3.4.68 \(\int \frac {1}{(d+e x)^{7/2} (b x+c x^2)} \, dx\) [368]

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

[Out]

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

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Rubi [A]
time = 0.25, 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 = {723, 842, 840, 1180, 214} \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 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 842

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 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)^{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 \text {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) \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^3}-\frac {\left (2 c^4\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)^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*}

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Mathematica [A]
time = 0.65, size = 182, normalized size = 0.97 \begin {gather*} \frac {2 e \left (-3 b c d e \left (22 d^2+35 d e x+15 e^2 x^2\right )+b^2 e^2 \left (23 d^2+35 d e x+15 e^2 x^2\right )+c^2 d^2 \left (58 d^2+100 d e x+45 e^2 x^2\right )\right )}{15 d^3 (-c d+b e)^3 (d+e x)^{5/2}}+\frac {2 c^{7/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{b (-c d+b e)^{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]

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

[Out]

(2*e*(-3*b*c*d*e*(22*d^2 + 35*d*e*x + 15*e^2*x^2) + b^2*e^2*(23*d^2 + 35*d*e*x + 15*e^2*x^2) + c^2*d^2*(58*d^2
+ 100*d*e*x + 45*e^2*x^2)))/(15*d^3*(-(c*d) + b*e)^3*(d + e*x)^(5/2)) + (2*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[d + e
*x])/Sqrt[-(c*d) + b*e]])/(b*(-(c*d) + b*e)^(7/2)) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(7/2))

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Maple [A]
time = 0.51, size = 180, normalized size = 0.96

method	result	size
derivativedivides	\(2 e \left (-\frac {\arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e b \,d^{\frac {7}{2}}}-\frac {-b e +2 c d}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {-b^{2} e^{2}+3 b c d e -3 d^{2} c^{2}}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {1}{5 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {c^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} b e \sqrt {\left (b e -c d \right ) c}}\right )\)	\(180\)
default	\(2 e \left (-\frac {\arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e b \,d^{\frac {7}{2}}}-\frac {-b e +2 c d}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {-b^{2} e^{2}+3 b c d e -3 d^{2} c^{2}}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {1}{5 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {c^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{3} b e \sqrt {\left (b e -c d \right ) c}}\right )\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e*(-1/e/b/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))-1/3*(-b*e+2*c*d)/d^2/(b*e-c*d)^2/(e*x+d)^(3/2)-1/d^3/(b*e-c
*d)^3*(-b^2*e^2+3*b*c*d*e-3*c^2*d^2)/(e*x+d)^(1/2)+1/5/d/(b*e-c*d)/(e*x+d)^(5/2)+1/(b*e-c*d)^3*c^4/b/e/((b*e-c
*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(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)^(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(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. 627 vs. \(2 (175) = 350\).
time = 2.30, size = 2541, normalized size = 13.59 \begin {gather*} \text {Too large to display} \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="fricas")

[Out]

