3.16.37 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{13}} \, dx\)

Optimal. Leaf size=438 \[ \frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{12 e^7 (a+b x) (d+e x)^{12}}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6} \]

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Rubi [A]  time = 0.30, antiderivative size = 438, 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 {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\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)}{8 e^7 (a+b x) (d+e x)^8}+\frac {10 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)}{9 e^7 (a+b x) (d+e x)^9}-\frac {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)}{2 e^7 (a+b x) (d+e x)^{10}}+\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)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{12 e^7 (a+b x) (d+e x)^{12}}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6} \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)^13,x]

[Out]

-((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^7*(a + b*x)*(d + e*x)^12) + ((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])/(11*e^7*(a + b*x)*(d + e*x)^11) - (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])/(2*e^7*(a + b*x)*(d + e*x)^10) + (10*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])/(9*e^7*(a + b*x)*(d + e*x)^9) - (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])/(8*e^7*(a + b*x)*(d + e*x)^8) + (b^4*(6*b*B*d - A*b
*e - 5*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^7) - (b^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(6*e^7*(a + b*x)*(d + e*x)^6)

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)^{13}} \, 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)^{13}} \, 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^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{13}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{12}}-\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)^{11}}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^{10}}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^9}+\frac {b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)^8}+\frac {b^{10} B}{e^6 (d+e x)^7}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{12 e^7 (a+b x) (d+e x)^{12}}+\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}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {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}}{2 e^7 (a+b x) (d+e x)^{10}}+\frac {10 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}}{9 e^7 (a+b x) (d+e x)^9}-\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}}{8 e^7 (a+b x) (d+e x)^8}+\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}}{7 e^7 (a+b x) (d+e x)^7}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 465, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (42 a^5 e^5 (11 A e+B (d+12 e x))+42 a^4 b e^4 \left (5 A e (d+12 e x)+B \left (d^2+12 d e x+66 e^2 x^2\right )\right )+28 a^3 b^2 e^3 \left (3 A e \left (d^2+12 d e x+66 e^2 x^2\right )+B \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )\right )+14 a^2 b^3 e^2 \left (2 A e \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+B \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )\right )+a b^4 e \left (7 A e \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+5 B \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+B \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )\right )\right )}{5544 e^7 (a+b x) (d+e x)^{12}} \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)^13,x]

[Out]

-1/5544*(Sqrt[(a + b*x)^2]*(42*a^5*e^5*(11*A*e + B*(d + 12*e*x)) + 42*a^4*b*e^4*(5*A*e*(d + 12*e*x) + B*(d^2 +
12*d*e*x + 66*e^2*x^2)) + 28*a^3*b^2*e^3*(3*A*e*(d^2 + 12*d*e*x + 66*e^2*x^2) + B*(d^3 + 12*d^2*e*x + 66*d*e^
2*x^2 + 220*e^3*x^3)) + 14*a^2*b^3*e^2*(2*A*e*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + B*(d^4 + 12*d^
3*e*x + 66*d^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4)) + a*b^4*e*(7*A*e*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 2
20*d*e^3*x^3 + 495*e^4*x^4) + 5*B*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e
^5*x^5)) + b^5*(A*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) + B*(d
^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6))))/(e^7*(a
+ b*x)*(d + e*x)^12)

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IntegrateAlgebraic [F]  time = 180.13, 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)^13,x]

[Out]

$Aborted

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fricas [A]  time = 0.44, size = 672, normalized size = 1.53 \begin {gather*} -\frac {924 \, B b^{5} e^{6} x^{6} + B b^{5} d^{6} + 462 \, A a^{5} e^{6} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 28 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 42 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 792 \, {\left (B b^{5} d e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 495 \, {\left (B b^{5} d^{2} e^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 220 \, {\left (B b^{5} d^{3} e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 28 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 66 \, {\left (B b^{5} d^{4} e^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 28 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 12 \, {\left (B b^{5} d^{5} e + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 7 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 28 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 42 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{5544 \, {\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} 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)^13,x, algorithm="fricas")

