3.11.61 \(\int \frac {(A+B x) (d+e x)^4}{(b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=341 \[ -\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {e \sqrt {b x+c x^2} \left (2 b^3 c e^2 (3 A e+7 B d)+4 b^2 c^2 d e (A e+2 B d)-16 b c^3 d^2 (3 A e+B d)+32 A c^4 d^3-15 b^4 B e^3\right )}{3 b^4 c^3}-\frac {2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (2 b^3 c e^2 (A e+3 B d)+4 b^2 c^2 d e (A e+B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{c^{7/2}} \]

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Rubi [A]  time = 0.46, antiderivative size = 341, 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 = {818, 640, 620, 206} \begin {gather*} -\frac {e \sqrt {b x+c x^2} \left (4 b^2 c^2 d e (A e+2 B d)+2 b^3 c e^2 (3 A e+7 B d)-16 b c^3 d^2 (3 A e+B d)+32 A c^4 d^3-15 b^4 B e^3\right )}{3 b^4 c^3}-\frac {2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (4 b^2 c^2 d e (A e+B d)+2 b^3 c e^2 (A e+3 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^3*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3*b^2*c*(b*x + c*x^2)^(3/2)) - (2*(d +
e*x)*(b*c*d^2*(4*b*B*c*d - 8*A*c^2*d - b^2*B*e + 10*A*b*c*e) - (16*A*c^4*d^3 - 5*b^4*B*e^3 + 4*b^2*c^2*d*e*(B
*d + A*e) + 2*b^3*c*e^2*(3*B*d + A*e) - 8*b*c^3*d^2*(B*d + 3*A*e))*x))/(3*b^4*c^2*Sqrt[b*x + c*x^2]) - (e*(32*
A*c^4*d^3 - 15*b^4*B*e^3 + 4*b^2*c^2*d*e*(2*B*d + A*e) - 16*b*c^3*d^2*(B*d + 3*A*e) + 2*b^3*c*e^2*(7*B*d + 3*A
*e))*Sqrt[b*x + c*x^2])/(3*b^4*c^3) + (e^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]
])/c^(7/2)

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 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {(d+e x)^2 \left (\frac {1}{2} d \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )+\frac {1}{2} e \left (4 A c^2 d+5 b^2 B e-2 b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 c}\\ &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x) \left (b c d^2 \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )-\left (16 A c^4 d^3-5 b^4 B e^3+4 b^2 c^2 d e (B d+A e)+2 b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} b d e \left (16 A c^3 d^2-5 b^3 B e^2+2 b^2 c e (2 B d+A e)-8 b c^2 d (B d+3 A e)\right )-\frac {1}{4} e \left (32 A c^4 d^3-15 b^4 B e^3+4 b^2 c^2 d e (2 B d+A e)-16 b c^3 d^2 (B d+3 A e)+2 b^3 c e^2 (7 B d+3 A e)\right ) x}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}\\ &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x) \left (b c d^2 \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )-\left (16 A c^4 d^3-5 b^4 B e^3+4 b^2 c^2 d e (B d+A e)+2 b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {e \left (32 A c^4 d^3-15 b^4 B e^3+4 b^2 c^2 d e (2 B d+A e)-16 b c^3 d^2 (B d+3 A e)+2 b^3 c e^2 (7 B d+3 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 c^3}+\frac {\left (e^3 (8 B c d-5 b B e+2 A c e)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2 c^3}\\ &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x) \left (b c d^2 \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )-\left (16 A c^4 d^3-5 b^4 B e^3+4 b^2 c^2 d e (B d+A e)+2 b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {e \left (32 A c^4 d^3-15 b^4 B e^3+4 b^2 c^2 d e (2 B d+A e)-16 b c^3 d^2 (B d+3 A e)+2 b^3 c e^2 (7 B d+3 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 c^3}+\frac {\left (e^3 (8 B c d-5 b B e+2 A c e)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c^3}\\ &=-\frac {2 (d+e x)^3 \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac {2 (d+e x) \left (b c d^2 \left (4 b B c d-8 A c^2 d-b^2 B e+10 A b c e\right )-\left (16 A c^4 d^3-5 b^4 B e^3+4 b^2 c^2 d e (B d+A e)+2 b^3 c e^2 (3 B d+A e)-8 b c^3 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {e \left (32 A c^4 d^3-15 b^4 B e^3+4 b^2 c^2 d e (2 B d+A e)-16 b c^3 d^2 (B d+3 A e)+2 b^3 c e^2 (7 B d+3 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 c^3}+\frac {e^3 (8 B c d-5 b B e+2 A c e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 5.37, size = 237, normalized size = 0.70 \begin {gather*} \frac {x^{5/2} \left (\frac {e^3 (b+c x)^{5/2} \log \left (\sqrt {c} \sqrt {b+c x}+c \sqrt {x}\right ) (2 A c e-5 b B e+8 B c d)}{c^{7/2}}-\frac {(b+c x) \left (2 x^2 (b+c x) (c d-b e)^3 \left (b c (5 B d-4 A e)-8 A c^2 d+7 b^2 B e\right )+2 c^3 d^3 x (b+c x)^2 (3 b (4 A e+B d)-8 A c d)+2 b x^2 (b B-A c) (c d-b e)^4+2 A b c^3 d^4 (b+c x)^2-3 b^4 B e^4 x^2 (b+c x)^2\right )}{3 b^4 c^3 x^{3/2}}\right )}{(x (b+c x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^(5/2),x]

