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

Optimal. Leaf size=446 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{3 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (a+b x)} \]

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Rubi [A]  time = 0.20, antiderivative size = 446, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} -\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{3 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}+\frac {2 \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)}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (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)^(5/2),x]

[Out]

(-2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^(3/2)) + (2*(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])/(e^7*(a + b*x)*Sqrt[d + e*x]) + (10*b*(b*d - a*
e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (20*b^2*(b*d -
a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (2*b^3*(b
*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (2*b^4*
(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) + (2*b^5*B*(d + e
*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*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)^{5/2}} \, 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)^{5/2}} \, 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 {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{5/2}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{3/2}}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 \sqrt {d+e x}}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) \sqrt {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)^{3/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{5/2}}{e^6}+\frac {b^{10} B (d+e x)^{7/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac {2 (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}}{e^7 (a+b x) \sqrt {d+e x}}+\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {2 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 239, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-9 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+63 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-210 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+315 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)+63 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)-21 (b d-a e)^5 (B d-A e)+7 b^5 B (d+e x)^6\right )}{63 e^7 (a+b x) (d+e x)^{3/2}} \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)^(5/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-21*(b*d - a*e)^5*(B*d - A*e) + 63*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x) +
315*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 210*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)
*(d + e*x)^3 + 63*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 9*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*
(d + e*x)^5 + 7*b^5*B*(d + e*x)^6))/(63*e^7*(a + b*x)*(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 30.08, size = 812, normalized size = 1.82 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (-21 b^5 B d^6+21 A b^5 e d^5+105 a b^4 B e d^5+378 b^5 B (d+e x) d^5-105 a A b^4 e^2 d^4-210 a^2 b^3 B e^2 d^4+945 b^5 B (d+e x)^2 d^4-315 A b^5 e (d+e x) d^4-1575 a b^4 B e (d+e x) d^4+210 a^2 A b^3 e^3 d^3+210 a^3 b^2 B e^3 d^3-420 b^5 B (d+e x)^3 d^3-630 A b^5 e (d+e x)^2 d^3-3150 a b^4 B e (d+e x)^2 d^3+1260 a A b^4 e^2 (d+e x) d^3+2520 a^2 b^3 B e^2 (d+e x) d^3-210 a^3 A b^2 e^4 d^2-105 a^4 b B e^4 d^2+189 b^5 B (d+e x)^4 d^2+210 A b^5 e (d+e x)^3 d^2+1050 a b^4 B e (d+e x)^3 d^2+1890 a A b^4 e^2 (d+e x)^2 d^2+3780 a^2 b^3 B e^2 (d+e x)^2 d^2-1890 a^2 A b^3 e^3 (d+e x) d^2-1890 a^3 b^2 B e^3 (d+e x) d^2+105 a^4 A b e^5 d+21 a^5 B e^5 d-54 b^5 B (d+e x)^5 d-63 A b^5 e (d+e x)^4 d-315 a b^4 B e (d+e x)^4 d-420 a A b^4 e^2 (d+e x)^3 d-840 a^2 b^3 B e^2 (d+e x)^3 d-1890 a^2 A b^3 e^3 (d+e x)^2 d-1890 a^3 b^2 B e^3 (d+e x)^2 d+1260 a^3 A b^2 e^4 (d+e x) d+630 a^4 b B e^4 (d+e x) d-21 a^5 A e^6+7 b^5 B (d+e x)^6+9 A b^5 e (d+e x)^5+45 a b^4 B e (d+e x)^5+63 a A b^4 e^2 (d+e x)^4+126 a^2 b^3 B e^2 (d+e x)^4+210 a^2 A b^3 e^3 (d+e x)^3+210 a^3 b^2 B e^3 (d+e x)^3+630 a^3 A b^2 e^4 (d+e x)^2+315 a^4 b B e^4 (d+e x)^2-315 a^4 A b e^5 (d+e x)-63 a^5 B e^5 (d+e x)\right )}{63 e^6 (d+e x)^{3/2} (a e+b x e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(-21*b^5*B*d^6 + 21*A*b^5*d^5*e + 105*a*b^4*B*d^5*e - 105*a*A*b^4*d^4*e^2 - 210*a
^2*b^3*B*d^4*e^2 + 210*a^2*A*b^3*d^3*e^3 + 210*a^3*b^2*B*d^3*e^3 - 210*a^3*A*b^2*d^2*e^4 - 105*a^4*b*B*d^2*e^4
+ 105*a^4*A*b*d*e^5 + 21*a^5*B*d*e^5 - 21*a^5*A*e^6 + 378*b^5*B*d^5*(d + e*x) - 315*A*b^5*d^4*e*(d + e*x) - 1
575*a*b^4*B*d^4*e*(d + e*x) + 1260*a*A*b^4*d^3*e^2*(d + e*x) + 2520*a^2*b^3*B*d^3*e^2*(d + e*x) - 1890*a^2*A*b
^3*d^2*e^3*(d + e*x) - 1890*a^3*b^2*B*d^2*e^3*(d + e*x) + 1260*a^3*A*b^2*d*e^4*(d + e*x) + 630*a^4*b*B*d*e^4*(
d + e*x) - 315*a^4*A*b*e^5*(d + e*x) - 63*a^5*B*e^5*(d + e*x) + 945*b^5*B*d^4*(d + e*x)^2 - 630*A*b^5*d^3*e*(d
+ e*x)^2 - 3150*a*b^4*B*d^3*e*(d + e*x)^2 + 1890*a*A*b^4*d^2*e^2*(d + e*x)^2 + 3780*a^2*b^3*B*d^2*e^2*(d + e*
x)^2 - 1890*a^2*A*b^3*d*e^3*(d + e*x)^2 - 1890*a^3*b^2*B*d*e^3*(d + e*x)^2 + 630*a^3*A*b^2*e^4*(d + e*x)^2 + 3
15*a^4*b*B*e^4*(d + e*x)^2 - 420*b^5*B*d^3*(d + e*x)^3 + 210*A*b^5*d^2*e*(d + e*x)^3 + 1050*a*b^4*B*d^2*e*(d +
e*x)^3 - 420*a*A*b^4*d*e^2*(d + e*x)^3 - 840*a^2*b^3*B*d*e^2*(d + e*x)^3 + 210*a^2*A*b^3*e^3*(d + e*x)^3 + 21
0*a^3*b^2*B*e^3*(d + e*x)^3 + 189*b^5*B*d^2*(d + e*x)^4 - 63*A*b^5*d*e*(d + e*x)^4 - 315*a*b^4*B*d*e*(d + e*x)
^4 + 63*a*A*b^4*e^2*(d + e*x)^4 + 126*a^2*b^3*B*e^2*(d + e*x)^4 - 54*b^5*B*d*(d + e*x)^5 + 9*A*b^5*e*(d + e*x)
^5 + 45*a*b^4*B*e*(d + e*x)^5 + 7*b^5*B*(d + e*x)^6))/(63*e^6*(d + e*x)^(3/2)*(a*e + b*e*x))

