3.16.30 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=422 \[ \frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)^2}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x) (d+e x)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{4 e^7 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^5}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x) (-5 a B e-A b e+6 b B d)}{e^7 (a+b x)}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x) (d+e x)}+\frac {b^5 B x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)} \]

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Rubi [A]  time = 0.36, antiderivative size = 422, 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} \begin {gather*} -\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x) (d+e x)}+\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)^2}-\frac {5 b \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)}{3 e^7 (a+b x) (d+e x)^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{4 e^7 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x) (d+e x)^5}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x) (-5 a B e-A b e+6 b B d)}{e^7 (a+b x)}+\frac {b^5 B x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^6,x]

[Out]

(b^5*B*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])/(5*e^7*(a + b*x)*(d + e*x)^5) + ((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]
)/(4*e^7*(a + b*x)*(d + e*x)^4) - (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(3*e^7*(a + b*x)*(d + e*x)^3) + (5*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(e^7*(a + b*x)*(d + e*x)^2) - (5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(e^7*(a + b*x)*(d + e*x)) - (b^4*(6*b*B*d - A*b*e - 5*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)^6} \, 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)^6} \, 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 {b^{10} B}{e^6}-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^6}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^5}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^4}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^3}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^2}+\frac {b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {b^5 B 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}}{5 e^7 (a+b x) (d+e x)^5}+\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}}{4 e^7 (a+b x) (d+e x)^4}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}+\frac {5 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {b^4 (6 b B d-A b e-5 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 = 0.29, size = 490, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3 a^5 e^5 (4 A e+B (d+5 e x))+5 a^4 b e^4 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+30 a^2 b^3 e^2 \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+5 a b^4 e \left (12 A e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-B d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 b^4 (d+e x)^5 \log (d+e x) (-5 a B e-A b e+6 b B d)+b^5 \left (6 B \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )-A d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )\right )}{60 e^7 (a+b x) (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^6,x]

[Out]

-1/60*(Sqrt[(a + b*x)^2]*(3*a^5*e^5*(4*A*e + B*(d + 5*e*x)) + 5*a^4*b*e^4*(3*A*e*(d + 5*e*x) + 2*B*(d^2 + 5*d*
e*x + 10*e^2*x^2)) + 10*a^3*b^2*e^3*(2*A*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*B*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2
+ 10*e^3*x^3)) + 30*a^2*b^3*e^2*(A*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*B*(d^4 + 5*d^3*e*x + 10
*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + 5*a*b^4*e*(12*A*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3
+ 5*e^4*x^4) - B*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + b^5*(-(A*d*e*(
137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 6*B*(87*d^6 + 375*d^5*e*x + 600*d^4
*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x^6)) + 60*b^4*(6*b*B*d - A*b*e - 5*a*B*e)
*(d + e*x)^5*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^5)

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IntegrateAlgebraic [F]  time = 180.07, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^6,x]

[Out]

$Aborted

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fricas [B]  time = 0.44, size = 802, normalized size = 1.90 \begin {gather*} \frac {60 \, B b^{5} e^{6} x^{6} + 300 \, B b^{5} d e^{5} x^{5} - 522 \, B b^{5} d^{6} - 12 \, A a^{5} e^{6} + 137 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 60 \, {\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} - 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 300 \, {\left (B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 300 \, {\left (8 \, B b^{5} d^{3} e^{3} - 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} - 100 \, {\left (36 \, B b^{5} d^{4} e^{2} - 11 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 6 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 3 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 5 \, {\left (450 \, B b^{5} d^{5} e - 125 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 60 \, {\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} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x - 60 \, {\left (6 \, B b^{5} d^{6} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + {\left (6 \, B b^{5} d e^{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} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5}\right )} x^{4} + 10 \, {\left (6 \, B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (6 \, B b^{5} d^{4} e^{2} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (6 \, B b^{5} d^{5} e - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \end {gather*}

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)^6,x, algorithm="fricas")

[Out]

