3.11.66 \(\int \frac {A+B x}{(d+e x) (b x+c x^2)^{5/2}} \, dx\)
Optimal. Leaf size=287 \[ -\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac {2 \left (b (c d-b e) \left (3 b^2 e (B d-A e)-4 b c d (A e+B d)+8 A c^2 d^2\right )+c x \left (-3 b^3 e^2 (B d-A e)+2 b^2 c d e (A e+7 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (c d-b e)^2}-\frac {e^3 (B d-A e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \]
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Rubi [A] time = 0.35, antiderivative size = 287, 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, 12, 724, 206} \begin {gather*} \frac {2 \left (c x \left (2 b^2 c d e (A e+7 B d)-3 b^3 e^2 (B d-A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )+b (c d-b e) \left (3 b^2 e (B d-A e)-4 b c d (A e+B d)+8 A c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (c d-b e)^2}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}-\frac {e^3 (B d-A e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)*(b*x + c*x^2)^(5/2)),x]
[Out]
(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d
- b*e)*(8*A*c^2*d^2 + 3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e)) + c*(16*A*c^3*d^3 - 3*b^3*e^2*(B*d - A*e) + 2
*b^2*c*d*e*(7*B*d + A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[b*x + c*x^2]) - (e^3*(
B*d - A*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(d^(5/2)*(c*d - b*e
)^(5/2))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 724
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
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])
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )-2 c e (b B d-2 A c d+A b e) x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )+c \left (16 A c^3 d^3-3 b^3 e^2 (B d-A e)+2 b^2 c d e (7 B d+A e)-8 b c^2 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {4 \int -\frac {3 b^4 e^3 (B d-A e)}{4 (d+e x) \sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )+c \left (16 A c^3 d^3-3 b^3 e^2 (B d-A e)+2 b^2 c d e (7 B d+A e)-8 b c^2 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {\left (e^3 (B d-A e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{d^2 (c d-b e)^2}\\ &=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )+c \left (16 A c^3 d^3-3 b^3 e^2 (B d-A e)+2 b^2 c d e (7 B d+A e)-8 b c^2 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (2 e^3 (B d-A e)\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{d^2 (c d-b e)^2}\\ &=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )+c \left (16 A c^3 d^3-3 b^3 e^2 (B d-A e)+2 b^2 c d e (7 B d+A e)-8 b c^2 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {e^3 (B d-A e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{d^{5/2} (c d-b e)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 277, normalized size = 0.97 \begin {gather*} \frac {2 \left (-\frac {3 c x^2 \left (3 b^2 e (B d-A e)-4 b c d (A e+B d)+8 A c^2 d^2\right )}{b^2 d (b e-c d)}+\frac {3 x^{3/2} (b+c x) \left (3 b^4 e^3 \sqrt {b+c x} (A e-B d) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )+c \sqrt {d} \sqrt {x} \sqrt {c d-b e} \left (3 b^3 e^2 (A e-B d)+2 b^2 c d e (A e+7 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{b^3 d^{3/2} (c d-b e)^{5/2}}+9 x \left (\frac {2 A c}{b}+\frac {A e}{d}-B\right )-3 A\right )}{9 b d (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)*(b*x + c*x^2)^(5/2)),x]
[Out]
(2*(-3*A + 9*(-B + (2*A*c)/b + (A*e)/d)*x - (3*c*(8*A*c^2*d^2 + 3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e))*x^2
)/(b^2*d*(-(c*d) + b*e)) + (3*x^(3/2)*(b + c*x)*(c*Sqrt[d]*Sqrt[c*d - b*e]*(16*A*c^3*d^3 + 3*b^3*e^2*(-(B*d) +
A*e) + 2*b^2*c*d*e*(7*B*d + A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*Sqrt[x] + 3*b^4*e^3*(-(B*d) + A*e)*Sqrt[b + c*x
]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(b^3*d^(3/2)*(c*d - b*e)^(5/2))))/(9*b*d*(x*(b
+ c*x))^(3/2))
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IntegrateAlgebraic [A] time = 1.81, size = 437, normalized size = 1.52 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (A b^5 d e^2-3 A b^5 e^3 x-2 A b^4 c d^2 e-6 A b^4 c e^3 x^2+A b^3 c^2 d^3+9 A b^3 c^2 d^2 e x-3 A b^3 c^2 d e^2 x^2-3 A b^3 c^2 e^3 x^3-6 A b^2 c^3 d^3 x+36 A b^2 c^3 d^2 e x^2-2 A b^2 c^3 d e^2 x^3-24 A b c^4 d^3 x^2+24 A b c^4 d^2 e x^3-16 A c^5 d^3 x^3+3 b^5 B d e^2 x-6 b^4 B c d^2 e x+6 b^4 B c d e^2 x^2+3 b^3 B c^2 d^3 x-21 b^3 B c^2 d^2 e x^2+3 b^3 B c^2 d e^2 x^3+12 b^2 B c^3 d^3 x^2-14 b^2 B c^3 d^2 e x^3+8 b B c^4 d^3 x^3\right )}{3 b^4 d^2 x^2 (b+c x)^2 (b e-c d)^2}-\frac {2 \left (B d e^3-A e^4\right ) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)*(b*x + c*x^2)^(5/2)),x]
[Out]
(-2*Sqrt[b*x + c*x^2]*(A*b^3*c^2*d^3 - 2*A*b^4*c*d^2*e + A*b^5*d*e^2 + 3*b^3*B*c^2*d^3*x - 6*A*b^2*c^3*d^3*x -
6*b^4*B*c*d^2*e*x + 9*A*b^3*c^2*d^2*e*x + 3*b^5*B*d*e^2*x - 3*A*b^5*e^3*x + 12*b^2*B*c^3*d^3*x^2 - 24*A*b*c^4
*d^3*x^2 - 21*b^3*B*c^2*d^2*e*x^2 + 36*A*b^2*c^3*d^2*e*x^2 + 6*b^4*B*c*d*e^2*x^2 - 3*A*b^3*c^2*d*e^2*x^2 - 6*A
*b^4*c*e^3*x^2 + 8*b*B*c^4*d^3*x^3 - 16*A*c^5*d^3*x^3 - 14*b^2*B*c^3*d^2*e*x^3 + 24*A*b*c^4*d^2*e*x^3 + 3*b^3*
B*c^2*d*e^2*x^3 - 2*A*b^2*c^3*d*e^2*x^3 - 3*A*b^3*c^2*e^3*x^3))/(3*b^4*d^2*(-(c*d) + b*e)^2*x^2*(b + c*x)^2) -
(2*(B*d*e^3 - A*e^4)*ArcTanh[(Sqrt[c]*d + Sqrt[c]*e*x - e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/(d^(
5/2)*(c*d - b*e)^(5/2))
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fricas [B] time = 0.46, size = 1390, normalized size = 4.84
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
[Out]
[-1/3*(3*((B*b^4*c^2*d*e^3 - A*b^4*c^2*e^4)*x^4 + 2*(B*b^5*c*d*e^3 - A*b^5*c*e^4)*x^3 + (B*b^6*d*e^3 - A*b^6*e
