3.1746 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=423 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) (d+e x)}-\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x) (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}+\frac {5 b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^6 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4 (-5 a B e-A b e+6 b B d)}{4 e^7 (a+b x)}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)} \]

[Out]

5*b*(-a*e+b*d)^3*(-2*A*b*e-B*a*e+3*B*b*d)*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-(-a*e+b*d)^5*(-A*e+B*d)*((b*x+a)^2)^
(1/2)/e^7/(b*x+a)/(e*x+d)-5*b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+5/
3*b^3*(-a*e+b*d)*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^3*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-1/4*b^4*(-A*b*e-5*B*a*e+6*B*
b*d)*(e*x+d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+1/5*b^5*B*(e*x+d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-(-a*e+b*d)^4*(-
5*A*b*e-B*a*e+6*B*b*d)*ln(e*x+d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

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Rubi [A]  time = 0.56, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) (d+e x)}+\frac {5 b x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^6 (a+b x)}-\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4 (-5 a B e-A b e+6 b B d)}{4 e^7 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x) (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) - ((b*d - a*e)
^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) - (5*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*
e - a*B*e)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e -
2*a*B*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d +
e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)) + (b^5*B*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5
*e^7*(a + b*x)) - ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*
(a + b*x))

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6}-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^2}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^2}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^3}{e^6}+\frac {b^{10} B (d+e x)^4}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {5 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {b^4 (6 b B d-A b e-5 a B e) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}+\frac {b^5 B (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}-\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 506, normalized size = 1.20 \[ \frac {\sqrt {(a+b x)^2} \left (60 a^5 e^5 (B d-A e)+300 a^4 b e^4 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+300 a^3 b^2 e^3 \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )+100 a^2 b^3 e^2 \left (3 A e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+2 B \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+25 a b^4 e \left (4 A e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+B \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )-60 (d+e x) (b d-a e)^4 \log (d+e x) (-a B e-5 A b e+6 b B d)+b^5 \left (5 A e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-6 B \left (10 d^6-50 d^5 e x-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4+3 d e^5 x^5-2 e^6 x^6\right )\right )\right )}{60 e^7 (a+b x) (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x)^2]*(60*a^5*e^5*(B*d - A*e) + 300*a^4*b*e^4*(A*d*e + B*(-d^2 + d*e*x + e^2*x^2)) + 300*a^3*b^2*
e^3*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + 100*a^2*b^3*e^2*(3*A*e*
(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)
) + 25*a*b^4*e*(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x -
30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + b^5*(5*A*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2
+ 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) - 6*B*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*
d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) - 60*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)*Log[d + e*x])
)/(60*e^7*(a + b*x)*(d + e*x))

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fricas [B]  time = 0.62, size = 797, normalized size = 1.88 \[ \frac {12 \, B b^{5} e^{6} x^{6} - 60 \, B b^{5} d^{6} - 60 \, A a^{5} e^{6} + 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 300 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 600 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 300 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 60 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 3 \, {\left (6 \, B b^{5} d e^{5} - 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \, {\left (6 \, B b^{5} d^{2} e^{4} - 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 20 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 10 \, {\left (6 \, B b^{5} d^{3} e^{3} - 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 20 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 30 \, {\left (6 \, B b^{5} d^{4} e^{2} - 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 20 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 60 \, {\left (5 \, B b^{5} d^{5} e - 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 15 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 20 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5}\right )} x - 60 \, {\left (6 \, B b^{5} d^{6} - 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 20 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + {\left (6 \, B b^{5} d^{5} e - 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 20 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{8} x + d e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*b^5*e^6*x^6 - 60*B*b^5*d^6 - 60*A*a^5*e^6 + 60*(5*B*a*b^4 + A*b^5)*d^5*e - 300*(2*B*a^2*b^3 + A*a*b
^4)*d^4*e^2 + 600*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 300*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 60*(B*a^5 + 5*A*a^4*
b)*d*e^5 - 3*(6*B*b^5*d*e^5 - 5*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(6*B*b^5*d^2*e^4 - 5*(5*B*a*b^4 + A*b^5)*d*e^
5 + 20*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 10*(6*B*b^5*d^3*e^3 - 5*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 20*(2*B*a^2*b^
3 + A*a*b^4)*d*e^5 - 30*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 30*(6*B*b^5*d^4*e^2 - 5*(5*B*a*b^4 + A*b^5)*d^3*e^3
+ 20*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 30*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 10*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2
+ 60*(5*B*b^5*d^5*e - 4*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 15*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 20*(B*a^3*b^2 + A*
a^2*b^3)*d^2*e^4 + 5*(B*a^4*b + 2*A*a^3*b^2)*d*e^5)*x - 60*(6*B*b^5*d^6 - 5*(5*B*a*b^4 + A*b^5)*d^5*e + 20*(2*
B*a^2*b^3 + A*a*b^4)*d^4*e^2 - 30*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 10*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - (B*a^
5 + 5*A*a^4*b)*d*e^5 + (6*B*b^5*d^5*e - 5*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 20*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 3
0*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 10*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - (B*a^5 + 5*A*a^4*b)*e^6)*x)*log(e*x + d
))/(e^8*x + d*e^7)