[Out]

(x^(5/2)*(-1/3*((b + c*x)*(2*b*(b*B - A*c)*(c*d - b*e)^4*x^2 + 2*(c*d - b*e)^3*(-8*A*c^2*d + 7*b^2*B*e + b*c*(
5*B*d - 4*A*e))*x^2*(b + c*x) + 2*A*b*c^3*d^4*(b + c*x)^2 + 2*c^3*d^3*(-8*A*c*d + 3*b*(B*d + 4*A*e))*x*(b + c*
x)^2 - 3*b^4*B*e^4*x^2*(b + c*x)^2))/(b^4*c^3*x^(3/2)) + (e^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*(b + c*x)^(5/2)*Lo
g[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/c^(7/2)))/(x*(b + c*x))^(5/2)

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IntegrateAlgebraic [A]  time = 1.03, size = 435, normalized size = 1.28 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-6 A b^5 c e^4 x^2-8 A b^4 c^2 e^4 x^3-2 A b^3 c^3 d^4-24 A b^3 c^3 d^3 e x+36 A b^3 c^3 d^2 e^2 x^2+8 A b^3 c^3 d e^3 x^3+12 A b^2 c^4 d^4 x-96 A b^2 c^4 d^3 e x^2+24 A b^2 c^4 d^2 e^2 x^3+48 A b c^5 d^4 x^2-64 A b c^5 d^3 e x^3+32 A c^6 d^4 x^3+15 b^6 B e^4 x^2-24 b^5 B c d e^3 x^2+20 b^5 B c e^4 x^3-32 b^4 B c^2 d e^3 x^3+3 b^4 B c^2 e^4 x^4-6 b^3 B c^3 d^4 x+24 b^3 B c^3 d^3 e x^2+12 b^3 B c^3 d^2 e^2 x^3-24 b^2 B c^4 d^4 x^2+16 b^2 B c^4 d^3 e x^3-16 b B c^5 d^4 x^3\right )}{3 b^4 c^3 x^2 (b+c x)^2}+\frac {\log \left (-2 c^{7/2} \sqrt {b x+c x^2}+b c^3+2 c^4 x\right ) \left (-2 A c e^4+5 b B e^4-8 B c d e^3\right )}{2 c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[b*x + c*x^2]*(-2*A*b^3*c^3*d^4 - 6*b^3*B*c^3*d^4*x + 12*A*b^2*c^4*d^4*x - 24*A*b^3*c^3*d^3*e*x - 24*b^2*
B*c^4*d^4*x^2 + 48*A*b*c^5*d^4*x^2 + 24*b^3*B*c^3*d^3*e*x^2 - 96*A*b^2*c^4*d^3*e*x^2 + 36*A*b^3*c^3*d^2*e^2*x^
2 - 24*b^5*B*c*d*e^3*x^2 + 15*b^6*B*e^4*x^2 - 6*A*b^5*c*e^4*x^2 - 16*b*B*c^5*d^4*x^3 + 32*A*c^6*d^4*x^3 + 16*b
^2*B*c^4*d^3*e*x^3 - 64*A*b*c^5*d^3*e*x^3 + 12*b^3*B*c^3*d^2*e^2*x^3 + 24*A*b^2*c^4*d^2*e^2*x^3 - 32*b^4*B*c^2
*d*e^3*x^3 + 8*A*b^3*c^3*d*e^3*x^3 + 20*b^5*B*c*e^4*x^3 - 8*A*b^4*c^2*e^4*x^3 + 3*b^4*B*c^2*e^4*x^4))/(3*b^4*c
^3*x^2*(b + c*x)^2) + ((-8*B*c*d*e^3 + 5*b*B*e^4 - 2*A*c*e^4)*Log[b*c^3 + 2*c^4*x - 2*c^(7/2)*Sqrt[b*x + c*x^2
]])/(2*c^(7/2))