^4)*x^2)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d))
+ 2*(A*b^3*c^3*d^5 - 3*A*b^4*c^2*d^4*e + 3*A*b^5*c*d^3*e^2 - A*b^6*d^2*e^3 + (3*A*b^4*c^2*d*e^4 + 8*(B*b*c^5 -
2*A*c^6)*d^5 - 2*(11*B*b^2*c^4 - 20*A*b*c^5)*d^4*e + (17*B*b^3*c^3 - 26*A*b^2*c^4)*d^3*e^2 - (3*B*b^4*c^2 + A
*b^3*c^3)*d^2*e^3)*x^3 + 3*(2*A*b^5*c*d*e^4 + 4*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - (11*B*b^3*c^3 - 20*A*b^2*c^4)*d^
4*e + (9*B*b^4*c^2 - 13*A*b^3*c^3)*d^3*e^2 - (2*B*b^5*c + A*b^4*c^2)*d^2*e^3)*x^2 + 3*(A*b^6*d*e^4 + (B*b^3*c^
3 - 2*A*b^2*c^4)*d^5 - (3*B*b^4*c^2 - 5*A*b^3*c^3)*d^4*e + 3*(B*b^5*c - A*b^4*c^2)*d^3*e^2 - (B*b^6 + A*b^5*c)
*d^2*e^3)*x)*sqrt(c*x^2 + b*x))/((b^4*c^5*d^6 - 3*b^5*c^4*d^5*e + 3*b^6*c^3*d^4*e^2 - b^7*c^2*d^3*e^3)*x^4 + 2
*(b^5*c^4*d^6 - 3*b^6*c^3*d^5*e + 3*b^7*c^2*d^4*e^2 - b^8*c*d^3*e^3)*x^3 + (b^6*c^3*d^6 - 3*b^7*c^2*d^5*e + 3*
b^8*c*d^4*e^2 - b^9*d^3*e^3)*x^2), -2/3*(3*((B*b^4*c^2*d*e^3 - A*b^4*c^2*e^4)*x^4 + 2*(B*b^5*c*d*e^3 - A*b^5*c
*e^4)*x^3 + (B*b^6*d*e^3 - A*b^6*e^4)*x^2)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)
/((c*d - b*e)*x)) + (A*b^3*c^3*d^5 - 3*A*b^4*c^2*d^4*e + 3*A*b^5*c*d^3*e^2 - A*b^6*d^2*e^3 + (3*A*b^4*c^2*d*e^
4 + 8*(B*b*c^5 - 2*A*c^6)*d^5 - 2*(11*B*b^2*c^4 - 20*A*b*c^5)*d^4*e + (17*B*b^3*c^3 - 26*A*b^2*c^4)*d^3*e^2 -
(3*B*b^4*c^2 + A*b^3*c^3)*d^2*e^3)*x^3 + 3*(2*A*b^5*c*d*e^4 + 4*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - (11*B*b^3*c^3 -
20*A*b^2*c^4)*d^4*e + (9*B*b^4*c^2 - 13*A*b^3*c^3)*d^3*e^2 - (2*B*b^5*c + A*b^4*c^2)*d^2*e^3)*x^2 + 3*(A*b^6*d
*e^4 + (B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - (3*B*b^4*c^2 - 5*A*b^3*c^3)*d^4*e + 3*(B*b^5*c - A*b^4*c^2)*d^3*e^2 - (
B*b^6 + A*b^5*c)*d^2*e^3)*x)*sqrt(c*x^2 + b*x))/((b^4*c^5*d^6 - 3*b^5*c^4*d^5*e + 3*b^6*c^3*d^4*e^2 - b^7*c^2*
d^3*e^3)*x^4 + 2*(b^5*c^4*d^6 - 3*b^6*c^3*d^5*e + 3*b^7*c^2*d^4*e^2 - b^8*c*d^3*e^3)*x^3 + (b^6*c^3*d^6 - 3*b^
7*c^2*d^5*e + 3*b^8*c*d^4*e^2 - b^9*d^3*e^3)*x^2)]
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giac [B] time = 0.28, size = 853, normalized size = 2.97 \begin {gather*} \frac {2 \, {\left (B d e^{3} - A e^{4}\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} - \frac {2 \, {\left ({\left ({\left (\frac {{\left (8 \, B b c^{6} d^{10} - 16 \, A c^{7} d^{10} - 30 \, B b^{2} c^{5} d^{9} e + 56 \, A b c^{6} d^{9} e + 39 \, B b^{3} c^{4} d^{8} e^{2} - 66 \, A b^{2} c^{5} d^{8} e^{2} - 20 \, B b^{4} c^{3} d^{7} e^{3} + 25 \, A b^{3} c^{4} d^{7} e^{3} + 3 \, B b^{5} c^{2} d^{6} e^{4} + 4 \, A b^{4} c^{3} d^{6} e^{4} - 3 \, A b^{5} c^{2} d^{5} e^{5}\right )} x}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}} + \frac {3 \, {\left (4 \, B b^{2} c^{5} d^{10} - 8 \, A b c^{6} d^{10} - 15 \, B b^{3} c^{4} d^{9} e + 28 \, A b^{2} c^{5} d^{9} e + 20 \, B b^{4} c^{3} d^{8} e^{2} - 33 \, A