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fricas [A]  time = 0.42, size = 581, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (7 \, B b^{5} e^{6} x^{6} + 1024 \, B b^{5} d^{6} - 21 \, A a^{5} e^{6} - 768 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 2688 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 3360 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 840 \, {\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} - 3 \, {\left (4 \, B b^{5} d e^{5} - 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 3 \, {\left (8 \, B b^{5} d^{2} e^{4} - 6 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 21 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 2 \, {\left (32 \, B b^{5} d^{3} e^{3} - 24 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 84 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 3 \, {\left (128 \, B b^{5} d^{4} e^{2} - 96 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 336 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 420 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 105 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 3 \, {\left (512 \, B b^{5} d^{5} e - 384 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 1344 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 1680 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 420 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 21 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/63*(7*B*b^5*e^6*x^6 + 1024*B*b^5*d^6 - 21*A*a^5*e^6 - 768*(5*B*a*b^4 + A*b^5)*d^5*e + 2688*(2*B*a^2*b^3 + A*
a*b^4)*d^4*e^2 - 3360*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 840*(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 - 3*(4*B*b^5*d*e^5 - 3*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 3*(8*B*b^5*d^2*e^4 - 6*(5*B*a*b^4 + A*b^5)*
d*e^5 + 21*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 2*(32*B*b^5*d^3*e^3 - 24*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 84*(2*B*a
^2*b^3 + A*a*b^4)*d*e^5 - 105*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 3*(128*B*b^5*d^4*e^2 - 96*(5*B*a*b^4 + A*b^5)
*d^3*e^3 + 336*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 420*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 105*(B*a^4*b + 2*A*a^3*b^
2)*e^6)*x^2 + 3*(512*B*b^5*d^5*e - 384*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 1344*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 16
80*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 420*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - 21*(B*a^5 + 5*A*a^4*b)*e^6)*x)*sqrt(e
*x + d)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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giac [B]  time = 0.40, size = 1101, normalized size = 2.47

