3.4.59 \(\int \frac {(d+e x)^{3/2}}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=246 \[ \frac {\sqrt {d+e x} (12 c x (2 c d-b e) (c d-b e)+b (12 c d-7 b e) (c d-b e))}{4 b^4 \left (b x+c x^2\right ) (c d-b e)}-\frac {\sqrt {d+e x} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {3 \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 \sqrt {d}}+\frac {3 \sqrt {c} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c d-b e}} \]

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Rubi [A]  time = 0.46, antiderivative size = 246, 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 = {738, 822, 826, 1166, 208} \begin {gather*} -\frac {3 \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 \sqrt {d}}+\frac {3 \sqrt {c} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c d-b e}}-\frac {\sqrt {d+e x} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (12 c x (2 c d-b e) (c d-b e)+b (12 c d-7 b e) (c d-b e))}{4 b^4 \left (b x+c x^2\right ) (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d + e*x]*(b*d + (2*c*d - b*e)*x))/(2*b^2*(b*x + c*x^2)^2) + (Sqrt[d + e*x]*(b*(12*c*d - 7*b*e)*(c*d - b
*e) + 12*c*(c*d - b*e)*(2*c*d - b*e)*x))/(4*b^4*(c*d - b*e)*(b*x + c*x^2)) - (3*(16*c^2*d^2 - 12*b*c*d*e + b^2
*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*Sqrt[d]) + (3*Sqrt[c]*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2)*ArcTa
nh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*Sqrt[c*d - b*e])

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 738

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

Rule 822

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])

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 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 {(d+e x)^{3/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} d (12 c d-7 b e)+\frac {5}{2} e (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} d (c d-b e) \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )+3 c d e (c d-b e) (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d (c d-b e)}\\ &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-3 c d^2 e (c d-b e) (2 c d-b e)+\frac {3}{4} d e (c d-b e) \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )+3 c d e (c d-b 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 )}{b^4 d (c d-b e)}\\ &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}+\frac {\left (3 c \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )\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 )}{4 b^5}-\frac {\left (3 c \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )\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 )}{4 b^5}\\ &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}-\frac {3 \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 \sqrt {d}}+\frac {3 \sqrt {c} \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c d-b e}}\\ \end {align*}

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Mathematica [A]  time = 0.53, size = 289, normalized size = 1.17 \begin {gather*} \frac {3 x^2 (b+c x)^2 \left (-b^3 e^3+13 b^2 c d e^2-28 b c^2 d^2 e+16 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-\sqrt {d} \left (3 \sqrt {c} x^2 (b+c x)^2 \sqrt {c d-b e} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )+b \sqrt {d+e x} \left (b^4 e (2 d+5 e x)+b^3 c \left (-2 d^2-13 d e x+19 e^2 x^2\right )+b^2 c^2 x \left (8 d^2-55 d e x+12 e^2 x^2\right )+36 b c^3 d x^2 (d-e x)+24 c^4 d^2 x^3\right )\right )}{4 b^5 \sqrt {d} x^2 (b+c x)^2 (b e-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(16*c^3*d^3 - 28*b*c^2*d^2*e + 13*b^2*c*d*e^2 - b^3*e^3)*x^2*(b + c*x)^2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] - S
qrt[d]*(b*Sqrt[d + e*x]*(24*c^4*d^2*x^3 + 36*b*c^3*d*x^2*(d - e*x) + b^4*e*(2*d + 5*e*x) + b^2*c^2*x*(8*d^2 -
55*d*e*x + 12*e^2*x^2) + b^3*c*(-2*d^2 - 13*d*e*x + 19*e^2*x^2)) + 3*Sqrt[c]*Sqrt[c*d - b*e]*(16*c^2*d^2 - 20*
b*c*d*e + 5*b^2*e^2)*x^2*(b + c*x)^2*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]]))/(4*b^5*Sqrt[d]*(-(c*d)
+ b*e)*x^2*(b + c*x)^2)