b^{3} c^{4} d^{8} e^{2} - 11 \, B b^{5} c^{2} d^{7} e^{3} + 12 \, A b^{4} c^{3} d^{7} e^{3} + 2 \, B b^{6} c d^{6} e^{4} + 3 \, A b^{5} c^{2} d^{6} e^{4} - 2 \, A b^{6} c d^{5} e^{5}\right )}}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}}\right )} x + \frac {3 \, {\left (B b^{3} c^{4} d^{10} - 2 \, A b^{2} c^{5} d^{10} - 4 \, B b^{4} c^{3} d^{9} e + 7 \, A b^{3} c^{4} d^{9} e + 6 \, B b^{5} c^{2} d^{8} e^{2} - 8 \, A b^{4} c^{3} d^{8} e^{2} - 4 \, B b^{6} c d^{7} e^{3} + 2 \, A b^{5} c^{2} d^{7} e^{3} + B b^{7} d^{6} e^{4} + 2 \, A b^{6} c d^{6} e^{4} - A b^{7} d^{5} e^{5}\right )}}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}}\right )} x + \frac {A b^{3} c^{4} d^{10} - 4 \, A b^{4} c^{3} d^{9} e + 6 \, A b^{5} c^{2} d^{8} e^{2} - 4 \, A b^{6} c d^{7} e^{3} + A b^{7} d^{6} e^{4}}{b^{4} c^{4} d^{11} - 4 \, b^{5} c^{3} d^{10} e + 6 \, b^{6} c^{2} d^{9} e^{2} - 4 \, b^{7} c d^{8} e^{3} + b^{8} d^{7} e^{4}}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="giac")
[Out]
2*(B*d*e^3 - A*e^4)*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^2*d^4 - 2
*b*c*d^3*e + b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) - 2/3*((((8*B*b*c^6*d^10 - 16*A*c^7*d^10 - 30*B*b^2*c^5*d^9*e
+ 56*A*b*c^6*d^9*e + 39*B*b^3*c^4*d^8*e^2 - 66*A*b^2*c^5*d^8*e^2 - 20*B*b^4*c^3*d^7*e^3 + 25*A*b^3*c^4*d^7*e^3
+ 3*B*b^5*c^2*d^6*e^4 + 4*A*b^4*c^3*d^6*e^4 - 3*A*b^5*c^2*d^5*e^5)*x/(b^4*c^4*d^11 - 4*b^5*c^3*d^10*e + 6*b^6
*c^2*d^9*e^2 - 4*b^7*c*d^8*e^3 + b^8*d^7*e^4) + 3*(4*B*b^2*c^5*d^10 - 8*A*b*c^6*d^10 - 15*B*b^3*c^4*d^9*e + 28
*A*b^2*c^5*d^9*e + 20*B*b^4*c^3*d^8*e^2 - 33*A*b^3*c^4*d^8*e^2 - 11*B*b^5*c^2*d^7*e^3 + 12*A*b^4*c^3*d^7*e^3 +
2*B*b^6*c*d^6*e^4 + 3*A*b^5*c^2*d^6*e^4 - 2*A*b^6*c*d^5*e^5)/(b^4*c^4*d^11 - 4*b^5*c^3*d^10*e + 6*b^6*c^2*d^9
*e^2 - 4*b^7*c*d^8*e^3 + b^8*d^7*e^4))*x + 3*(B*b^3*c^4*d^10 - 2*A*b^2*c^5*d^10 - 4*B*b^4*c^3*d^9*e + 7*A*b^3*
c^4*d^9*e + 6*B*b^5*c^2*d^8*e^2 - 8*A*b^4*c^3*d^8*e^2 - 4*B*b^6*c*d^7*e^3 + 2*A*b^5*c^2*d^7*e^3 + B*b^7*d^6*e^
4 + 2*A*b^6*c*d^6*e^4 - A*b^7*d^5*e^5)/(b^4*c^4*d^11 - 4*b^5*c^3*d^10*e + 6*b^6*c^2*d^9*e^2 - 4*b^7*c*d^8*e^3
+ b^8*d^7*e^4))*x + (A*b^3*c^4*d^10 - 4*A*b^4*c^3*d^9*e + 6*A*b^5*c^2*d^8*e^2 - 4*A*b^6*c*d^7*e^3 + A*b^7*d^6*
e^4)/(b^4*c^4*d^11 - 4*b^5*c^3*d^10*e + 6*b^6*c^2*d^9*e^2 - 4*b^7*c*d^8*e^3 + b^8*d^7*e^4))/(c*x^2 + b*x)^(3/2
)
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maple [B] time = 0.06, size = 2000, normalized size = 6.97
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)/(c*x^2+b*x)^(5/2),x)
[Out]
32/3/e/(b*e-c*d)*c^3/b^4/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*B*d+16/3*e/(b*e-c*d)/d*c^
2/b^3/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*A-2*e^2/(b*e-c*d)^2/d/((x+d/e)^2*c-(b*e-c*d)
*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*B+2*e^3/(b*e-c*d)^2/d^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e