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)^(5/2),x, algorithm="giac")

[Out]

2/63*(7*(x*e + d)^(9/2)*B*b^5*e^56*sgn(b*x + a) - 54*(x*e + d)^(7/2)*B*b^5*d*e^56*sgn(b*x + a) + 189*(x*e + d)
^(5/2)*B*b^5*d^2*e^56*sgn(b*x + a) - 420*(x*e + d)^(3/2)*B*b^5*d^3*e^56*sgn(b*x + a) + 945*sqrt(x*e + d)*B*b^5
*d^4*e^56*sgn(b*x + a) + 45*(x*e + d)^(7/2)*B*a*b^4*e^57*sgn(b*x + a) + 9*(x*e + d)^(7/2)*A*b^5*e^57*sgn(b*x +
a) - 315*(x*e + d)^(5/2)*B*a*b^4*d*e^57*sgn(b*x + a) - 63*(x*e + d)^(5/2)*A*b^5*d*e^57*sgn(b*x + a) + 1050*(x
*e + d)^(3/2)*B*a*b^4*d^2*e^57*sgn(b*x + a) + 210*(x*e + d)^(3/2)*A*b^5*d^2*e^57*sgn(b*x + a) - 3150*sqrt(x*e
+ d)*B*a*b^4*d^3*e^57*sgn(b*x + a) - 630*sqrt(x*e + d)*A*b^5*d^3*e^57*sgn(b*x + a) + 126*(x*e + d)^(5/2)*B*a^2
*b^3*e^58*sgn(b*x + a) + 63*(x*e + d)^(5/2)*A*a*b^4*e^58*sgn(b*x + a) - 840*(x*e + d)^(3/2)*B*a^2*b^3*d*e^58*s
gn(b*x + a) - 420*(x*e + d)^(3/2)*A*a*b^4*d*e^58*sgn(b*x + a) + 3780*sqrt(x*e + d)*B*a^2*b^3*d^2*e^58*sgn(b*x
+ a) + 1890*sqrt(x*e + d)*A*a*b^4*d^2*e^58*sgn(b*x + a) + 210*(x*e + d)^(3/2)*B*a^3*b^2*e^59*sgn(b*x + a) + 21
0*(x*e + d)^(3/2)*A*a^2*b^3*e^59*sgn(b*x + a) - 1890*sqrt(x*e + d)*B*a^3*b^2*d*e^59*sgn(b*x + a) - 1890*sqrt(x
*e + d)*A*a^2*b^3*d*e^59*sgn(b*x + a) + 315*sqrt(x*e + d)*B*a^4*b*e^60*sgn(b*x + a) + 630*sqrt(x*e + d)*A*a^3*
b^2*e^60*sgn(b*x + a))*e^(-63) + 2/3*(18*(x*e + d)*B*b^5*d^5*sgn(b*x + a) - B*b^5*d^6*sgn(b*x + a) - 75*(x*e +
d)*B*a*b^4*d^4*e*sgn(b*x + a) - 15*(x*e + d)*A*b^5*d^4*e*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) + 120*(x*e + d)*B*a^2*b^3*d^3*e^2*sgn(b*x + a) + 60*(x*e + d)*A*a*b^4*d^3*e^2*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) - 90*(x*e + d)*B*a^3*b^2*d^2*e^3*sgn(b*x
+ a) - 90*(x*e + d)*A*a^2*b^3*d^2*e^3*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) + 30*(x*e + d)*B*a^4*b*d*e^4*sgn(b*x + a) + 60*(x*e + d)*A*a^3*b^2*d*e^4*sgn(b*x + a) - 5*B*a^4*b
*d^2*e^4*sgn(b*x + a) - 10*A*a^3*b^2*d^2*e^4*sgn(b*x + a) - 3*(x*e + d)*B*a^5*e^5*sgn(b*x + a) - 15*(x*e + d)*
A*a^4*b*e^5*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)^(3/2)