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IntegrateAlgebraic [A]  time = 1.33, size = 350, normalized size = 1.42 \begin {gather*} \frac {3 \left (5 b^2 \sqrt {c} e^2-20 b c^{3/2} d e+16 c^{5/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{4 b^5 \sqrt {b e-c d}}-\frac {3 \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 \sqrt {d}}-\frac {\sqrt {d+e x} \left (5 b^3 e^3 (d+e x)-3 b^3 d e^3+27 b^2 c d^2 e^2-46 b^2 c d e^2 (d+e x)+19 b^2 c e^2 (d+e x)^2-48 b c^2 d^3 e+108 b c^2 d^2 e (d+e x)-72 b c^2 d e (d+e x)^2+12 b c^2 e (d+e x)^3+24 c^3 d^4-72 c^3 d^3 (d+e x)+72 c^3 d^2 (d+e x)^2-24 c^3 d (d+e x)^3\right )}{4 b^4 e x^2 (b e+c (d+e x)-c d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[d + e*x]*(24*c^3*d^4 - 48*b*c^2*d^3*e + 27*b^2*c*d^2*e^2 - 3*b^3*d*e^3 - 72*c^3*d^3*(d + e*x) + 108
*b*c^2*d^2*e*(d + e*x) - 46*b^2*c*d*e^2*(d + e*x) + 5*b^3*e^3*(d + e*x) + 72*c^3*d^2*(d + e*x)^2 - 72*b*c^2*d*
e*(d + e*x)^2 + 19*b^2*c*e^2*(d + e*x)^2 - 24*c^3*d*(d + e*x)^3 + 12*b*c^2*e*(d + e*x)^3))/(b^4*e*x^2*(-(c*d)
+ b*e + c*(d + e*x))^2) + (3*(16*c^(5/2)*d^2 - 20*b*c^(3/2)*d*e + 5*b^2*Sqrt[c]*e^2)*ArcTan[(Sqrt[c]*Sqrt[-(c*
d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4*b^5*Sqrt[-(c*d) + b*e]) - (3*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*Arc
Tanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*Sqrt[d])

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fricas [A]  time = 0.54, size = 1673, normalized size = 6.80

