3.11.98 \(\int \frac {A+B x}{\sqrt {d+e x} (b x+c x^2)^2} \, dx\)

Optimal. Leaf size=188 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (-A b e-4 A c d+2 b B d)}{b^3 d^{3/2}}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)}+\frac {\sqrt {c} \left (5 A b c e-4 A c^2 d-3 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{3/2}} \]

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Rubi [A]  time = 0.34, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {822, 826, 1166, 208} \begin {gather*} \frac {\sqrt {c} \left (5 A b c e-4 A c^2 d-3 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (-A b e-4 A c d+2 b B d)}{b^3 d^{3/2}}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[d + e*x]*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(b*x + c*x^2))) - ((2*b
*B*d - 4*A*c*d - A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(3/2)) + (Sqrt[c]*(2*b*B*c*d - 4*A*c^2*d - 3*b^
2*B*e + 5*A*b*c*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d - b*e)^(3/2))

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

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Mathematica [A]  time = 0.43, size = 231, normalized size = 1.23 \begin {gather*} \frac {-\frac {2 \sqrt {c} d \left (-b c (5 A e+2 B d)+4 A c^2 d+3 b^2 B e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^2 (c d-b e)^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (-A b e-4 A c d+2 b B d)}{b^2 \sqrt {d}}+\frac {c \sqrt {d+e x} (A b e+4 A c d-2 b B d)}{b (b+c x) (b e-c d)}+\frac {3 A c e \sqrt {d+e x}}{(b+c x) (c d-b e)}-\frac {2 A \sqrt {d+e x}}{x (b+c x)}}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*A*c*e*Sqrt[d + e*x])/((c*d - b*e)*(b + c*x)) + (c*(-2*b*B*d + 4*A*c*d + A*b*e)*Sqrt[d + e*x])/(b*(-(c*d) +
b*e)*(b + c*x)) - (2*A*Sqrt[d + e*x])/(x*(b + c*x)) - (2*(2*b*B*d - 4*A*c*d - A*b*e)*ArcTanh[Sqrt[d + e*x]/Sq
rt[d]])/(b^2*Sqrt[d]) - (2*Sqrt[c]*d*(4*A*c^2*d + 3*b^2*B*e - b*c*(2*B*d + 5*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e
*x])/Sqrt[c*d - b*e]])/(b^2*(c*d - b*e)^(3/2)))/(2*b*d)

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IntegrateAlgebraic [A]  time = 1.30, size = 249, normalized size = 1.32 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (A b e+4 A c d-2 b B d)}{b^3 d^{3/2}}-\frac {\sqrt {d+e x} \left (A b^2 e^2+A b c e (d+e x)-2 A b c d e+2 A c^2 d^2-2 A c^2 d (d+e x)-b B c d^2+b B c d (d+e x)\right )}{b^2 d x (b e-c d) (b e+c (d+e x)-c d)}+\frac {\left (-5 A b c^{3/2} e+4 A c^{5/2} d+3 b^2 B \sqrt {c} e-2 b B c^{3/2} d\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 (b e-c d)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[d + e*x]*(-(b*B*c*d^2) + 2*A*c^2*d^2 - 2*A*b*c*d*e + A*b^2*e^2 + b*B*c*d*(d + e*x) - 2*A*c^2*d*(d + e*
x) + A*b*c*e*(d + e*x)))/(b^2*d*(-(c*d) + b*e)*x*(-(c*d) + b*e + c*(d + e*x)))) + ((-2*b*B*c^(3/2)*d + 4*A*c^(
5/2)*d + 3*b^2*B*Sqrt[c]*e - 5*A*b*c^(3/2)*e)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/
(b^3*(-(c*d) + b*e)^(3/2)) + ((-2*b*B*d + 4*A*c*d + A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(3/2))

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fricas [B]  time = 2.92, size = 1540, normalized size = 8.19