[Out]

-1/5544*(924*B*b^5*e^6*x^6 + B*b^5*d^6 + 462*A*a^5*e^6 + (5*B*a*b^4 + A*b^5)*d^5*e + 7*(2*B*a^2*b^3 + A*a*b^4)
*d^4*e^2 + 28*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 42*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 42*(B*a^5 + 5*A*a^4*b)*d*
e^5 + 792*(B*b^5*d*e^5 + (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 495*(B*b^5*d^2*e^4 + (5*B*a*b^4 + A*b^5)*d*e^5 + 7*(2*
B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 220*(B*b^5*d^3*e^3 + (5*B*a*b^4 + A*b^5)*d^2*e^4 + 7*(2*B*a^2*b^3 + A*a*b^4)*d
*e^5 + 28*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 66*(B*b^5*d^4*e^2 + (5*B*a*b^4 + A*b^5)*d^3*e^3 + 7*(2*B*a^2*b^3
+ A*a*b^4)*d^2*e^4 + 28*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 42*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 12*(B*b^5*d^5*e
+ (5*B*a*b^4 + A*b^5)*d^4*e^2 + 7*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 28*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 42*(B
*a^4*b + 2*A*a^3*b^2)*d*e^5 + 42*(B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^19*x^12 + 12*d*e^18*x^11 + 66*d^2*e^17*x^10 +
220*d^3*e^16*x^9 + 495*d^4*e^15*x^8 + 792*d^5*e^14*x^7 + 924*d^6*e^13*x^6 + 792*d^7*e^12*x^5 + 495*d^8*e^11*x^
4 + 220*d^9*e^10*x^3 + 66*d^10*e^9*x^2 + 12*d^11*e^8*x + d^12*e^7)

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giac [B]  time = 0.22, size = 917, normalized size = 2.09

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)^13,x, algorithm="giac")

[Out]

-1/5544*(924*B*b^5*x^6*e^6*sgn(b*x + a) + 792*B*b^5*d*x^5*e^5*sgn(b*x + a) + 495*B*b^5*d^2*x^4*e^4*sgn(b*x + a
) + 220*B*b^5*d^3*x^3*e^3*sgn(b*x + a) + 66*B*b^5*d^4*x^2*e^2*sgn(b*x + a) + 12*B*b^5*d^5*x*e*sgn(b*x + a) + B
*b^5*d^6*sgn(b*x + a) + 3960*B*a*b^4*x^5*e^6*sgn(b*x + a) + 792*A*b^5*x^5*e^6*sgn(b*x + a) + 2475*B*a*b^4*d*x^
4*e^5*sgn(b*x + a) + 495*A*b^5*d*x^4*e^5*sgn(b*x + a) + 1100*B*a*b^4*d^2*x^3*e^4*sgn(b*x + a) + 220*A*b^5*d^2*
x^3*e^4*sgn(b*x + a) + 330*B*a*b^4*d^3*x^2*e^3*sgn(b*x + a) + 66*A*b^5*d^3*x^2*e^3*sgn(b*x + a) + 60*B*a*b^4*d
^4*x*e^2*sgn(b*x + a) + 12*A*b^5*d^4*x*e^2*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) + 6930*B*a^2*b^3*x^4*e^6*sgn(b*x + a) + 3465*A*a*b^4*x^4*e^6*sgn(b*x + a) + 3080*B*a^2*b^3*d*x^3*e^5*sgn(b
*x + a) + 1540*A*a*b^4*d*x^3*e^5*sgn(b*x + a) + 924*B*a^2*b^3*d^2*x^2*e^4*sgn(b*x + a) + 462*A*a*b^4*d^2*x^2*e
^4*sgn(b*x + a) + 168*B*a^2*b^3*d^3*x*e^3*sgn(b*x + a) + 84*A*a*b^4*d^3*x*e^3*sgn(b*x + a) + 14*B*a^2*b^3*d^4*
e^2*sgn(b*x + a) + 7*A*a*b^4*d^4*e^2*sgn(b*x + a) + 6160*B*a^3*b^2*x^3*e^6*sgn(b*x + a) + 6160*A*a^2*b^3*x^3*e
^6*sgn(b*x + a) + 1848*B*a^3*b^2*d*x^2*e^5*sgn(b*x + a) + 1848*A*a^2*b^3*d*x^2*e^5*sgn(b*x + a) + 336*B*a^3*b^
2*d^2*x*e^4*sgn(b*x + a) + 336*A*a^2*b^3*d^2*x*e^4*sgn(b*x + a) + 28*B*a^3*b^2*d^3*e^3*sgn(b*x + a) + 28*A*a^2
*b^3*d^3*e^3*sgn(b*x + a) + 2772*B*a^4*b*x^2*e^6*sgn(b*x + a) + 5544*A*a^3*b^2*x^2*e^6*sgn(b*x + a) + 504*B*a^
4*b*d*x*e^5*sgn(b*x + a) + 1008*A*a^3*b^2*d*x*e^5*sgn(b*x + a) + 42*B*a^4*b*d^2*e^4*sgn(b*x + a) + 84*A*a^3*b^
2*d^2*e^4*sgn(b*x + a) + 504*B*a^5*x*e^6*sgn(b*x + a) + 2520*A*a^4*b*x*e^6*sgn(b*x + a) + 42*B*a^5*d*e^5*sgn(b
*x + a) + 210*A*a^4*b*d*e^5*sgn(b*x + a) + 462*A*a^5*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^12