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(3*((16*c^4*d^3 - 20*b*c^3*d^2*e + 5*b^2*c^2*d*e^2)*x^4 + 2*(16*b*c^3*d^3 - 20*b^2*c^2*d^2*e + 5*b^3*c*d*
e^2)*x^3 + (16*b^2*c^2*d^3 - 20*b^3*c*d^2*e + 5*b^4*d*e^2)*x^2)*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*((16*c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^
4 + 2*(16*b*c^3*d^2 - 12*b^2*c^2*d*e + b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*x^2)*sqrt(d)
*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*d^2 - 12*(2*b*c^3*d^2 - b^2*c^2*d*e)*x^3 - (36*b^2*c^
2*d^2 - 19*b^3*c*d*e)*x^2 - (8*b^3*c*d^2 - 5*b^4*d*e)*x)*sqrt(e*x + d))/(b^5*c^2*d*x^4 + 2*b^6*c*d*x^3 + b^7*d
*x^2), 1/8*(6*((16*c^4*d^3 - 20*b*c^3*d^2*e + 5*b^2*c^2*d*e^2)*x^4 + 2*(16*b*c^3*d^3 - 20*b^2*c^2*d^2*e + 5*b^
3*c*d*e^2)*x^3 + (16*b^2*c^2*d^3 - 20*b^3*c*d^2*e + 5*b^4*d*e^2)*x^2)*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*((16*c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*
b*c^3*d^2 - 12*b^2*c^2*d*e + b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*x^2)*sqrt(d)*log((e*x
- 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*d^2 - 12*(2*b*c^3*d^2 - b^2*c^2*d*e)*x^3 - (36*b^2*c^2*d^2 - 19
*b^3*c*d*e)*x^2 - (8*b^3*c*d^2 - 5*b^4*d*e)*x)*sqrt(e*x + d))/(b^5*c^2*d*x^4 + 2*b^6*c*d*x^3 + b^7*d*x^2), 1/8
*(6*((16*c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 12*b^2*c^2*d*e + b^3*c*e^2)*x^3 + (16*b
^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 3*((16*c^4*d^3 - 20*b*c^
3*d^2*e + 5*b^2*c^2*d*e^2)*x^4 + 2*(16*b*c^3*d^3 - 20*b^2*c^2*d^2*e + 5*b^3*c*d*e^2)*x^3 + (16*b^2*c^2*d^3 - 2
0*b^3*c*d^2*e + 5*b^4*d*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*s
qrt(c/(c*d - b*e)))/(c*x + b)) - 2*(2*b^4*d^2 - 12*(2*b*c^3*d^2 - b^2*c^2*d*e)*x^3 - (36*b^2*c^2*d^2 - 19*b^3*
c*d*e)*x^2 - (8*b^3*c*d^2 - 5*b^4*d*e)*x)*sqrt(e*x + d))/(b^5*c^2*d*x^4 + 2*b^6*c*d*x^3 + b^7*d*x^2), 1/4*(3*(
(16*c^4*d^3 - 20*b*c^3*d^2*e + 5*b^2*c^2*d*e^2)*x^4 + 2*(16*b*c^3*d^3 - 20*b^2*c^2*d^2*e + 5*b^3*c*d*e^2)*x^3
+ (16*b^2*c^2*d^3 - 20*b^3*c*d^2*e + 5*b^4*d*e^2)*x^2)*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*((16*c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 12*
b^2*c^2*d*e + b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sq
rt(-d)/d) - (2*b^4*d^2 - 12*(2*b*c^3*d^2 - b^2*c^2*d*e)*x^3 - (36*b^2*c^2*d^2 - 19*b^3*c*d*e)*x^2 - (8*b^3*c*d
^2 - 5*b^4*d*e)*x)*sqrt(e*x + d))/(b^5*c^2*d*x^4 + 2*b^6*c*d*x^3 + b^7*d*x^2)]

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giac [A]  time = 0.23, size = 392, normalized size = 1.59 \begin {gather*} -\frac {3 \, {\left (16 \, c^{3} d^{2} - 20 \, b c^{2} d e + 5 \, b^{2} c e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, \sqrt {-c^{2} d + b c e} b^{5}} + \frac {3 \, {\left (16 \, c^{2} d^{2} - 12 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{2} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e - 24 \, \sqrt {x e + d} c^{3} d^{4} e - 12 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{2} e^{2} + 72 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{2} d e^{2} - 108 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{2} + 48 \, \sqrt {x e + d} b c^{2} d^{3} e^{2} - 19 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c e^{3} + 46 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{3} - 27 \, \sqrt {x e + d} b^{2} c d^{2} e^{3} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{4} + 3 \, \sqrt {x e + d} b^{3} d e^{4}}{4 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(16*c^3*d^2 - 20*b*c^2*d*e + 5*b^2*c*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c
*e)*b^5) + 3/4*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) + 1/4*(24*(x*
e + d)^(7/2)*c^3*d*e - 72*(x*e + d)^(5/2)*c^3*d^2*e + 72*(x*e + d)^(3/2)*c^3*d^3*e - 24*sqrt(x*e + d)*c^3*d^4*
e - 12*(x*e + d)^(7/2)*b*c^2*e^2 + 72*(x*e + d)^(5/2)*b*c^2*d*e^2 - 108*(x*e + d)^(3/2)*b*c^2*d^2*e^2 + 48*sqr
t(x*e + d)*b*c^2*d^3*e^2 - 19*(x*e + d)^(5/2)*b^2*c*e^3 + 46*(x*e + d)^(3/2)*b^2*c*d*e^3 - 27*sqrt(x*e + d)*b^
2*c*d^2*e^3 - 5*(x*e + d)^(3/2)*b^3*e^4 + 3*sqrt(x*e + d)*b^3*d*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2
+ (x*e + d)*b*e - b*d*e)^2*b^4)