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giac [B]  time = 0.22, size = 905, normalized size = 2.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*B*b^5*d^5*sgn(b*x + a) - 25*B*a*b^4*d^4*e*sgn(b*x + a) - 5*A*b^5*d^4*e*sgn(b*x + a) + 40*B*a^2*b^3*d^3*e^2
*sgn(b*x + a) + 20*A*a*b^4*d^3*e^2*sgn(b*x + a) - 30*B*a^3*b^2*d^2*e^3*sgn(b*x + a) - 30*A*a^2*b^3*d^2*e^3*sgn
(b*x + a) + 10*B*a^4*b*d*e^4*sgn(b*x + a) + 20*A*a^3*b^2*d*e^4*sgn(b*x + a) - B*a^5*e^5*sgn(b*x + a) - 5*A*a^4
*b*e^5*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/60*(12*B*b^5*x^5*e^8*sgn(b*x + a) - 30*B*b^5*d*x^4*e^7*sgn(b
*x + a) + 60*B*b^5*d^2*x^3*e^6*sgn(b*x + a) - 120*B*b^5*d^3*x^2*e^5*sgn(b*x + a) + 300*B*b^5*d^4*x*e^4*sgn(b*x
+ a) + 75*B*a*b^4*x^4*e^8*sgn(b*x + a) + 15*A*b^5*x^4*e^8*sgn(b*x + a) - 200*B*a*b^4*d*x^3*e^7*sgn(b*x + a) -
40*A*b^5*d*x^3*e^7*sgn(b*x + a) + 450*B*a*b^4*d^2*x^2*e^6*sgn(b*x + a) + 90*A*b^5*d^2*x^2*e^6*sgn(b*x + a) -
1200*B*a*b^4*d^3*x*e^5*sgn(b*x + a) - 240*A*b^5*d^3*x*e^5*sgn(b*x + a) + 200*B*a^2*b^3*x^3*e^8*sgn(b*x + a) +
100*A*a*b^4*x^3*e^8*sgn(b*x + a) - 600*B*a^2*b^3*d*x^2*e^7*sgn(b*x + a) - 300*A*a*b^4*d*x^2*e^7*sgn(b*x + a) +
1800*B*a^2*b^3*d^2*x*e^6*sgn(b*x + a) + 900*A*a*b^4*d^2*x*e^6*sgn(b*x + a) + 300*B*a^3*b^2*x^2*e^8*sgn(b*x +
a) + 300*A*a^2*b^3*x^2*e^8*sgn(b*x + a) - 1200*B*a^3*b^2*d*x*e^7*sgn(b*x + a) - 1200*A*a^2*b^3*d*x*e^7*sgn(b*x
+ a) + 300*B*a^4*b*x*e^8*sgn(b*x + a) + 600*A*a^3*b^2*x*e^8*sgn(b*x + a))*e^(-10) - (B*b^5*d^6*sgn(b*x + a) -
5*B*a*b^4*d^5*e*sgn(b*x + a) - A*b^5*d^5*e*sgn(b*x + a) + 10*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 5*A*a*b^4*d^4*e
^2*sgn(b*x + a) - 10*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^3*e^3*sgn(b*x + a) + 5*B*a^4*b*d^2*e^4*sg
n(b*x + a) + 10*A*a^3*b^2*d^2*e^4*sgn(b*x + a) - B*a^5*d*e^5*sgn(b*x + a) - 5*A*a^4*b*d*e^5*sgn(b*x + a) + A*a
^5*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)