1/60*(60*B*b^5*e^6*x^6 + 300*B*b^5*d*e^5*x^5 - 522*B*b^5*d^6 - 12*A*a^5*e^6 + 137*(5*B*a*b^4 + A*b^5)*d^5*e -
60*(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 -
3*(B*a^5 + 5*A*a^4*b)*d*e^5 - 300*(B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5 + (2*B*a^2*b^3 + A*a*b^4)*e^6)*x
^4 - 300*(8*B*b^5*d^3*e^3 - 3*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 2*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + (B*a^3*b^2 + A*a
^2*b^3)*e^6)*x^3 - 100*(36*B*b^5*d^4*e^2 - 11*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 6*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4
+ 3*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + (B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - 5*(450*B*b^5*d^5*e - 125*(5*B*a*b^4 + A
*b^5)*d^4*e^2 + 60*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 30*(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 + 3*(B*a^5 + 5*A*a^4*b)*e^6)*x - 60*(6*B*b^5*d^6 - (5*B*a*b^4 + A*b^5)*d^5*e + (6*B*b^5*d*e^5 - (
5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(6*B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5)*x^4 + 10*(6*B*b^5*d^3*e^3 - (5*B
*a*b^4 + A*b^5)*d^2*e^4)*x^3 + 10*(6*B*b^5*d^4*e^2 - (5*B*a*b^4 + A*b^5)*d^3*e^3)*x^2 + 5*(6*B*b^5*d^5*e - (5*
B*a*b^4 + A*b^5)*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4
*e^8*x + d^5*e^7)

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giac [B]  time = 0.27, size = 863, normalized size = 2.05

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)^6,x, algorithm="giac")

[Out]

B*b^5*x*e^(-6)*sgn(b*x + a) - (6*B*b^5*d*sgn(b*x + a) - 5*B*a*b^4*e*sgn(b*x + a) - A*b^5*e*sgn(b*x + a))*e^(-7
)*log(abs(x*e + d)) - 1/60*(522*B*b^5*d^6*sgn(b*x + a) - 685*B*a*b^4*d^5*e*sgn(b*x + a) - 137*A*b^5*d^5*e*sgn(
b*x + a) + 120*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 60*A*a*b^4*d^4*e^2*sgn(b*x + a) + 30*B*a^3*b^2*d^3*e^3*sgn(b*x
+ a) + 30*A*a^2*b^3*d^3*e^3*sgn(b*x + a) + 10*B*a^4*b*d^2*e^4*sgn(b*x + a) + 20*A*a^3*b^2*d^2*e^4*sgn(b*x + a
) + 3*B*a^5*d*e^5*sgn(b*x + a) + 15*A*a^4*b*d*e^5*sgn(b*x + a) + 12*A*a^5*e^6*sgn(b*x + a) + 300*(3*B*b^5*d^2*
e^4*sgn(b*x + a) - 5*B*a*b^4*d*e^5*sgn(b*x + a) - A*b^5*d*e^5*sgn(b*x + a) + 2*B*a^2*b^3*e^6*sgn(b*x + a) + A*
a*b^4*e^6*sgn(b*x + a))*x^4 + 300*(10*B*b^5*d^3*e^3*sgn(b*x + a) - 15*B*a*b^4*d^2*e^4*sgn(b*x + a) - 3*A*b^5*d
^2*e^4*sgn(b*x + a) + 4*B*a^2*b^3*d*e^5*sgn(b*x + a) + 2*A*a*b^4*d*e^5*sgn(b*x + a) + B*a^3*b^2*e^6*sgn(b*x +
a) + A*a^2*b^3*e^6*sgn(b*x + a))*x^3 + 100*(39*B*b^5*d^4*e^2*sgn(b*x + a) - 55*B*a*b^4*d^3*e^3*sgn(b*x + a) -
11*A*b^5*d^3*e^3*sgn(b*x + a) + 12*B*a^2*b^3*d^2*e^4*sgn(b*x + a) + 6*A*a*b^4*d^2*e^4*sgn(b*x + a) + 3*B*a^3*b
^2*d*e^5*sgn(b*x + a) + 3*A*a^2*b^3*d*e^5*sgn(b*x + a) + B*a^4*b*e^6*sgn(b*x + a) + 2*A*a^3*b^2*e^6*sgn(b*x +
a))*x^2 + 5*(462*B*b^5*d^5*e*sgn(b*x + a) - 625*B*a*b^4*d^4*e^2*sgn(b*x + a) - 125*A*b^5*d^4*e^2*sgn(b*x + a)
+ 120*B*a^2*b^3*d^3*e^3*sgn(b*x + a) + 60*A*a*b^4*d^3*e^3*sgn(b*x + a) + 30*B*a^3*b^2*d^2*e^4*sgn(b*x + a) + 3
0*A*a^2*b^3*d^2*e^4*sgn(b*x + a) + 10*B*a^4*b*d*e^5*sgn(b*x + a) + 20*A*a^3*b^2*d*e^5*sgn(b*x + a) + 3*B*a^5*e
^6*sgn(b*x + a) + 15*A*a^4*b*e^6*sgn(b*x + a))*x)*e^(-7)/(x*e + d)^5