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fricas [A]  time = 0.46, size = 974, normalized size = 2.86 \begin {gather*} \left [\frac {3 \, {\left ({\left (8 \, B b^{4} c^{3} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{4} + 2 \, {\left (8 \, B b^{5} c^{2} d e^{3} - {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{3} + {\left (8 \, B b^{6} c d e^{3} - {\left (5 \, B b^{7} - 2 \, A b^{6} c\right )} e^{4}\right )} x^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (3 \, B b^{4} c^{3} e^{4} x^{4} - 2 \, A b^{3} c^{4} d^{4} - 4 \, {\left (4 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} - 4 \, {\left (B b^{2} c^{5} - 4 \, A b c^{6}\right )} d^{3} e - 3 \, {\left (B b^{3} c^{4} + 2 \, A b^{2} c^{5}\right )} d^{2} e^{2} + 2 \, {\left (4 \, B b^{4} c^{3} - A b^{3} c^{4}\right )} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (12 \, A b^{3} c^{4} d^{2} e^{2} - 8 \, B b^{5} c^{2} d e^{3} - 8 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} + 8 \, {\left (B b^{3} c^{4} - 4 \, A b^{2} c^{5}\right )} d^{3} e + {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{2} - 6 \, {\left (4 \, A b^{3} c^{4} d^{3} e + {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{6 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}}, -\frac {3 \, {\left ({\left (8 \, B b^{4} c^{3} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{4} + 2 \, {\left (8 \, B b^{5} c^{2} d e^{3} - {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{3} + {\left (8 \, B b^{6} c d e^{3} - {\left (5 \, B b^{7} - 2 \, A b^{6} c\right )} e^{4}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (3 \, B b^{4} c^{3} e^{4} x^{4} - 2 \, A b^{3} c^{4} d^{4} - 4 \, {\left (4 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} - 4 \, {\left (B b^{2} c^{5} - 4 \, A b c^{6}\right )} d^{3} e - 3 \, {\left (B b^{3} c^{4} + 2 \, A b^{2} c^{5}\right )} d^{2} e^{2} + 2 \, {\left (4 \, B b^{4} c^{3} - A b^{3} c^{4}\right )} d e^{3} - {\left (5 \, B b^{5} c^{2} - 2 \, A b^{4} c^{3}\right )} e^{4}\right )} x^{3} + 3 \, {\left (12 \, A b^{3} c^{4} d^{2} e^{2} - 8 \, B b^{5} c^{2} d e^{3} - 8 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} + 8 \, {\left (B b^{3} c^{4} - 4 \, A b^{2} c^{5}\right )} d^{3} e + {\left (5 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{4}\right )} x^{2} - 6 \, {\left (4 \, A b^{3} c^{4} d^{3} e + {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*((8*B*b^4*c^3*d*e^3 - (5*B*b^5*c^2 - 2*A*b^4*c^3)*e^4)*x^4 + 2*(8*B*b^5*c^2*d*e^3 - (5*B*b^6*c - 2*A*b
^5*c^2)*e^4)*x^3 + (8*B*b^6*c*d*e^3 - (5*B*b^7 - 2*A*b^6*c)*e^4)*x^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b
*x)*sqrt(c)) + 2*(3*B*b^4*c^3*e^4*x^4 - 2*A*b^3*c^4*d^4 - 4*(4*(B*b*c^6 - 2*A*c^7)*d^4 - 4*(B*b^2*c^5 - 4*A*b*
c^6)*d^3*e - 3*(B*b^3*c^4 + 2*A*b^2*c^5)*d^2*e^2 + 2*(4*B*b^4*c^3 - A*b^3*c^4)*d*e^3 - (5*B*b^5*c^2 - 2*A*b^4*
c^3)*e^4)*x^3 + 3*(12*A*b^3*c^4*d^2*e^2 - 8*B*b^5*c^2*d*e^3 - 8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 8*(B*b^3*c^4 - 4
*A*b^2*c^5)*d^3*e + (5*B*b^6*c - 2*A*b^5*c^2)*e^4)*x^2 - 6*(4*A*b^3*c^4*d^3*e + (B*b^3*c^4 - 2*A*b^2*c^5)*d^4)
*x)*sqrt(c*x^2 + b*x))/(b^4*c^6*x^4 + 2*b^5*c^5*x^3 + b^6*c^4*x^2), -1/3*(3*((8*B*b^4*c^3*d*e^3 - (5*B*b^5*c^2
- 2*A*b^4*c^3)*e^4)*x^4 + 2*(8*B*b^5*c^2*d*e^3 - (5*B*b^6*c - 2*A*b^5*c^2)*e^4)*x^3 + (8*B*b^6*c*d*e^3 - (5*B
*b^7 - 2*A*b^6*c)*e^4)*x^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (3*B*b^4*c^3*e^4*x^4 - 2*A*b^3
*c^4*d^4 - 4*(4*(B*b*c^6 - 2*A*c^7)*d^4 - 4*(B*b^2*c^5 - 4*A*b*c^6)*d^3*e - 3*(B*b^3*c^4 + 2*A*b^2*c^5)*d^2*e^
2 + 2*(4*B*b^4*c^3 - A*b^3*c^4)*d*e^3 - (5*B*b^5*c^2 - 2*A*b^4*c^3)*e^4)*x^3 + 3*(12*A*b^3*c^4*d^2*e^2 - 8*B*b
^5*c^2*d*e^3 - 8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 8*(B*b^3*c^4 - 4*A*b^2*c^5)*d^3*e + (5*B*b^6*c - 2*A*b^5*c^2)*e
^4)*x^2 - 6*(4*A*b^3*c^4*d^3*e + (B*b^3*c^4 - 2*A*b^2*c^5)*d^4)*x)*sqrt(c*x^2 + b*x))/(b^4*c^6*x^4 + 2*b^5*c^5
*x^3 + b^6*c^4*x^2)]