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maple [A]  time = 0.05, size = 689, normalized size = 1.54 \begin {gather*} -\frac {2 \left (-7 B \,b^{5} e^{6} x^{6}-9 A \,b^{5} e^{6} x^{5}-45 B a \,b^{4} e^{6} x^{5}+12 B \,b^{5} d \,e^{5} x^{5}-63 A a \,b^{4} e^{6} x^{4}+18 A \,b^{5} d \,e^{5} x^{4}-126 B \,a^{2} b^{3} e^{6} x^{4}+90 B a \,b^{4} d \,e^{5} x^{4}-24 B \,b^{5} d^{2} e^{4} x^{4}-210 A \,a^{2} b^{3} e^{6} x^{3}+168 A a \,b^{4} d \,e^{5} x^{3}-48 A \,b^{5} d^{2} e^{4} x^{3}-210 B \,a^{3} b^{2} e^{6} x^{3}+336 B \,a^{2} b^{3} d \,e^{5} x^{3}-240 B a \,b^{4} d^{2} e^{4} x^{3}+64 B \,b^{5} d^{3} e^{3} x^{3}-630 A \,a^{3} b^{2} e^{6} x^{2}+1260 A \,a^{2} b^{3} d \,e^{5} x^{2}-1008 A a \,b^{4} d^{2} e^{4} x^{2}+288 A \,b^{5} d^{3} e^{3} x^{2}-315 B \,a^{4} b \,e^{6} x^{2}+1260 B \,a^{3} b^{2} d \,e^{5} x^{2}-2016 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1440 B a \,b^{4} d^{3} e^{3} x^{2}-384 B \,b^{5} d^{4} e^{2} x^{2}+315 A \,a^{4} b \,e^{6} x -2520 A \,a^{3} b^{2} d \,e^{5} x +5040 A \,a^{2} b^{3} d^{2} e^{4} x -4032 A a \,b^{4} d^{3} e^{3} x +1152 A \,b^{5} d^{4} e^{2} x +63 B \,a^{5} e^{6} x -1260 B \,a^{4} b d \,e^{5} x +5040 B \,a^{3} b^{2} d^{2} e^{4} x -8064 B \,a^{2} b^{3} d^{3} e^{3} x +5760 B a \,b^{4} d^{4} e^{2} x -1536 B \,b^{5} d^{5} e x +21 A \,a^{5} e^{6}+210 A \,a^{4} b d \,e^{5}-1680 A \,a^{3} b^{2} d^{2} e^{4}+3360 A \,a^{2} b^{3} d^{3} e^{3}-2688 A a \,b^{4} d^{4} e^{2}+768 A \,b^{5} d^{5} e +42 B \,a^{5} d \,e^{5}-840 B \,a^{4} b \,d^{2} e^{4}+3360 B \,a^{3} b^{2} d^{3} e^{3}-5376 B \,a^{2} b^{3} d^{4} e^{2}+3840 B a \,b^{4} d^{5} e -1024 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {3}{2}} \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)^(5/2),x)

[Out]

