^13*d^23*e^7 + 56
672*b^7*c^12*d^22*e^8 - 85184*b^8*c^11*d^21*e^9 + 99000*b^9*c^10*d^20*e^10 - 89760*b^10*c^9*d^19*e^11 + 63536*
b^11*c^8*d^18*e^12 - 34848*b^12*c^7*d^17*e^13 + 14552*b^13*c^6*d^16*e^14 - 4480*b^14*c^5*d^15*e^15 + 960*b^15*
c^4*d^14*e^16 - 128*b^16*c^3*d^13*e^17 + 8*b^17*c^2*d^12*e^18 + ((-c^7*(b*e - c*d)^7)^(1/2)*(d + e*x)^(1/2)*(1
6*b^2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 + 1800*b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480*b^6*c^14
*d^27*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b^8*c^12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^10*c^10*d
^23*e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b^12*c^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^14*c^6*d^
19*e^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*c^4*d^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^15*e^18))/
(b*(b*e - c*d)^7)))/(b*(b*e - c*d)^7)))/(b*(b*e - c*d)^7) - ((-c^7*(b*e - c*d)^7)^(1/2)*((d + e*x)^(1/2)*(16*c
^18*d^24*e^2 - 192*b*c^17*d^23*e^3 + 1128*b^2*c^16*d^22*e^4 - 4312*b^3*c^15*d^21*e^5 + 11928*b^4*c^14*d^20*e^6
- 25032*b^5*c^13*d^19*e^7 + 40712*b^6*c^12*d^18*e^8 - 51768*b^7*c^11*d^17*e^9 + 51552*b^8*c^10*d^16*e^10 - 40
048*b^9*c^9*d^15*e^11 + 24024*b^10*c^8*d^14*e^12 - 10920*b^11*c^7*d^13*e^13 + 3640*b^12*c^6*d^12*e^14 - 840*b^
13*c^5*d^11*e^15 + 120*b^14*c^4*d^10*e^16 - 8*b^15*c^3*d^9*e^17) - ((-c^7*(b*e - c*d)^7)^(1/2)*(32*b^2*c^17*d^
27*e^3 - 432*b^3*c^16*d^26*e^4 + 2720*b^4*c^15*d^25*e^5 - 10600*b^5*c^14*d^24*e^6 + 28608*b^6*c^13*d^23*e^7 -
56672*b^7*c^12*d^22*e^8 + 85184*b^8*c^11*d^21*e^9 - 99000*b^9*c^10*d^20*e^10 + 89760*b^10*c^9*d^19*e^11 - 6353
6*b^11*c^8*d^18*e^12 + 34848*b^12*c^7*d^17*e^13 - 14552*b^13*c^6*d^16*e^14 + 4480*b^14*c^5*d^15*e^15 - 960*b^1
5*c^4*d^14*e^16 + 128*b^16*c^3*d^13*e^17 - 8*b^17*c^2*d^12*e^18 + ((-c^7*(b*e - c*d)^7)^(1/2)*(d + e*x)^(1/2)*
(16*b^2*c^18*d^31*e^2 - 248*b^3*c^17*d^30*e^3 + 1800*b^4*c^16*d^29*e^4 - 8120*b^5*c^15*d^28*e^5 + 25480*b^6*c^
14*d^27*e^6 - 58968*b^7*c^13*d^26*e^7 + 104104*b^8*c^12*d^25*e^8 - 143000*b^9*c^11*d^24*e^9 + 154440*b^10*c^10
*d^23*e^10 - 131560*b^11*c^9*d^22*e^11 + 88088*b^12*c^8*d^21*e^12 - 45864*b^13*c^7*d^20*e^13 + 18200*b^14*c^6*
d^19*e^14 - 5320*b^15*c^5*d^18*e^15 + 1080*b^16*c^4*d^17*e^16 - 136*b^17*c^3*d^16*e^17 + 8*b^18*c^2*d^15*e^18)
)/(b*(b*e - c*d)^7)))/(b*(b*e - c*d)^7)))/(b*(b*e - c*d)^7) + 2176*b^2*c^15*d^18*e^5 - 5904*b^3*c^14*d^17*e^6
+ 10656*b^4*c^13*d^16*e^7 - 13440*b^5*c^12*d^15*e^8 + 12096*b^6*c^11*d^14*e^9 - 7776*b^7*c^10*d^13*e^10 + 3504
*b^8*c^9*d^12*e^11 - 1056*b^9*c^8*d^11*e^12 + 192*b^10*c^7*d^10*e^13 - 16*b^11*c^6*d^9*e^14))*(-c^7*(b*e - c*d
)^7)^(1/2)*2i)/(b*(b*e - c*d)^7) - (atan((b^16*d^15*e^16*(d + e*x)^(1/2)*1i - b*c^15*d^30*e*(d + e*x)^(1/2)*7i
- b^15*c*d^16*e^15*(d + e*x)^(1/2)*16i + b^2*c^14*d^29*e^2*(d + e*x)^(1/2)*84i - b^3*c^13*d^28*e^3*(d + e*x)^
(1/2)*476i + b^4*c^12*d^27*e^4*(d + e*x)^(1/2)*1694i - b^5*c^11*d^26*e^5*(d + e*x)^(1/2)*4242i + b^6*c^10*d^25
*e^6*(d + e*x)^(1/2)*7924i - b^7*c^9*d^24*e^7*(d + e*x)^(1/2)*11404i + b^8*c^8*d^23*e^8*(d + e*x)^(1/2)*12861i
- b^9*c^7*d^22*e^9*(d + e*x)^(1/2)*11439i + b^10*c^6*d^21*e^10*(d + e*x)^(1/2)*8008i - b^11*c^5*d^20*e^11*(d
+ e*x)^(1/2)*4368i + b^12*c^4*d^19*e^12*(d + e*x)^(1/2)*1820i - b^13*c^3*d^18*e^13*(d + e*x)^(1/2)*560i + b^14
*c^2*d^17*e^14*(d + e*x)^(1/2)*120i)/(d^7*(d^7)^(1/2)*(d^7*(d^7*(11404*b^7*c^9*e^7 - 7924*b^6*c^10*d*e^6 - 84*
b^2*c^14*d^5*e^2 + 476*b^3*c^13*d^4*e^3 - 1694*b^4*c^12*d^3*e^4 + 4242*b^5*c^11*d^2*e^5 + 7*b*c^15*d^6*e) - 12
0*b^14*c^2*e^14 + 560*b^13*c^3*d*e^13 - 12861*b^8*c^8*d^6*e^8 + 11439*b^9*c^7*d^5*e^9 - 8008*b^10*c^6*d^4*e^10
+ 4368*b^11*c^5*d^3*e^11 - 1820*b^12*c^4*d^2*e^12) - b^16*d^5*e^16 + 16*b^15*c*d^6*e^15)))*2i)/(b*(d^7)^(1/2)
) - ((2*e)/(5*(c*d^2 - b*d*e)) + (2*e*(d + e*x)^2*(b^2*e^2 + 3*c^2*d^2 - 3*b*c*d*e))/(c*d^2 - b*d*e)^3 - (2*e*
(b*e - 2*c*d)*(d + e*x))/(3*(c*d^2 - b*d*e)^2))/(d + e*x)^(5/2)