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(((2*(B*b*c^2 - 2*A*c^3)*d^3 - (3*B*b^2*c - 5*A*b*c^2)*d^2*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d^3 - (3*B*
b^3 - 5*A*b^2*c)*d^2*e)*x)*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)) + ((A*b^2*c*e^2 + 2*(B*b*c^2 - 2*A*c^3)*d^2 - (2*B*b^2*c - 3*A*b*c^2)*d*e)*x^2 + (A*b^
3*e^2 + 2*(B*b^2*c - 2*A*b*c^2)*d^2 - (2*B*b^3 - 3*A*b^2*c)*d*e)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d)
+ 2*d)/x) + 2*(A*b^2*c*d^2 - A*b^3*d*e - (A*b^2*c*d*e + (B*b^2*c - 2*A*b*c^2)*d^2)*x)*sqrt(e*x + d))/((b^3*c^
2*d^3 - b^4*c*d^2*e)*x^2 + (b^4*c*d^3 - b^5*d^2*e)*x), 1/2*(2*((2*(B*b*c^2 - 2*A*c^3)*d^3 - (3*B*b^2*c - 5*A*b
*c^2)*d^2*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d^3 - (3*B*b^3 - 5*A*b^2*c)*d^2*e)*x)*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)) - ((A*b^2*c*e^2 + 2*(B*b*c^2 - 2*A*c^3)*d^2 - (
2*B*b^2*c - 3*A*b*c^2)*d*e)*x^2 + (A*b^3*e^2 + 2*(B*b^2*c - 2*A*b*c^2)*d^2 - (2*B*b^3 - 3*A*b^2*c)*d*e)*x)*sqr
t(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(A*b^2*c*d^2 - A*b^3*d*e - (A*b^2*c*d*e + (B*b^2*c - 2*A
*b*c^2)*d^2)*x)*sqrt(e*x + d))/((b^3*c^2*d^3 - b^4*c*d^2*e)*x^2 + (b^4*c*d^3 - b^5*d^2*e)*x), 1/2*(2*((A*b^2*c
*e^2 + 2*(B*b*c^2 - 2*A*c^3)*d^2 - (2*B*b^2*c - 3*A*b*c^2)*d*e)*x^2 + (A*b^3*e^2 + 2*(B*b^2*c - 2*A*b*c^2)*d^2
- (2*B*b^3 - 3*A*b^2*c)*d*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - ((2*(B*b*c^2 - 2*A*c^3)*d^3 - (3*
B*b^2*c - 5*A*b*c^2)*d^2*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d^3 - (3*B*b^3 - 5*A*b^2*c)*d^2*e)*x)*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*(A*b^2*c*d^2
- A*b^3*d*e - (A*b^2*c*d*e + (B*b^2*c - 2*A*b*c^2)*d^2)*x)*sqrt(e*x + d))/((b^3*c^2*d^3 - b^4*c*d^2*e)*x^2 +
(b^4*c*d^3 - b^5*d^2*e)*x), (((2*(B*b*c^2 - 2*A*c^3)*d^3 - (3*B*b^2*c - 5*A*b*c^2)*d^2*e)*x^2 + (2*(B*b^2*c -
2*A*b*c^2)*d^3 - (3*B*b^3 - 5*A*b^2*c)*d^2*e)*x)*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)) + ((A*b^2*c*e^2 + 2*(B*b*c^2 - 2*A*c^3)*d^2 - (2*B*b^2*c - 3*A*b*c^2)*d*e)*x^2 +
(A*b^3*e^2 + 2*(B*b^2*c - 2*A*b*c^2)*d^2 - (2*B*b^3 - 3*A*b^2*c)*d*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-
d)/d) - (A*b^2*c*d^2 - A*b^3*d*e - (A*b^2*c*d*e + (B*b^2*c - 2*A*b*c^2)*d^2)*x)*sqrt(e*x + d))/((b^3*c^2*d^3 -
b^4*c*d^2*e)*x^2 + (b^4*c*d^3 - b^5*d^2*e)*x)]