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giac [A]  time = 0.30, size = 370, normalized size = 1.09 \begin {gather*} -\frac {\frac {2 \, A d^{4}}{b} - {\left ({\left ({\left (\frac {3 \, B x e^{4}}{c} - \frac {4 \, {\left (4 \, B b c^{5} d^{4} - 8 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e + 16 \, A b c^{5} d^{3} e - 3 \, B b^{3} c^{3} d^{2} e^{2} - 6 \, A b^{2} c^{4} d^{2} e^{2} + 8 \, B b^{4} c^{2} d e^{3} - 2 \, A b^{3} c^{3} d e^{3} - 5 \, B b^{5} c e^{4} + 2 \, A b^{4} c^{2} e^{4}\right )}}{b^{4} c^{3}}\right )} x - \frac {3 \, {\left (8 \, B b^{2} c^{4} d^{4} - 16 \, A b c^{5} d^{4} - 8 \, B b^{3} c^{3} d^{3} e + 32 \, A b^{2} c^{4} d^{3} e - 12 \, A b^{3} c^{3} d^{2} e^{2} + 8 \, B b^{5} c d e^{3} - 5 \, B b^{6} e^{4} + 2 \, A b^{5} c e^{4}\right )}}{b^{4} c^{3}}\right )} x - \frac {6 \, {\left (B b^{3} c^{3} d^{4} - 2 \, A b^{2} c^{4} d^{4} + 4 \, A b^{3} c^{3} d^{3} e\right )}}{b^{4} c^{3}}\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} - \frac {{\left (8 \, B c d e^{3} - 5 \, B b e^{4} + 2 \, A c e^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