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maple [A]  time = 0.06, size = 414, normalized size = 1.68 \begin {gather*} -\frac {9 \sqrt {e x +d}\, c \,e^{3}}{4 \left (c e x +b e \right )^{2} b^{2}}+\frac {21 \sqrt {e x +d}\, c^{2} d \,e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {3 \sqrt {e x +d}\, c^{3} d^{2} e}{\left (c e x +b e \right )^{2} b^{4}}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} c^{2} e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {15 c \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c^{3} d e}{\left (c e x +b e \right )^{2} b^{4}}+\frac {15 c^{2} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{4}}-\frac {12 c^{3} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{5}}-\frac {3 e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} \sqrt {d}}+\frac {9 c \sqrt {d}\, e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4}}-\frac {12 c^{2} d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5}}+\frac {3 \sqrt {e x +d}\, d}{4 b^{3} x^{2}}-\frac {3 \sqrt {e x +d}\, c \,d^{2}}{b^{4} e \,x^{2}}-\frac {5 \left (e x +d \right )^{\frac {3}{2}}}{4 b^{3} x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c d}{b^{4} e \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/4*e^2*c^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)+3*e*c^3/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d-9/4*e^3*c/b^2/(c*e*x+b*
e)^2*(e*x+d)^(1/2)+21/4*e^2*c^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d-3*e*c^3/b^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^2-
15/4*e^2*c/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)+15*e*c^2/b^4/((b*e-c*d)*c)^(1/2
)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d-12*c^3/b^5/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)
*c)^(1/2)*c)*d^2-5/4/b^3/x^2*(e*x+d)^(3/2)+3/e/b^4/x^2*(e*x+d)^(3/2)*c*d-3/e/b^4/x^2*(e*x+d)^(1/2)*c*d^2+3/4/b
^3/x^2*(e*x+d)^(1/2)*d-3/4*e^2/b^3/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))+9*e/b^4*d^(1/2)*arctanh((e*x+d)^(1/2
)/d^(1/2))*c-12/b^5*d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*c^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((e*x+d)^(3/2)/(c*x^2+b*x)^3,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 = 0.79, size = 1880, normalized size = 7.64