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giac [A]  time = 0.24, size = 315, normalized size = 1.68 \begin {gather*} -\frac {{\left (2 \, B b c^{2} d - 4 \, A c^{3} d - 3 \, B b^{2} c e + 5 \, A b c^{2} e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c d - b^{4} e\right )} \sqrt {-c^{2} d + b c e}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} B b c d e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{2} d e - \sqrt {x e + d} B b c d^{2} e + 2 \, \sqrt {x e + d} A c^{2} d^{2} e + {\left (x e + d\right )}^{\frac {3}{2}} A b c e^{2} - 2 \, \sqrt {x e + d} A b c d e^{2} + \sqrt {x e + d} A b^{2} e^{3}}{{\left (b^{2} c d^{2} - b^{3} d e\right )} {\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 )}} + \frac {{\left (2 \, B b d - 4 \, A c d - A b e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*B*b*c^2*d - 4*A*c^3*d - 3*B*b^2*c*e + 5*A*b*c^2*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c*d
- b^4*e)*sqrt(-c^2*d + b*c*e)) + ((x*e + d)^(3/2)*B*b*c*d*e - 2*(x*e + d)^(3/2)*A*c^2*d*e - sqrt(x*e + d)*B*b*
c*d^2*e + 2*sqrt(x*e + d)*A*c^2*d^2*e + (x*e + d)^(3/2)*A*b*c*e^2 - 2*sqrt(x*e + d)*A*b*c*d*e^2 + sqrt(x*e + d
)*A*b^2*e^3)/((b^2*c*d^2 - b^3*d*e)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)) + (2*B*
b*d - 4*A*c*d - A*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)*d)

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maple [B]  time = 0.07, size = 370, normalized size = 1.97 \begin {gather*} \frac {5 A \,c^{2} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {4 A \,c^{3} d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{3}}-\frac {3 B c e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b}+\frac {2 B \,c^{2} d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {\sqrt {e x +d}\, A \,c^{2} e}{\left (b e -c d \right ) \left (c e x +b e \right ) b^{2}}-\frac {\sqrt {e x +d}\, B c e}{\left (b e -c d \right ) \left (c e x +b e \right ) b}+\frac {A e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} d^{\frac {3}{2}}}+\frac {4 A c \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3} \sqrt {d}}-\frac {2 B \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} \sqrt {d}}-\frac {\sqrt {e x +d}\, A}{b^{2} d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x)^2,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 = 4.28, size = 5828, normalized size = 31.00