-1/3*(2*A*d^4/b - (((3*B*x*e^4/c - 4*(4*B*b*c^5*d^4 - 8*A*c^6*d^4 - 4*B*b^2*c^4*d^3*e + 16*A*b*c^5*d^3*e - 3*B
*b^3*c^3*d^2*e^2 - 6*A*b^2*c^4*d^2*e^2 + 8*B*b^4*c^2*d*e^3 - 2*A*b^3*c^3*d*e^3 - 5*B*b^5*c*e^4 + 2*A*b^4*c^2*e
^4)/(b^4*c^3))*x - 3*(8*B*b^2*c^4*d^4 - 16*A*b*c^5*d^4 - 8*B*b^3*c^3*d^3*e + 32*A*b^2*c^4*d^3*e - 12*A*b^3*c^3
*d^2*e^2 + 8*B*b^5*c*d*e^3 - 5*B*b^6*e^4 + 2*A*b^5*c*e^4)/(b^4*c^3))*x - 6*(B*b^3*c^3*d^4 - 2*A*b^2*c^4*d^4 +
4*A*b^3*c^3*d^3*e)/(b^4*c^3))*x)/(c*x^2 + b*x)^(3/2) - 1/2*(8*B*c*d*e^3 - 5*B*b*e^4 + 2*A*c*e^4)*log(abs(-2*(s
qrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(7/2)

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maple [B]  time = 0.06, size = 1026, normalized size = 3.01 \begin {gather*} \frac {B \,e^{4} x^{4}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {A \,e^{4} x^{3}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {5 B b \,e^{4} x^{3}}{6 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}-\frac {4 B d \,e^{3} x^{3}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {A b \,e^{4} x^{2}}{2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}-\frac {4 A d \,e^{3} x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {5 B \,b^{2} e^{4} x^{2}}{4 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}+\frac {2 B b d \,e^{3} x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}-\frac {6 B \,d^{2} e^{2} x^{2}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {A \,b^{2} e^{4} x}{6 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}-\frac {4 A b d \,e^{3} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}+\frac {8 A \,d^{3} e x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b}-\frac {4 A c \,d^{4} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2}}-\frac {4 A \,d^{2} e^{2} x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}-\frac {5 B \,b^{3} e^{4} x}{12 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{4}}+\frac {2 B \,b^{2} d \,e^{3} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{3}}-\frac {2 B b \,d^{2} e^{2} x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}}+\frac {2 B \,d^{4} x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b}-\frac {8 B \,d^{3} e x}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} c}+\frac {8 A d \,e^{3} x}{3 \sqrt {c \,x^{2}+b x}\, b c}-\frac {2 A \,d^{4}}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b}+\frac {8 A \,d^{2} e^{2} x}{\sqrt {c \,x^{2}+b x}\, b^{2}}-\frac {64 A c \,d^{3} e x}{3 \sqrt {c \,x^{2}+b x}\, b^{3}}+\frac {32 A \,c^{2} d^{4} x}{3 \sqrt {c \,x^{2}+b x}\, b^{4}}-\frac {7 A \,e^{4} x}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {35 B b \,e^{4} x}{6 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {4 B \,d^{2} e^{2} x}{\sqrt {c \,x^{2}+b x}\, b c}+\frac {16 B \,d^{3} e x}{3 \sqrt {c \,x^{2}+b x}\, b^{2}}-\frac {16 B c \,d^{4} x}{3 \sqrt {c \,x^{2}+b x}\, b^{3}}-\frac {28 B d \,e^{3} x}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {A \,e^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}-\frac {5 B b \,e^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {7}{2}}}+\frac {4 B d \,e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {5}{2}}}-\frac {A b \,e^{4}}{6 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {4 A \,d^{2} e^{2}}{\sqrt {c \,x^{2}+b x}\, b c}-\frac {32 A \,d^{3} e}{3 \sqrt {c \,x^{2}+b x}\, b^{2}}+\frac {16 A c \,d^{4}}{3 \sqrt {c \,x^{2}+b x}\, b^{3}}+\frac {4 A d \,e^{3}}{3 \sqrt {c \,x^{2}+b x}\, c^{2}}+\frac {5 B \,b^{2} e^{4}}{12 \sqrt {c \,x^{2}+b x}\, c^{4}}-\frac {2 B b d \,e^{3}}{3 \sqrt {c \,x^{2}+b x}\, c^{3}}+\frac {8 B \,d^{3} e}{3 \sqrt {c \,x^{2}+b x}\, b c}-\frac {8 B \,d^{4}}{3 \sqrt {c \,x^{2}+b x}\, b^{2}}+\frac {2 B \,d^{2} e^{2}}{\sqrt {c \,x^{2}+b x}\, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^(5/2),x)

[Out]