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maple [B]  time = 0.07, size = 1012, normalized size = 2.40 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 A \,b^{5} e^{6} x^{5} \ln \left (e x +d \right )+300 B a \,b^{4} e^{6} x^{5} \ln \left (e x +d \right )-360 B \,b^{5} d \,e^{5} x^{5} \ln \left (e x +d \right )+60 B \,b^{5} e^{6} x^{6}+300 A \,b^{5} d \,e^{5} x^{4} \ln \left (e x +d \right )+1500 B a \,b^{4} d \,e^{5} x^{4} \ln \left (e x +d \right )-1800 B \,b^{5} d^{2} e^{4} x^{4} \ln \left (e x +d \right )+300 B \,b^{5} d \,e^{5} x^{5}-300 A a \,b^{4} e^{6} x^{4}+600 A \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+300 A \,b^{5} d \,e^{5} x^{4}-600 B \,a^{2} b^{3} e^{6} x^{4}+3000 B a \,b^{4} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+1500 B a \,b^{4} d \,e^{5} x^{4}-3600 B \,b^{5} d^{3} e^{3} x^{3} \ln \left (e x +d \right )-300 B \,b^{5} d^{2} e^{4} x^{4}-300 A \,a^{2} b^{3} e^{6} x^{3}-600 A a \,b^{4} d \,e^{5} x^{3}+600 A \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )+900 A \,b^{5} d^{2} e^{4} x^{3}-300 B \,a^{3} b^{2} e^{6} x^{3}-1200 B \,a^{2} b^{3} d \,e^{5} x^{3}+3000 B a \,b^{4} d^{3} e^{3} x^{2} \ln \left (e x +d \right )+4500 B a \,b^{4} d^{2} e^{4} x^{3}-3600 B \,b^{5} d^{4} e^{2} x^{2} \ln \left (e x +d \right )-2400 B \,b^{5} d^{3} e^{3} x^{3}-200 A \,a^{3} b^{2} e^{6} x^{2}-300 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^{4} e^{2} x \ln \left (e x +d \right )+1100 A \,b^{5} d^{3} e^{3} x^{2}-100 B \,a^{4} b \,e^{6} x^{2}-300 B \,a^{3} b^{2} d \,e^{5} x^{2}-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 )+5500 B a \,b^{4} d^{3} e^{3} x^{2}-1800 B \,b^{5} d^{5} e x \ln \left (e x +d \right )-3600 B \,b^{5} d^{4} e^{2} x^{2}-75 A \,a^{4} b \,e^{6} x -100 A \,a^{3} b^{2} d \,e^{5} x -150 A \,a^{2} b^{3} d^{2} e^{4} x -300 A a \,b^{4} d^{3} e^{3} x +60 A \,b^{5} d^{5} e \ln \left (e x +d \right )+625 A \,b^{5} d^{4} e^{2} x -15 B \,a^{5} e^{6} x -50 B \,a^{4} b d \,e^{5} x -150 B \,a^{3} b^{2} d^{2} e^{4} x -600 B \,a^{2} b^{3} d^{3} e^{3} x +300 B a \,b^{4} d^{5} e \ln \left (e x +d \right )+3125 B a \,b^{4} d^{4} e^{2} x -360 B \,b^{5} d^{6} \ln \left (e x +d \right )-2250 B \,b^{5} d^{5} e x -12 A \,a^{5} e^{6}-15 A \,a^{4} b d \,e^{5}-20 A \,a^{3} b^{2} d^{2} e^{4}-30 A \,a^{2} b^{3} d^{3} e^{3}-60 A a \,b^{4} d^{4} e^{2}+137 A \,b^{5} d^{5} e -3 B \,a^{5} d \,e^{5}-10 B \,a^{4} b \,d^{2} e^{4}-30 B \,a^{3} b^{2} d^{3} e^{3}-120 B \,a^{2} b^{3} d^{4} e^{2}+685 B a \,b^{4} d^{5} e -522 B \,b^{5} d^{6}\right )}{60 \left (b x +a \right )^{5} \left (e x +d \right )^{5} e^{7}} \end {gather*}