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d + e*x)^(1/2)*(A*b^2*e^3 + 2*A*c^2*d^2*e - 2*A*b*c*d*e^2 - B*b*c*d^2*e))/(b^2*(c*d^2 - b*d*e)) + (c*(d + e
*x)^(3/2)*(A*b*e^2 - 2*A*c*d*e + B*b*d*e))/(b^2*(c*d^2 - b*d*e)))/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + c
*d^2 - b*d*e) + (atan(((((2*(d + e*x)^(1/2)*(A^2*b^4*c^3*e^6 + 32*A^2*c^7*d^4*e^2 + 26*A^2*b^2*c^5*d^2*e^4 + 8
*B^2*b^2*c^5*d^4*e^2 - 20*B^2*b^3*c^4*d^3*e^3 + 13*B^2*b^4*c^3*d^2*e^4 - 64*A^2*b*c^6*d^3*e^3 + 6*A^2*b^3*c^4*
d*e^5 - 32*A*B*b*c^6*d^4*e^2 - 4*A*B*b^4*c^3*d*e^5 + 72*A*B*b^2*c^5*d^3*e^3 - 38*A*B*b^3*c^4*d^2*e^4))/(b^4*c^
2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) - (((4*A*b^9*c^2*d*e^6 + 8*A*b^6*c^5*d^4*e^3 - 16*A*b^7*c^4*d^3*e^4 + 4*A
*b^8*c^3*d^2*e^5 - 4*B*b^7*c^4*d^4*e^3 + 12*B*b^8*c^3*d^3*e^4 - 8*B*b^9*c^2*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^
2 - 2*b^7*c*d^3*e) + ((-c*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^
8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d))/((b^4*c^2*d^4 + b^6*d^2*e^
2 - 2*b^5*c*d^3*e)*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(-c*(b*e - c*d)^3)^(1/2)*(4*A*c
^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d))/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(-c*
(b*e - c*d)^3)^(1/2)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d)*1i)/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2
*d^2*e - 3*b^5*c*d*e^2)) + (((2*(d + e*x)^(1/2)*(A^2*b^4*c^3*e^6 + 32*A^2*c^7*d^4*e^2 + 26*A^2*b^2*c^5*d^2*e^4
+ 8*B^2*b^2*c^5*d^4*e^2 - 20*B^2*b^3*c^4*d^3*e^3 + 13*B^2*b^4*c^3*d^2*e^4 - 64*A^2*b*c^6*d^3*e^3 + 6*A^2*b^3*
c^4*d*e^5 - 32*A*B*b*c^6*d^4*e^2 - 4*A*B*b^4*c^3*d*e^5 + 72*A*B*b^2*c^5*d^3*e^3 - 38*A*B*b^3*c^4*d^2*e^4))/(b^
4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) + (((4*A*b^9*c^2*d*e^6 + 8*A*b^6*c^5*d^4*e^3 - 16*A*b^7*c^4*d^3*e^4 +
4*A*b^8*c^3*d^2*e^5 - 4*B*b^7*c^4*d^4*e^3 + 12*B*b^8*c^3*d^3*e^4 - 8*B*b^9*c^2*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^
2*e^2 - 2*b^7*c*d^3*e) - ((-c*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 1
6*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d))/((b^4*c^2*d^4 + b^6*d^
2*e^2 - 2*b^5*c*d^3*e)*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(-c*(b*e - c*d)^3)^(1/2)*(4
*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d))/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*
(-c*(b*e - c*d)^3)^(1/2)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d)*1i)/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4
*c^2*d^2*e - 3*b^5*c*d*e^2)))/((2*(5*A^3*b^3*c^4*e^6 + 32*A^3*c^7*d^3*e^3 - 4*B^3*b^3*c^4*d^3*e^3 + 6*B^3*b^4*
c^3*d^2*e^4 - 3*A^2*B*b^4*c^3*e^6 - 48*A^3*b*c^6*d^2*e^4 + 6*A^3*b^2*c^5*d*e^5 + 24*A*B^2*b^2*c^5*d^3*e^3 - 36
*A*B^2*b^3*c^4*d^2*e^4 + 72*A^2*B*b^2*c^5*d^2*e^4 + 3*A*B^2*b^4*c^3*d*e^5 - 48*A^2*B*b*c^6*d^3*e^3 - 9*A^2*B*b
^3*c^4*d*e^5))/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + (((2*(d + e*x)^(1/2)*(A^2*b^4*c^3*e^6 + 32*A^2*c^
7*d^4*e^2 + 26*A^2*b^2*c^5*d^2*e^4 + 8*B^2*b^2*c^5*d^4*e^2 - 20*B^2*b^3*c^4*d^3*e^3 + 13*B^2*b^4*c^3*d^2*e^4 -