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maple [A]  time = 0.06, size = 687, normalized size = 1.57 \begin {gather*} -\frac {\left (924 B \,b^{5} e^{6} x^{6}+792 A \,b^{5} e^{6} x^{5}+3960 B a \,b^{4} e^{6} x^{5}+792 B \,b^{5} d \,e^{5} x^{5}+3465 A a \,b^{4} e^{6} x^{4}+495 A \,b^{5} d \,e^{5} x^{4}+6930 B \,a^{2} b^{3} e^{6} x^{4}+2475 B a \,b^{4} d \,e^{5} x^{4}+495 B \,b^{5} d^{2} e^{4} x^{4}+6160 A \,a^{2} b^{3} e^{6} x^{3}+1540 A a \,b^{4} d \,e^{5} x^{3}+220 A \,b^{5} d^{2} e^{4} x^{3}+6160 B \,a^{3} b^{2} e^{6} x^{3}+3080 B \,a^{2} b^{3} d \,e^{5} x^{3}+1100 B a \,b^{4} d^{2} e^{4} x^{3}+220 B \,b^{5} d^{3} e^{3} x^{3}+5544 A \,a^{3} b^{2} e^{6} x^{2}+1848 A \,a^{2} b^{3} d \,e^{5} x^{2}+462 A a \,b^{4} d^{2} e^{4} x^{2}+66 A \,b^{5} d^{3} e^{3} x^{2}+2772 B \,a^{4} b \,e^{6} x^{2}+1848 B \,a^{3} b^{2} d \,e^{5} x^{2}+924 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+330 B a \,b^{4} d^{3} e^{3} x^{2}+66 B \,b^{5} d^{4} e^{2} x^{2}+2520 A \,a^{4} b \,e^{6} x +1008 A \,a^{3} b^{2} d \,e^{5} x +336 A \,a^{2} b^{3} d^{2} e^{4} x +84 A a \,b^{4} d^{3} e^{3} x +12 A \,b^{5} d^{4} e^{2} x +504 B \,a^{5} e^{6} x +504 B \,a^{4} b d \,e^{5} x +336 B \,a^{3} b^{2} d^{2} e^{4} x +168 B \,a^{2} b^{3} d^{3} e^{3} x +60 B a \,b^{4} d^{4} e^{2} x +12 B \,b^{5} d^{5} e x +462 A \,a^{5} e^{6}+210 A \,a^{4} b d \,e^{5}+84 A \,a^{3} b^{2} d^{2} e^{4}+28 A \,a^{2} b^{3} d^{3} e^{3}+7 A a \,b^{4} d^{4} e^{2}+A \,b^{5} d^{5} e +42 B \,a^{5} d \,e^{5}+42 B \,a^{4} b \,d^{2} e^{4}+28 B \,a^{3} b^{2} d^{3} e^{3}+14 B \,a^{2} b^{3} d^{4} e^{2}+5 B a \,b^{4} d^{5} e +B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{5544 \left (e x +d \right )^{12} \left (b x +a \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)^13,x)