________________________________________________________________________________________

sympy [A]  time = 25.17, size = 182, normalized size = 0.97 \begin {gather*} \frac {2 e}{5 d \left (d + e x\right )^{\frac {5}{2}} \left (b e - c d\right )} + \frac {2 e \left (b e - 2 c d\right )}{3 d^{2} \left (d + e x\right )^{\frac {3}{2}} \left (b e - c d\right )^{2}} + \frac {2 e \left (b^{2} e^{2} - 3 b c d e + 3 c^{2} d^{2}\right )}{d^{3} \sqrt {d + e x} \left (b e - c d\right )^{3}} + \frac {2 c^{3} \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 )^{3}} + \frac {2 \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b d^{3} \sqrt {- d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(7/2)/(c*x**2+b*x),x)

[Out]

2*e/(5*d*(d + e*x)**(5/2)*(b*e - c*d)) + 2*e*(b*e - 2*c*d)/(3*d**2*(d + e*x)**(3/2)*(b*e - c*d)**2) + 2*e*(b**
2*e**2 - 3*b*c*d*e + 3*c**2*d**2)/(d**3*sqrt(d + e*x)*(b*e - c*d)**3) + 2*c**3*atan(sqrt(d + e*x)/sqrt((b*e -
c*d)/c))/(b*sqrt((b*e - c*d)/c)*(b*e - c*d)**3) + 2*atan(sqrt(d + e*x)/sqrt(-d))/(b*d**3*sqrt(-d))

________________________________________________________________________________________