-8/3/b^2/(c*x^2+b*x)^(1/2)*B*d^4-2/3*A*d^4/b/(c*x^2+b*x)^(3/2)+1/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1
/2))*A*e^4+5/6*B*e^4*b/c^2*x^3/(c*x^2+b*x)^(3/2)-5/4*B*e^4*b^2/c^3*x^2/(c*x^2+b*x)^(3/2)-6*x^2/c/(c*x^2+b*x)^(
3/2)*B*d^2*e^2+8/3/b/(c*x^2+b*x)^(3/2)*x*A*d^3*e-16/3/b^3*c/(c*x^2+b*x)^(1/2)*x*B*d^4+8/3/b/c/(c*x^2+b*x)^(1/2
)*B*d^3*e-5/12*B*e^4*b^3/c^4/(c*x^2+b*x)^(3/2)*x+35/6*B*e^4*b/c^3/(c*x^2+b*x)^(1/2)*x+8/3/b/c/(c*x^2+b*x)^(1/2
)*x*A*d*e^3+4/b/c/(c*x^2+b*x)^(1/2)*x*B*d^2*e^2-64/3/b^3*c/(c*x^2+b*x)^(1/2)*x*A*d^3*e-4/3*b/c^2/(c*x^2+b*x)^(
3/2)*x*A*d*e^3+2*b/c^2*x^2/(c*x^2+b*x)^(3/2)*B*d*e^3+2/3*b^2/c^3/(c*x^2+b*x)^(3/2)*x*B*d*e^3-4/3*x^3/c/(c*x^2+
b*x)^(3/2)*B*d*e^3+1/2*b/c^2*x^2/(c*x^2+b*x)^(3/2)*A*e^4+1/6*b^2/c^3/(c*x^2+b*x)^(3/2)*x*A*e^4-4/c/(c*x^2+b*x)
^(3/2)*x*A*d^2*e^2-8/3/c/(c*x^2+b*x)^(3/2)*x*B*d^3*e-4/3*A*d^4/b^2/(c*x^2+b*x)^(3/2)*c*x+32/3*A*d^4*c^2/b^4/(c
*x^2+b*x)^(1/2)*x-2*b/c^2/(c*x^2+b*x)^(3/2)*x*B*d^2*e^2+4/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*
d*e^3-7/3/c^2/(c*x^2+b*x)^(1/2)*x*A*e^4+8/b^2/(c*x^2+b*x)^(1/2)*x*A*d^2*e^2+16/3/b^2/(c*x^2+b*x)^(1/2)*x*B*d^3
*e+4/b/c/(c*x^2+b*x)^(1/2)*A*d^2*e^2-28/3/c^2/(c*x^2+b*x)^(1/2)*x*B*d*e^3-2/3*b/c^3/(c*x^2+b*x)^(1/2)*B*d*e^3-
4*x^2/c/(c*x^2+b*x)^(3/2)*A*d*e^3+16/3*A*d^4*c/b^3/(c*x^2+b*x)^(1/2)+2/3/b/(c*x^2+b*x)^(3/2)*x*B*d^4-1/3*x^3/c
/(c*x^2+b*x)^(3/2)*A*e^4+B*e^4*x^4/c/(c*x^2+b*x)^(3/2)-1/6*b/c^3/(c*x^2+b*x)^(1/2)*A*e^4-32/3/b^2/(c*x^2+b*x)^
(1/2)*A*d^3*e+5/12*B*e^4*b^2/c^4/(c*x^2+b*x)^(1/2)-5/2*B*e^4*b/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2
))+4/3/c^2/(c*x^2+b*x)^(1/2)*A*d*e^3+2/c^2/(c*x^2+b*x)^(1/2)*B*d^2*e^2