64*A^2*b*c^6*d^3*e^3 + 6*A^2*b^3*c^4*d*e^5 - 32*A*B*b*c^6*d^4*e^2 - 4*A*B*b^4*c^3*d*e^5 + 72*A*B*b^2*c^5*d^3*
e^3 - 38*A*B*b^3*c^4*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) - (((4*A*b^9*c^2*d*e^6 + 8*A*b^6*c^
5*d^4*e^3 - 16*A*b^7*c^4*d^3*e^4 + 4*A*b^8*c^3*d^2*e^5 - 4*B*b^7*c^4*d^4*e^3 + 12*B*b^8*c^3*d^3*e^4 - 8*B*b^9*
c^2*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + ((-c*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(8*b^6*c^
5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e -
2*B*b*c*d))/((b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*
e^2)))*(-c*(b*e - c*d)^3)^(1/2)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d))/(2*(b^6*e^3 - b^3*c^3*d^3 + 3
*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(-c*(b*e - c*d)^3)^(1/2)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d))/(2
*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)) - (((2*(d + e*x)^(1/2)*(A^2*b^4*c^3*e^6 + 32*A^2*c
^7*d^4*e^2 + 26*A^2*b^2*c^5*d^2*e^4 + 8*B^2*b^2*c^5*d^4*e^2 - 20*B^2*b^3*c^4*d^3*e^3 + 13*B^2*b^4*c^3*d^2*e^4
- 64*A^2*b*c^6*d^3*e^3 + 6*A^2*b^3*c^4*d*e^5 - 32*A*B*b*c^6*d^4*e^2 - 4*A*B*b^4*c^3*d*e^5 + 72*A*B*b^2*c^5*d^3
*e^3 - 38*A*B*b^3*c^4*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) + (((4*A*b^9*c^2*d*e^6 + 8*A*b^6*c
^5*d^4*e^3 - 16*A*b^7*c^4*d^3*e^4 + 4*A*b^8*c^3*d^2*e^5 - 4*B*b^7*c^4*d^4*e^3 + 12*B*b^8*c^3*d^3*e^4 - 8*B*b^9
*c^2*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) - ((-c*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(8*b^6*c
^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e -
2*B*b*c*d))/((b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d
*e^2)))*(-c*(b*e - c*d)^3)^(1/2)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d))/(2*(b^6*e^3 - b^3*c^3*d^3 +
3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(-c*(b*e - c*d)^3)^(1/2)*(4*A*c^2*d + 3*B*b^2*e - 5*A*b*c*e - 2*B*b*c*d))/(
2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2))))*(-c*(b*e - c*d)^3)^(1/2)*(4*A*c^2*d + 3*B*b^2*e
- 5*A*b*c*e - 2*B*b*c*d)*1i)/(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2) + (atan(((((2*(d + e*x
)^(1/2)*(A^2*b^4*c^3*e^6 + 32*A^2*c^7*d^4*e^2 + 26*A^2*b^2*c^5*d^2*e^4 + 8*B^2*b^2*c^5*d^4*e^2 - 20*B^2*b^3*c^
4*d^3*e^3 + 13*B^2*b^4*c^3*d^2*e^4 - 64*A^2*b*c^6*d^3*e^3 + 6*A^2*b^3*c^4*d*e^5 - 32*A*B*b*c^6*d^4*e^2 - 4*A*B
*b^4*c^3*d*e^5 + 72*A*B*b^2*c^5*d^3*e^3 - 38*A*B*b^3*c^4*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)
- (((4*A*b^9*c^2*d*e^6 + 8*A*b^6*c^5*d^4*e^3 - 16*A*b^7*c^4*d^3*e^4 + 4*A*b^8*c^3*d^2*e^5 - 4*B*b^7*c^4*d^4*e
^3 + 12*B*b^8*c^3*d^3*e^4 - 8*B*b^9*c^2*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + ((d + e*x)^(1/2
)*(A*b*e + 4*A*c*d - 2*B*b*d)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5
))/(b^3*(d^3)^(1/2)*(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)))*(A*b*e + 4*A*c*d - 2*B*b*d))/(2*b^3*(d^3)^(1
/2)))*(A*b*e + 4*A*c*d - 2*B*b*d)*1i)/(2*b^3*(d^3)^(1/2)) + (((2*(d + e*x)^(1/2)*(A^2*b^4*c^3*e^6 + 32*A^2*c^7
*d^4*e^2 + 26*A^2*b^2*c^5*d^2*e^4 + 8*B^2*b^2*c^5*d^4*e^2 - 20*B^2*b^3*c^4*d^3*e^3 + 13*B^2*b^4*c^3*d^2*e^4 -