[Out]

-1/5544/e^7*(924*B*b^5*e^6*x^6+792*A*b^5*e^6*x^5+3960*B*a*b^4*e^6*x^5+792*B*b^5*d*e^5*x^5+3465*A*a*b^4*e^6*x^4
+495*A*b^5*d*e^5*x^4+6930*B*a^2*b^3*e^6*x^4+2475*B*a*b^4*d*e^5*x^4+495*B*b^5*d^2*e^4*x^4+6160*A*a^2*b^3*e^6*x^
3+1540*A*a*b^4*d*e^5*x^3+220*A*b^5*d^2*e^4*x^3+6160*B*a^3*b^2*e^6*x^3+3080*B*a^2*b^3*d*e^5*x^3+1100*B*a*b^4*d^
2*e^4*x^3+220*B*b^5*d^3*e^3*x^3+5544*A*a^3*b^2*e^6*x^2+1848*A*a^2*b^3*d*e^5*x^2+462*A*a*b^4*d^2*e^4*x^2+66*A*b
^5*d^3*e^3*x^2+2772*B*a^4*b*e^6*x^2+1848*B*a^3*b^2*d*e^5*x^2+924*B*a^2*b^3*d^2*e^4*x^2+330*B*a*b^4*d^3*e^3*x^2
+66*B*b^5*d^4*e^2*x^2+2520*A*a^4*b*e^6*x+1008*A*a^3*b^2*d*e^5*x+336*A*a^2*b^3*d^2*e^4*x+84*A*a*b^4*d^3*e^3*x+1
2*A*b^5*d^4*e^2*x+504*B*a^5*e^6*x+504*B*a^4*b*d*e^5*x+336*B*a^3*b^2*d^2*e^4*x+168*B*a^2*b^3*d^3*e^3*x+60*B*a*b
^4*d^4*e^2*x+12*B*b^5*d^5*e*x+462*A*a^5*e^6+210*A*a^4*b*d*e^5+84*A*a^3*b^2*d^2*e^4+28*A*a^2*b^3*d^3*e^3+7*A*a*
b^4*d^4*e^2+A*b^5*d^5*e+42*B*a^5*d*e^5+42*B*a^4*b*d^2*e^4+28*B*a^3*b^2*d^3*e^3+14*B*a^2*b^3*d^4*e^2+5*B*a*b^4*
d^5*e+B*b^5*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^12/(b*x+a)^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)^13,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 [B]  time = 2.63, size = 1489, normalized size = 3.40

result too large to display

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)^13,x)

[Out]