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maple [B]  time = 0.07, size = 1084, normalized size = 2.56 \[ \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (12 B \,b^{5} e^{6} x^{6}+15 A \,b^{5} e^{6} x^{5}+75 B a \,b^{4} e^{6} x^{5}-18 B \,b^{5} d \,e^{5} x^{5}+100 A a \,b^{4} e^{6} x^{4}-25 A \,b^{5} d \,e^{5} x^{4}+200 B \,a^{2} b^{3} e^{6} x^{4}-125 B a \,b^{4} d \,e^{5} x^{4}+30 B \,b^{5} d^{2} e^{4} x^{4}+300 A \,a^{2} b^{3} e^{6} x^{3}-200 A a \,b^{4} d \,e^{5} x^{3}+50 A \,b^{5} d^{2} e^{4} x^{3}+300 B \,a^{3} b^{2} e^{6} x^{3}-400 B \,a^{2} b^{3} d \,e^{5} x^{3}+250 B a \,b^{4} d^{2} e^{4} x^{3}-60 B \,b^{5} d^{3} e^{3} x^{3}+300 A \,a^{4} b \,e^{6} x \ln \left (e x +d \right )-1200 A \,a^{3} b^{2} d \,e^{5} x \ln \left (e x +d \right )+600 A \,a^{3} b^{2} e^{6} x^{2}+1800 A \,a^{2} b^{3} d^{2} e^{4} x \ln \left (e x +d \right )-900 A \,a^{2} b^{3} d \,e^{5} x^{2}-1200 A a \,b^{4} d^{3} e^{3} x \ln \left (e x +d \right )+600 A a \,b^{4} d^{2} e^{4} x^{2}+300 A \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )-150 A \,b^{5} d^{3} e^{3} x^{2}+60 B \,a^{5} e^{6} x \ln \left (e x +d \right )-600 B \,a^{4} b d \,e^{5} x \ln \left (e x +d \right )+300 B \,a^{4} b \,e^{6} x^{2}+1800 B \,a^{3} b^{2} d^{2} e^{4} x \ln \left (e x +d \right )-900 B \,a^{3} b^{2} d \,e^{5} x^{2}-2400 B \,a^{2} b^{3} d^{3} e^{3} x \ln \left (e x +d \right )+1200 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1500 B a \,b^{4} d^{4} e^{2} x \ln \left (e x +d \right )-750 B a \,b^{4} d^{3} e^{3} x^{2}-360 B \,b^{5} d^{5} e x \ln \left (e x +d \right )+180 B \,b^{5} d^{4} e^{2} x^{2}+300 A \,a^{4} b d \,e^{5} \ln \left (e x +d \right )-1200 A \,a^{3} b^{2} d^{2} e^{4} \ln \left (e x +d \right )+600 A \,a^{3} b^{2} d \,e^{5} x +1800 A \,a^{2} b^{3} d^{3} e^{3} \ln \left (e x +d \right )-1200 A \,a^{2} b^{3} d^{2} e^{4} x -1200 A a \,b^{4} d^{4} e^{2} \ln \left (e x +d \right )+900 A a \,b^{4} d^{3} e^{3} x +300 A \,b^{5} d^{5} e \ln \left (e x +d \right )-240 A \,b^{5} d^{4} e^{2} x +60 B \,a^{5} d \,e^{5} \ln \left (e x +d \right )-600 B \,a^{4} b \,d^{2} e^{4} \ln \left (e x +d \right )+300 B \,a^{4} b d \,e^{5} x +1800 B \,a^{3} b^{2} d^{3} e^{3} \ln \left (e x +d \right )-1200 B \,a^{3} b^{2} d^{2} e^{4} x -2400 B \,a^{2} b^{3} d^{4} e^{2} \ln \left (e x +d \right )+1800 B \,a^{2} b^{3} d^{3} e^{3} x +1500 B a \,b^{4} d^{5} e \ln \left (e x +d \right )-1200 B a \,b^{4} d^{4} e^{2} x -360 B \,b^{5} d^{6} \ln \left (e x +d \right )+300 B \,b^{5} d^{5} e x -60 A \,a^{5} e^{6}+300 A \,a^{4} b d \,e^{5}-600 A \,a^{3} b^{2} d^{2} e^{4}+600 A \,a^{2} b^{3} d^{3} e^{3}-300 A a \,b^{4} d^{4} e^{2}+60 A \,b^{5} d^{5} e +60 B \,a^{5} d \,e^{5}-300 B \,a^{4} b \,d^{2} e^{4}+600 B \,a^{3} b^{2} d^{3} e^{3}-600 B \,a^{2} b^{3} d^{4} e^{2}+300 B a \,b^{4} d^{5} e -60 B \,b^{5} d^{6}\right )}{60 \left (b x +a \right )^{5} \left (e x +d \right ) e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*((b*x+a)^2)^(5/2)*(-1200*A*a*b^4*d^4*e^2*ln(e*x+d)+300*B*a^4*b*d*e^5*x-1200*B*a^3*b^2*d^2*e^4*x+1800*B*a^
2*b^3*d^3*e^3*x-1200*B*a*b^4*d^4*e^2*x+60*A*b^5*d^5*e-2400*B*a^2*b^3*d^4*e^2*ln(e*x+d)+1500*B*a*b^4*d^5*e*ln(e
*x+d)+300*A*a^4*b*d*e^5*ln(e*x+d)-300*A*a*b^4*d^4*e^2-300*B*a^4*b*d^2*e^4-600*A*a^3*b^2*d^2*e^4+600*A*a^2*b^3*
d^3*e^3+60*B*d*e^5*a^5+600*B*a^3*b^2*d^3*e^3-600*B*a^2*b^3*d^4*e^2+300*B*a*b^4*d^5*e-600*B*ln(e*x+d)*x*a^4*b*d
*e^5+1800*B*ln(e*x+d)*x*a^3*b^2*d^2*e^4-2400*B*ln(e*x+d)*x*a^2*b^3*d^3*e^3+1500*B*ln(e*x+d)*x*a*b^4*d^4*e^2-90
0*A*a^2*b^3*d*e^5*x^2+600*A*a*b^4*d^2*e^4*x^2+300*A*b^5*d^5*e*ln(e*x+d)+60*B*a^5*d*e^5*ln(e*x+d)-240*A*b^5*d^4
*e^2*x+300*A*a^2*b^3*e^6*x^3-1200*A*ln(e*x+d)*x*a^3*b^2*d*e^5+1800*A*ln(e*x+d)*x*a^2*b^3*d^2*e^4-1200*A*ln(e*x
+d)*x*a*b^4*d^3*e^3-600*B*a^4*b*d^2*e^4*ln(e*x+d)+1800*B*a^3*b^2*d^3*e^3*ln(e*x+d)+300*A*ln(e*x+d)*x*a^4*b*e^6
+300*A*ln(e*x+d)*x*b^5*d^4*e^2-360*B*ln(e*x+d)*x*b^5*d^5*e-60*A*a^5*e^6-60*B*b^5*d^6-1200*A*a^3*b^2*d^2*e^4*ln
(e*x+d)+1800*A*a^2*b^3*d^3*e^3*ln(e*x+d)-125*B*a*b^4*d*e^5*x^4+300*A*d*e^5*a^4*b-400*B*a^2*b^3*d*e^5*x^3+250*B
*a*b^4*d^2*e^4*x^3+600*A*a^3*b^2*d*e^5*x-1200*A*a^2*b^3*d^2*e^4*x+900*A*a*b^4*d^3*e^3*x-900*B*a^3*b^2*d*e^5*x^
2+50*A*b^5*d^2*e^4*x^3+300*B*a^3*b^2*e^6*x^3-60*B*b^5*d^3*e^3*x^3+600*A*a^3*b^2*e^6*x^2-150*A*b^5*d^3*e^3*x^2+
300*B*a^4*b*e^6*x^2+180*B*b^5*d^4*e^2*x^2+75*B*a*b^4*e^6*x^5+12*B*b^5*e^6*x^6+15*A*b^5*e^6*x^5-360*B*b^5*d^6*l
n(e*x+d)-200*A*a*b^4*d*e^5*x^3-750*B*a*b^4*d^3*e^3*x^2+1200*B*a^2*b^3*d^2*e^4*x^2+300*B*b^5*d^5*e*x-18*B*b^5*d
*e^5*x^5+100*A*a*b^4*e^6*x^4-25*A*b^5*d*e^5*x^4+200*B*a^2*b^3*e^6*x^4+30*B*b^5*d^2*e^4*x^4+60*B*ln(e*x+d)*x*a^
5*e^6)/(b*x+a)^5/e^7/(e*x+d)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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