64*A^2*b*c^6*d^3*e^3 + 6*A^2*b^3*c^4*d*e^5 - 32*A*B*b*c^6*d^4*e^2 - 4*A*B*b^4*c^3*d*e^5 + 72*A*B*b^2*c^5*d^3*e
^3 - 38*A*B*b^3*c^4*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) + (((4*A*b^9*c^2*d*e^6 + 8*A*b^6*c^5
*d^4*e^3 - 16*A*b^7*c^4*d^3*e^4 + 4*A*b^8*c^3*d^2*e^5 - 4*B*b^7*c^4*d^4*e^3 + 12*B*b^8*c^3*d^3*e^4 - 8*B*b^9*c
^2*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) - ((d + e*x)^(1/2)*(A*b*e + 4*A*c*d - 2*B*b*d)*(8*b^6*
c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5))/(b^3*(d^3)^(1/2)*(b^4*c^2*d^4 + b^
6*d^2*e^2 - 2*b^5*c*d^3*e)))*(A*b*e + 4*A*c*d - 2*B*b*d))/(2*b^3*(d^3)^(1/2)))*(A*b*e + 4*A*c*d - 2*B*b*d)*1i)
/(2*b^3*(d^3)^(1/2)))/((2*(5*A^3*b^3*c^4*e^6 + 32*A^3*c^7*d^3*e^3 - 4*B^3*b^3*c^4*d^3*e^3 + 6*B^3*b^4*c^3*d^2*
e^4 - 3*A^2*B*b^4*c^3*e^6 - 48*A^3*b*c^6*d^2*e^4 + 6*A^3*b^2*c^5*d*e^5 + 24*A*B^2*b^2*c^5*d^3*e^3 - 36*A*B^2*b
^3*c^4*d^2*e^4 + 72*A^2*B*b^2*c^5*d^2*e^4 + 3*A*B^2*b^4*c^3*d*e^5 - 48*A^2*B*b*c^6*d^3*e^3 - 9*A^2*B*b^3*c^4*d
*e^5))/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + (((2*(d + e*x)^(1/2)*(A^2*b^4*c^3*e^6 + 32*A^2*c^7*d^4*e^
2 + 26*A^2*b^2*c^5*d^2*e^4 + 8*B^2*b^2*c^5*d^4*e^2 - 20*B^2*b^3*c^4*d^3*e^3 + 13*B^2*b^4*c^3*d^2*e^4 - 64*A^2*
b*c^6*d^3*e^3 + 6*A^2*b^3*c^4*d*e^5 - 32*A*B*b*c^6*d^4*e^2 - 4*A*B*b^4*c^3*d*e^5 + 72*A*B*b^2*c^5*d^3*e^3 - 38
*A*B*b^3*c^4*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e) - (((4*A*b^9*c^2*d*e^6 + 8*A*b^6*c^5*d^4*e^
3 - 16*A*b^7*c^4*d^3*e^4 + 4*A*b^8*c^3*d^2*e^5 - 4*B*b^7*c^4*d^4*e^3 + 12*B*b^8*c^3*d^3*e^4 - 8*B*b^9*c^2*d^2*
e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + ((d + e*x)^(1/2)*(A*b*e + 4*A*c*d - 2*B*b*d)*(8*b^6*c^5*d^5
*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5))/(b^3*(d^3)^(1/2)*(b^4*c^2*d^4 + b^6*d^2*e
^2 - 2*b^5*c*d^3*e)))*(A*b*e + 4*A*c*d - 2*B*b*d))/(2*b^3*(d^3)^(1/2)))*(A*b*e + 4*A*c*d - 2*B*b*d))/(2*b^3*(d
^3)^(1/2)) - (((2*(d + e*x)^(1/2)*(A^2*b^4*c^3*e^6 + 32*A^2*c^7*d^4*e^2 + 26*A^2*b^2*c^5*d^2*e^4 + 8*B^2*b^2*c
^5*d^4*e^2 - 20*B^2*b^3*c^4*d^3*e^3 + 13*B^2*b^4*c^3*d^2*e^4 - 64*A^2*b*c^6*d^3*e^3 + 6*A^2*b^3*c^4*d*e^5 - 32
*A*B*b*c^6*d^4*e^2 - 4*A*B*b^4*c^3*d*e^5 + 72*A*B*b^2*c^5*d^3*e^3 - 38*A*B*b^3*c^4*d^2*e^4))/(b^4*c^2*d^4 + b^
6*d^2*e^2 - 2*b^5*c*d^3*e) + (((4*A*b^9*c^2*d*e^6 + 8*A*b^6*c^5*d^4*e^3 - 16*A*b^7*c^4*d^3*e^4 + 4*A*b^8*c^3*d
^2*e^5 - 4*B*b^7*c^4*d^4*e^3 + 12*B*b^8*c^3*d^3*e^4 - 8*B*b^9*c^2*d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*
c*d^3*e) - ((d + e*x)^(1/2)*(A*b*e + 4*A*c*d - 2*B*b*d)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 16*b^8*c^3*d
^3*e^4 - 4*b^9*c^2*d^2*e^5))/(b^3*(d^3)^(1/2)*(b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^5*c*d^3*e)))*(A*b*e + 4*A*c*d -
2*B*b*d))/(2*b^3*(d^3)^(1/2)))*(A*b*e + 4*A*c*d - 2*B*b*d))/(2*b^3*(d^3)^(1/2))))*(A*b*e + 4*A*c*d - 2*B*b*d)
*1i)/(b^3*(d^3)^(1/2))

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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((B*x+A)/(e*x+d)**(1/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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