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)^6,x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(1500*B*a*b^4*d*e^5*x^4*ln(e*x+d)-50*B*a^4*b*d*e^5*x-150*B*a^3*b^2*d^2*e^4*x-600*B*a^2*
b^3*d^3*e^3*x+3125*B*a*b^4*d^4*e^2*x+300*B*ln(e*x+d)*x^5*a*b^4*e^6+137*A*b^5*d^5*e+300*B*a*b^4*d^5*e*ln(e*x+d)
-60*A*a*b^4*d^4*e^2-10*B*a^4*b*d^2*e^4-20*A*a^3*b^2*d^2*e^4-30*A*a^2*b^3*d^3*e^3-3*B*a^5*d*e^5-30*B*a^3*b^2*d^
3*e^3-120*B*a^2*b^3*d^4*e^2+685*B*a*b^4*d^5*e+1500*B*a*b^4*d^4*e^2*x*ln(e*x+d)-300*A*a^2*b^3*d*e^5*x^2-600*A*a
*b^4*d^2*e^4*x^2+60*A*b^5*d^5*e*ln(e*x+d)-75*A*a^4*b*e^6*x+625*A*b^5*d^4*e^2*x-300*A*a^2*b^3*e^6*x^3-360*B*ln(
e*x+d)*x^5*b^5*d*e^5-3600*B*b^5*d^3*e^3*x^3*ln(e*x+d)+300*A*b^5*d^4*e^2*x*ln(e*x+d)-1800*B*b^5*d^5*e*x*ln(e*x+
d)+300*A*b^5*d*e^5*x^4*ln(e*x+d)-1800*B*b^5*d^2*e^4*x^4*ln(e*x+d)-12*A*a^5*e^6-522*B*b^5*d^6+1500*B*a*b^4*d*e^
5*x^4-15*A*a^4*b*d*e^5-1200*B*a^2*b^3*d*e^5*x^3+4500*B*a*b^4*d^2*e^4*x^3-100*A*a^3*b^2*d*e^5*x-150*A*a^2*b^3*d
^2*e^4*x-300*A*a*b^4*d^3*e^3*x-300*B*a^3*b^2*d*e^5*x^2+900*A*b^5*d^2*e^4*x^3-300*B*a^3*b^2*e^6*x^3-2400*B*b^5*
d^3*e^3*x^3-200*A*a^3*b^2*e^6*x^2+1100*A*b^5*d^3*e^3*x^2-100*B*a^4*b*e^6*x^2-3600*B*b^5*d^4*e^2*x^2+60*B*b^5*e
^6*x^6-15*B*a^5*e^6*x-360*B*b^5*d^6*ln(e*x+d)-600*A*a*b^4*d*e^5*x^3+3000*B*a*b^4*d^2*e^4*x^3*ln(e*x+d)+600*A*b
^5*d^3*e^3*x^2*ln(e*x+d)-3600*B*b^5*d^4*e^2*x^2*ln(e*x+d)+600*A*b^5*d^2*e^4*x^3*ln(e*x+d)+5500*B*a*b^4*d^3*e^3
*x^2-1200*B*a^2*b^3*d^2*e^4*x^2+3000*B*a*b^4*d^3*e^3*x^2*ln(e*x+d)-2250*B*b^5*d^5*e*x+300*B*b^5*d*e^5*x^5-300*
A*a*b^4*e^6*x^4+300*A*b^5*d*e^5*x^4-600*B*a^2*b^3*e^6*x^4-300*B*b^5*d^2*e^4*x^4+60*A*ln(e*x+d)*x^5*b^5*e^6)/(b
*x+a)^5/e^7/(e*x+d)^5

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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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^6,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 \begin {gather*} \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 )}^6} \,d x \end {gather*}

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)^6,x)

[Out]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^6, 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 (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}

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)**6,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/(d + e*x)**6, x)

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