)^(1/2)*A+16/3*B/e*c/b^3/(c*x^2+b*x)^(1/2)+2/3/(b*e-c*d)/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)
^(3/2)*c*A-2/3*B/e/b/(c*x^2+b*x)^(3/2)-8/3/(b*e-c*d)*c/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)
^(1/2)*B-e^3/(b*e-c*d)^2/d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*
d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*A+e^2/(b*e-c*d)^2/d/(-(b*e
-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*B+2*e/(b*e-c*d)^2/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*
(x+d/e)/e)^(1/2)*c*B+32/3*B/e*c^2/b^4/(c*x^2+b*x)^(1/2)*x-4/3*B/e/b^2/(c*x^2+b*x)^(3/2)*c*x-4/3/e/(b*e-c*d)/b^
2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c^2*x*B*d-4*e^2/(b*e-c*d)^2/d/b^2/((x+d/e)^2*c-(b*
e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^2*A-2*e^2/(b*e-c*d)^2/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*
d)*(x+d/e)/e)^(1/2)*x*c*B+4*e/(b*e-c*d)^2/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^2*
B-2*e^2/(b*e-c*d)^2/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c*A-2/3/e/(b*e-c*d)/b/((x+d/
e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c*B*d+2*e^3/(b*e-c*d)^2/d^2/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2
+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c*A-2/3*e/(b*e-c*d)/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3
/2)*c*x*A-2/3*e/(b*e-c*d)/d/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*A-16/3/(b*e-c*d)*c^2/b^3
/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*B+2/3/(b*e-c*d)/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b
*e-2*c*d)*(x+d/e)/e)^(3/2)*c*x*B-32/3/(b*e-c*d)*c^3/b^4/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1
/2)*x*A+2/3/(b*e-c*d)/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*B+4/3/(b*e-c*d)/b^2/((x+d/e)^2
*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c^2*x*A-16/3/(b*e-c*d)*c^2/b^3/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b
*e-2*c*d)*(x+d/e)/e)^(1/2)*A+16/3/e/(b*e-c*d)*c^2/b^3/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2
)*B*d+8/3*e/(b*e-c*d)/d*c/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*A
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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)/(c*x^2+b*x)^(5/2),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-(2*c*d)/e^2)^2>0)', see `
assume?` for more details)Is (b/e-(2*c*d)/e^2)^2 -(4*c *((c*d^2)/e^2 -(b*d)/e)) /e^2 zero
or nonzero?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/((b*x + c*x^2)^(5/2)*(d + e*x)),x)
[Out]
int((A + B*x)/((b*x + c*x^2)^(5/2)*(d + e*x)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)/(c*x**2+b*x)**(5/2),x)
[Out]
Integral((A + B*x)/((x*(b + c*x))**(5/2)*(d + e*x)), x)
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