- (((10*B*b^5*d^2 - 4*A*b^5*d*e + 5*A*a*b^4*e^2 + 10*B*a^2*b^3*e^2 - 20*B*a*b^4*d*e)/(8*e^7) - (d*((b^4*(A*b*e
+ 5*B*a*e - 4*B*b*d))/(8*e^6) - (B*b^5*d)/(8*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^
8) - (((A*b^5*e - 5*B*b^5*d + 5*B*a*b^4*e)/(7*e^7) - (B*b^5*d)/(7*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a +
b*x)*(d + e*x)^7) - (((A*a^5)/(12*e) - (d*((B*a^5 + 5*A*a^4*b)/(12*e) + (d*((d*((d*((d*((A*b^5 + 5*B*a*b^4)/(
12*e) - (B*b^5*d)/(12*e^2)))/e - (5*a*b^3*(A*b + 2*B*a))/(12*e)))/e + (5*a^2*b^2*(A*b + B*a))/(6*e)))/e - (5*a
^3*b*(2*A*b + B*a))/(12*e)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^12) - (((6*A*b^5*d^2
*e - 10*B*b^5*d^3 + 10*A*a^2*b^3*e^3 + 10*B*a^3*b^2*e^3 - 30*B*a^2*b^3*d*e^2 - 15*A*a*b^4*d*e^2 + 30*B*a*b^4*d
^2*e)/(9*e^7) - (d*((5*A*a*b^4*e^3 - 3*A*b^5*d*e^2 + 6*B*b^5*d^2*e + 10*B*a^2*b^3*e^3 - 15*B*a*b^4*d*e^2)/(9*e
^7) - (d*((b^4*(A*b*e + 5*B*a*e - 3*B*b*d))/(9*e^5) - (B*b^5*d)/(9*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/
2))/((a + b*x)*(d + e*x)^9) - (((B*a^5*e^5 - B*b^5*d^5 + 5*A*a^4*b*e^5 + A*b^5*d^4*e - 5*A*a*b^4*d^3*e^2 - 10*
A*a^3*b^2*d*e^4 + 10*A*a^2*b^3*d^2*e^3 - 10*B*a^2*b^3*d^3*e^2 + 10*B*a^3*b^2*d^2*e^3 + 5*B*a*b^4*d^4*e - 5*B*a
^4*b*d*e^4)/(11*e^7) - (d*((5*B*a^4*b*e^5 + B*b^5*d^4*e + 10*A*a^3*b^2*e^5 - A*b^5*d^3*e^2 + 5*A*a*b^4*d^2*e^3
- 10*A*a^2*b^3*d*e^4 - 5*B*a*b^4*d^3*e^2 - 10*B*a^3*b^2*d*e^4 + 10*B*a^2*b^3*d^2*e^3)/(11*e^7) - (d*((10*A*a^
2*b^3*e^5 + 10*B*a^3*b^2*e^5 + A*b^5*d^2*e^3 - B*b^5*d^3*e^2 + 5*B*a*b^4*d^2*e^3 - 10*B*a^2*b^3*d*e^4 - 5*A*a*
b^4*d*e^4)/(11*e^7) - (d*((5*A*a*b^4*e^5 - A*b^5*d*e^4 + 10*B*a^2*b^3*e^5 + B*b^5*d^2*e^3 - 5*B*a*b^4*d*e^4)/(
11*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - B*b*d))/(11*e^3) - (B*b^5*d)/(11*e^3)))/e))/e))/e))/e)*(a^2 + b^2*x^2 +
2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^11) - (((5*B*b^5*d^4 + 5*B*a^4*b*e^4 - 4*A*b^5*d^3*e + 10*A*a^3*b^2*e^4 +
15*A*a*b^4*d^2*e^2 - 20*A*a^2*b^3*d*e^3 - 20*B*a^3*b^2*d*e^3 + 30*B*a^2*b^3*d^2*e^2 - 20*B*a*b^4*d^3*e)/(10*e
^7) - (d*((10*A*a^2*b^3*e^4 - 4*B*b^5*d^3*e + 10*B*a^3*b^2*e^4 + 3*A*b^5*d^2*e^2 + 15*B*a*b^4*d^2*e^2 - 20*B*a
^2*b^3*d*e^3 - 10*A*a*b^4*d*e^3)/(10*e^7) - (d*((5*A*a*b^4*e^4 - 2*A*b^5*d*e^3 + 10*B*a^2*b^3*e^4 + 3*B*b^5*d^
2*e^2 - 10*B*a*b^4*d*e^3)/(10*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - 2*B*b*d))/(10*e^4) - (B*b^5*d)/(10*e^4)))/e))
/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6
*e^7*(a + b*x)*(d + e*x)^6)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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)**13,x)

[Out]

Exception raised: HeuristicGCDFailed

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