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maxima [B]  time = 0.65, size = 795, normalized size = 2.33 \begin {gather*} \frac {5 \, B b e^{4} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )}}{6 \, c} + \frac {B e^{4} x^{4}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} - \frac {4 \, A c d^{4} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, A c^{2} d^{4} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {10 \, B b e^{4} x}{3 \, \sqrt {c x^{2} + b x} c^{3}} - \frac {5 \, B b e^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {7}{2}}} - \frac {1}{3} \, {\left (4 \, B d e^{3} + A e^{4}\right )} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {b x}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, x}{\sqrt {c x^{2} + b x} b c} - \frac {1}{\sqrt {c x^{2} + b x} c^{2}}\right )} - \frac {2 \, A d^{4}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, A c d^{4}}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {5 \, \sqrt {c x^{2} + b x} B e^{4}}{3 \, c^{3}} - \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {8 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} x}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (B d^{4} + 4 \, A d^{3} e\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} - \frac {4 \, {\left (4 \, B d e^{3} + A e^{4}\right )} x}{3 \, \sqrt {c x^{2} + b x} c^{2}} - \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {4 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {4 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x}{3 \, \sqrt {c x^{2} + b x} b c} - \frac {16 \, {\left (B d^{4} + 4 \, A d^{3} e\right )} c x}{3 \, \sqrt {c x^{2} + b x} b^{3}} + \frac {{\left (4 \, B d e^{3} + A e^{4}\right )} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {5}{2}}} - \frac {8 \, {\left (B d^{4} + 4 \, A d^{3} e\right )}}{3 \, \sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )}}{3 \, \sqrt {c x^{2} + b x} c^{2}} - \frac {2 \, {\left (4 \, B d e^{3} + A e^{4}\right )} \sqrt {c x^{2} + b x}}{3 \, b c^{2}} + \frac {4 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )}}{3 \, \sqrt {c x^{2} + b x} b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

5/6*B*b*e^4*x*(3*x^2/((c*x^2 + b*x)^(3/2)*c) + b*x/((c*x^2 + b*x)^(3/2)*c^2) - 2*x/(sqrt(c*x^2 + b*x)*b*c) - 1
/(sqrt(c*x^2 + b*x)*c^2))/c + B*e^4*x^4/((c*x^2 + b*x)^(3/2)*c) - 4/3*A*c*d^4*x/((c*x^2 + b*x)^(3/2)*b^2) + 32
/3*A*c^2*d^4*x/(sqrt(c*x^2 + b*x)*b^4) + 10/3*B*b*e^4*x/(sqrt(c*x^2 + b*x)*c^3) - 5/2*B*b*e^4*log(2*c*x + b +
2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) - 1/3*(4*B*d*e^3 + A*e^4)*x*(3*x^2/((c*x^2 + b*x)^(3/2)*c) + b*x/((c*x^2
+ b*x)^(3/2)*c^2) - 2*x/(sqrt(c*x^2 + b*x)*b*c) - 1/(sqrt(c*x^2 + b*x)*c^2)) - 2/3*A*d^4/((c*x^2 + b*x)^(3/2)*
b) + 16/3*A*c*d^4/(sqrt(c*x^2 + b*x)*b^3) + 5/3*sqrt(c*x^2 + b*x)*B*e^4/c^3 - 2*(3*B*d^2*e^2 + 2*A*d*e^3)*x^2/
((c*x^2 + b*x)^(3/2)*c) + 8/3*(2*B*d^3*e + 3*A*d^2*e^2)*x/(sqrt(c*x^2 + b*x)*b^2) + 2/3*(B*d^4 + 4*A*d^3*e)*x/
((c*x^2 + b*x)^(3/2)*b) - 4/3*(4*B*d*e^3 + A*e^4)*x/(sqrt(c*x^2 + b*x)*c^2) - 2/3*(3*B*d^2*e^2 + 2*A*d*e^3)*b*
x/((c*x^2 + b*x)^(3/2)*c^2) - 4/3*(2*B*d^3*e + 3*A*d^2*e^2)*x/((c*x^2 + b*x)^(3/2)*c) + 4/3*(3*B*d^2*e^2 + 2*A
*d*e^3)*x/(sqrt(c*x^2 + b*x)*b*c) - 16/3*(B*d^4 + 4*A*d^3*e)*c*x/(sqrt(c*x^2 + b*x)*b^3) + (4*B*d*e^3 + A*e^4)
*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 8/3*(B*d^4 + 4*A*d^3*e)/(sqrt(c*x^2 + b*x)*b^2) + 2/3*
(3*B*d^2*e^2 + 2*A*d*e^3)/(sqrt(c*x^2 + b*x)*c^2) - 2/3*(4*B*d*e^3 + A*e^4)*sqrt(c*x^2 + b*x)/(b*c^2) + 4/3*(2
*B*d^3*e + 3*A*d^2*e^2)/(sqrt(c*x^2 + b*x)*b*c)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (d + e x\right )^{4}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**4/(x*(b + c*x))**(5/2), x)

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