[-1/15*(15*(c^3*d^4*x^3*e^3 + 3*c^3*d^5*x^2*e^2 + 3*c^3*d^6*x*e + c^3*d^7)*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)) - 15*(c^3*d^6 - b^3*x^3*e^6 + 3*(b^2*c
*d*x^3 - b^3*d*x^2)*e^5 - 3*(b*c^2*d^2*x^3 - 3*b^2*c*d^2*x^2 + b^3*d^2*x)*e^4 + (c^3*d^3*x^3 - 9*b*c^2*d^3*x^2
+ 9*b^2*c*d^3*x - b^3*d^3)*e^3 + 3*(c^3*d^4*x^2 - 3*b*c^2*d^4*x + b^2*c*d^4)*e^2 + 3*(c^3*d^5*x - b*c^2*d^5)*
e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(58*b*c^2*d^5*e + 15*b^3*d*x^2*e^5 - 5*(9*b^2*c*d^
2*x^2 - 7*b^3*d^2*x)*e^4 + (45*b*c^2*d^3*x^2 - 105*b^2*c*d^3*x + 23*b^3*d^3)*e^3 + 2*(50*b*c^2*d^4*x - 33*b^2*
c*d^4)*e^2)*sqrt(x*e + d))/(b*c^3*d^10 - b^4*d^4*x^3*e^6 + 3*(b^3*c*d^5*x^3 - b^4*d^5*x^2)*e^5 - 3*(b^2*c^2*d^
6*x^3 - 3*b^3*c*d^6*x^2 + b^4*d^6*x)*e^4 + (b*c^3*d^7*x^3 - 9*b^2*c^2*d^7*x^2 + 9*b^3*c*d^7*x - b^4*d^7)*e^3 +
3*(b*c^3*d^8*x^2 - 3*b^2*c^2*d^8*x + b^3*c*d^8)*e^2 + 3*(b*c^3*d^9*x - b^2*c^2*d^9)*e), 1/15*(30*(c^3*d^4*x^3
*e^3 + 3*c^3*d^5*x^2*e^2 + 3*c^3*d^6*x*e + c^3*d^7)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(x*e + d)*sqr
t(-c/(c*d - b*e))/(c*x*e + c*d)) + 15*(c^3*d^6 - b^3*x^3*e^6 + 3*(b^2*c*d*x^3 - b^3*d*x^2)*e^5 - 3*(b*c^2*d^2*
x^3 - 3*b^2*c*d^2*x^2 + b^3*d^2*x)*e^4 + (c^3*d^3*x^3 - 9*b*c^2*d^3*x^2 + 9*b^2*c*d^3*x - b^3*d^3)*e^3 + 3*(c^
3*d^4*x^2 - 3*b*c^2*d^4*x + b^2*c*d^4)*e^2 + 3*(c^3*d^5*x - b*c^2*d^5)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*s
qrt(d) + 2*d)/x) - 2*(58*b*c^2*d^5*e + 15*b^3*d*x^2*e^5 - 5*(9*b^2*c*d^2*x^2 - 7*b^3*d^2*x)*e^4 + (45*b*c^2*d^
3*x^2 - 105*b^2*c*d^3*x + 23*b^3*d^3)*e^3 + 2*(50*b*c^2*d^4*x - 33*b^2*c*d^4)*e^2)*sqrt(x*e + d))/(b*c^3*d^10
- b^4*d^4*x^3*e^6 + 3*(b^3*c*d^5*x^3 - b^4*d^5*x^2)*e^5 - 3*(b^2*c^2*d^6*x^3 - 3*b^3*c*d^6*x^2 + b^4*d^6*x)*e^
4 + (b*c^3*d^7*x^3 - 9*b^2*c^2*d^7*x^2 + 9*b^3*c*d^7*x - b^4*d^7)*e^3 + 3*(b*c^3*d^8*x^2 - 3*b^2*c^2*d^8*x + b
^3*c*d^8)*e^2 + 3*(b*c^3*d^9*x - b^2*c^2*d^9)*e), 1/15*(30*(c^3*d^6 - b^3*x^3*e^6 + 3*(b^2*c*d*x^3 - b^3*d*x^2
)*e^5 - 3*(b*c^2*d^2*x^3 - 3*b^2*c*d^2*x^2 + b^3*d^2*x)*e^4 + (c^3*d^3*x^3 - 9*b*c^2*d^3*x^2 + 9*b^2*c*d^3*x -
b^3*d^3)*e^3 + 3*(c^3*d^4*x^2 - 3*b*c^2*d^4*x + b^2*c*d^4)*e^2 + 3*(c^3*d^5*x - b*c^2*d^5)*e)*sqrt(-d)*arctan
(sqrt(x*e + d)*sqrt(-d)/d) - 15*(c^3*d^4*x^3*e^3 + 3*c^3*d^5*x^2*e^2 + 3*c^3*d^6*x*e + c^3*d^7)*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)) - 2*(58*b*c^2*d^5
*e + 15*b^3*d*x^2*e^5 - 5*(9*b^2*c*d^2*x^2 - 7*b^3*d^2*x)*e^4 + (45*b*c^2*d^3*x^2 - 105*b^2*c*d^3*x + 23*b^3*d
^3)*e^3 + 2*(50*b*c^2*d^4*x - 33*b^2*c*d^4)*e^2)*sqrt(x*e + d))/(b*c^3*d^10 - b^4*d^4*x^3*e^6 + 3*(b^3*c*d^5*x
^3 - b^4*d^5*x^2)*e^5 - 3*(b^2*c^2*d^6*x^3 - 3*b^3*c*d^6*x^2 + b^4*d^6*x)*e^4 + (b*c^3*d^7*x^3 - 9*b^2*c^2*d^7
*x^2 + 9*b^3*c*d^7*x - b^4*d^7)*e^3 + 3*(b*c^3*d^8*x^2 - 3*b^2*c^2*d^8*x + b^3*c*d^8)*e^2 + 3*(b*c^3*d^9*x - b
^2*c^2*d^9)*e), 2/15*(15*(c^3*d^4*x^3*e^3 + 3*c^3*d^5*x^2*e^2 + 3*c^3*d^6*x*e + c^3*d^7)*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)) + 15*(c^3*d^6 - b^3*x^3*e^6 + 3*(b^2*c*d
*x^3 - b^3*d*x^2)*e^5 - 3*(b*c^2*d^2*x^3 - 3*b^2*c*d^2*x^2 + b^3*d^2*x)*e^4 + (c^3*d^3*x^3 - 9*b*c^2*d^3*x^2 +
9*b^2*c*d^3*x - b^3*d^3)*e^3 + 3*(c^3*d^4*x^2 - 3*b*c^2*d^4*x + b^2*c*d^4)*e^2 + 3*(c^3*d^5*x - b*c^2*d^5)*e)
*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) - (58*b*c^2*d^5*e + 15*b^3*d*x^2*e^5 - 5*(9*b^2*c*d^2*x^2 - 7*b^3*d
^2*x)*e^4 + (45*b*c^2*d^3*x^2 - 105*b^2*c*d^3*x + 23*b^3*d^3)*e^3 + 2*(50*b*c^2*d^4*x - 33*b^2*c*d^4)*e^2)*sqr
t(x*e + d))/(b*c^3*d^10 - b^4*d^4*x^3*e^6 + 3*(b^3*c*d^5*x^3 - b^4*d^5*x^2)*e^5 - 3*(b^2*c^2*d^6*x^3 - 3*b^3*c
*d^6*x^2 + b^4*d^6*x)*e^4 + (b*c^3*d^7*x^3 - 9*b^2*c^2*d^7*x^2 + 9*b^3*c*d^7*x - b^4*d^7)*e^3 + 3*(b*c^3*d^8*x
^2 - 3*b^2*c^2*d^8*x + b^3*c*d^8)*e^2 + 3*(b*c^3*d^9*x - b^2*c^2*d^9)*e)]

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Sympy [A]
time = 8.80, 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))

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Giac [A]
time = 1.26, 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)

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Mupad [B]
time = 2.25, size = 2500, normalized size = 13.37 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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