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*(d + e*x)^(1/2)*(b^3*d*e^4 - 8*c^3*d^4*e + 16*b*c^2*d^3*e^2 - 9*b^2*c*d^2*e^3))/(4*b^4) - ((d + e*x)^(3/2)
*(5*b^3*e^4 - 72*c^3*d^3*e + 108*b*c^2*d^2*e^2 - 46*b^2*c*d*e^3))/(4*b^4) - (e*(d + e*x)^(5/2)*(72*c^3*d^2 + 1
9*b^2*c*e^2 - 72*b*c^2*d*e))/(4*b^4) + (3*c*e*(2*c^2*d - b*c*e)*(d + e*x)^(7/2))/b^4)/(c^2*(d + e*x)^4 - (d +
e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*
d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) - (3*atanh((27*c^2*e^9*(d + e*x)^(1/2))/(32*d^(3/2)*((
27*c^2*e^9)/(32*d) - (81*c^3*e^8)/(8*b) + (27*c^4*d*e^7)/(2*b^2))) - (81*c^3*e^8*(d + e*x)^(1/2))/(8*d^(1/2)*(
(27*b*c^2*e^9)/(32*d) - (81*c^3*e^8)/8 + (27*c^4*d*e^7)/(2*b))) + (27*c^4*d^(1/2)*e^7*(d + e*x)^(1/2))/(2*((27
*c^4*d*e^7)/2 - (81*b*c^3*e^8)/8 + (27*b^2*c^2*e^9)/(32*d))))*(b^2*e^2 + 16*c^2*d^2 - 12*b*c*d*e))/(4*b^5*d^(1
/2)) + (atan((((((d + e*x)^(1/2)*(117*b^4*c^3*e^6 + 2304*c^7*d^4*e^2 - 4608*b*c^6*d^3*e^3 - 1008*b^3*c^4*d*e^5
+ 3312*b^2*c^5*d^2*e^4))/(4*b^8) - (3*(-c*(b*e - c*d))^(1/2)*((3*b^12*c^2*e^5 - 24*b^11*c^3*d*e^4 + 24*b^10*c
^4*d^2*e^3)/b^12 - (3*(32*b^11*c^2*e^3 - 64*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(5*b^2*e^2
+ 16*c^2*d^2 - 20*b*c*d*e))/(32*b^8*(b^6*e - b^5*c*d)))*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5
*c*d)))*(-c*(b*e - c*d))^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e)*3i)/(8*(b^6*e - b^5*c*d)) + ((((d + e*x)^
(1/2)*(117*b^4*c^3*e^6 + 2304*c^7*d^4*e^2 - 4608*b*c^6*d^3*e^3 - 1008*b^3*c^4*d*e^5 + 3312*b^2*c^5*d^2*e^4))/(
4*b^8) + (3*(-c*(b*e - c*d))^(1/2)*((3*b^12*c^2*e^5 - 24*b^11*c^3*d*e^4 + 24*b^10*c^4*d^2*e^3)/b^12 + (3*(32*b
^11*c^2*e^3 - 64*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))
/(32*b^8*(b^6*e - b^5*c*d)))*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d)))*(-c*(b*e - c*d))^(1
/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e)*3i)/(8*(b^6*e - b^5*c*d)))/(((135*b^5*c^3*e^8)/8 - 1728*c^8*d^5*e^3
+ 4320*b*c^7*d^4*e^4 - (1215*b^4*c^4*d*e^7)/4 - 3996*b^2*c^6*d^3*e^5 + 1674*b^3*c^5*d^2*e^6)/b^12 - (3*(((d +
e*x)^(1/2)*(117*b^4*c^3*e^6 + 2304*c^7*d^4*e^2 - 4608*b*c^6*d^3*e^3 - 1008*b^3*c^4*d*e^5 + 3312*b^2*c^5*d^2*e^
4))/(4*b^8) - (3*(-c*(b*e - c*d))^(1/2)*((3*b^12*c^2*e^5 - 24*b^11*c^3*d*e^4 + 24*b^10*c^4*d^2*e^3)/b^12 - (3*
(32*b^11*c^2*e^3 - 64*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*
d*e))/(32*b^8*(b^6*e - b^5*c*d)))*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d)))*(-c*(b*e - c*d
))^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d)) + (3*(((d + e*x)^(1/2)*(117*b^4*c^3*e^6
+ 2304*c^7*d^4*e^2 - 4608*b*c^6*d^3*e^3 - 1008*b^3*c^4*d*e^5 + 3312*b^2*c^5*d^2*e^4))/(4*b^8) + (3*(-c*(b*e -
c*d))^(1/2)*((3*b^12*c^2*e^5 - 24*b^11*c^3*d*e^4 + 24*b^10*c^4*d^2*e^3)/b^12 + (3*(32*b^11*c^2*e^3 - 64*b^10*c
^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(32*b^8*(b^6*e - b^5*c
*d)))*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d)))*(-c*(b*e - c*d))^(1/2)*(5*b^2*e^2 + 16*c^2
*d^2 - 20*b*c*d*e))/(8*(b^6*e - b^5*c*d))))*(-c*(b*e - c*d))^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 20*b*c*d*e)*3i)/(
4*(b^6*e - b^5*c*d))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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