-2/63/(e*x+d)^(3/2)*(-7*B*b^5*e^6*x^6-9*A*b^5*e^6*x^5-45*B*a*b^4*e^6*x^5+12*B*b^5*d*e^5*x^5-63*A*a*b^4*e^6*x^4
+18*A*b^5*d*e^5*x^4-126*B*a^2*b^3*e^6*x^4+90*B*a*b^4*d*e^5*x^4-24*B*b^5*d^2*e^4*x^4-210*A*a^2*b^3*e^6*x^3+168*
A*a*b^4*d*e^5*x^3-48*A*b^5*d^2*e^4*x^3-210*B*a^3*b^2*e^6*x^3+336*B*a^2*b^3*d*e^5*x^3-240*B*a*b^4*d^2*e^4*x^3+6
4*B*b^5*d^3*e^3*x^3-630*A*a^3*b^2*e^6*x^2+1260*A*a^2*b^3*d*e^5*x^2-1008*A*a*b^4*d^2*e^4*x^2+288*A*b^5*d^3*e^3*
x^2-315*B*a^4*b*e^6*x^2+1260*B*a^3*b^2*d*e^5*x^2-2016*B*a^2*b^3*d^2*e^4*x^2+1440*B*a*b^4*d^3*e^3*x^2-384*B*b^5
*d^4*e^2*x^2+315*A*a^4*b*e^6*x-2520*A*a^3*b^2*d*e^5*x+5040*A*a^2*b^3*d^2*e^4*x-4032*A*a*b^4*d^3*e^3*x+1152*A*b
^5*d^4*e^2*x+63*B*a^5*e^6*x-1260*B*a^4*b*d*e^5*x+5040*B*a^3*b^2*d^2*e^4*x-8064*B*a^2*b^3*d^3*e^3*x+5760*B*a*b^
4*d^4*e^2*x-1536*B*b^5*d^5*e*x+21*A*a^5*e^6+210*A*a^4*b*d*e^5-1680*A*a^3*b^2*d^2*e^4+3360*A*a^2*b^3*d^3*e^3-26
88*A*a*b^4*d^4*e^2+768*A*b^5*d^5*e+42*B*a^5*d*e^5-840*B*a^4*b*d^2*e^4+3360*B*a^3*b^2*d^3*e^3-5376*B*a^2*b^3*d^
4*e^2+3840*B*a*b^4*d^5*e-1024*B*b^5*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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maxima [A]  time = 0.87, size = 625, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \, {\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} A}{21 \, {\left (e^{7} x + d e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (7 \, b^{5} e^{6} x^{6} + 1024 \, b^{5} d^{6} - 3840 \, a b^{4} d^{5} e + 5376 \, a^{2} b^{3} d^{4} e^{2} - 3360 \, a^{3} b^{2} d^{3} e^{3} + 840 \, a^{4} b d^{2} e^{4} - 42 \, a^{5} d e^{5} - 3 \, {\left (4 \, b^{5} d e^{5} - 15 \, a b^{4} e^{6}\right )} x^{5} + 6 \, {\left (4 \, b^{5} d^{2} e^{4} - 15 \, a b^{4} d e^{5} + 21 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (32 \, b^{5} d^{3} e^{3} - 120 \, a b^{4} d^{2} e^{4} + 168 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{4} e^{2} - 480 \, a b^{4} d^{3} e^{3} + 672 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 105 \, a^{4} b e^{6}\right )} x^{2} + 3 \, {\left (512 \, b^{5} d^{5} e - 1920 \, a b^{4} d^{4} e^{2} + 2688 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 420 \, a^{4} b d e^{5} - 21 \, a^{5} e^{6}\right )} x\right )} B}{63 \, {\left (e^{8} x + d e^{7}\right )} \sqrt {e x + d}} \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)^(5/2),x, algorithm="maxima")

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*
e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3
- 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e - 448*a*b^4
*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*A/((e^7*x + d*e^6)*sqrt(e*x + d)) + 2/63
*(7*b^5*e^6*x^6 + 1024*b^5*d^6 - 3840*a*b^4*d^5*e + 5376*a^2*b^3*d^4*e^2 - 3360*a^3*b^2*d^3*e^3 + 840*a^4*b*d^
2*e^4 - 42*a^5*d*e^5 - 3*(4*b^5*d*e^5 - 15*a*b^4*e^6)*x^5 + 6*(4*b^5*d^2*e^4 - 15*a*b^4*d*e^5 + 21*a^2*b^3*e^6
)*x^4 - 2*(32*b^5*d^3*e^3 - 120*a*b^4*d^2*e^4 + 168*a^2*b^3*d*e^5 - 105*a^3*b^2*e^6)*x^3 + 3*(128*b^5*d^4*e^2
- 480*a*b^4*d^3*e^3 + 672*a^2*b^3*d^2*e^4 - 420*a^3*b^2*d*e^5 + 105*a^4*b*e^6)*x^2 + 3*(512*b^5*d^5*e - 1920*a
*b^4*d^4*e^2 + 2688*a^2*b^3*d^3*e^3 - 1680*a^3*b^2*d^2*e^4 + 420*a^4*b*d*e^5 - 21*a^5*e^6)*x)*B/((e^8*x + d*e^
7)*sqrt(e*x + d))

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mupad [B]  time = 3.89, size = 695, normalized size = 1.56 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {x^2\,\left (630\,B\,a^4\,b\,e^6-2520\,B\,a^3\,b^2\,d\,e^5+1260\,A\,a^3\,b^2\,e^6+4032\,B\,a^2\,b^3\,d^2\,e^4-2520\,A\,a^2\,b^3\,d\,e^5-2880\,B\,a\,b^4\,d^3\,e^3+2016\,A\,a\,b^4\,d^2\,e^4+768\,B\,b^5\,d^4\,e^2-576\,A\,b^5\,d^3\,e^3\right )}{63\,b\,e^8}-\frac {84\,B\,a^5\,d\,e^5+42\,A\,a^5\,e^6-1680\,B\,a^4\,b\,d^2\,e^4+420\,A\,a^4\,b\,d\,e^5+6720\,B\,a^3\,b^2\,d^3\,e^3-3360\,A\,a^3\,b^2\,d^2\,e^4-10752\,B\,a^2\,b^3\,d^4\,e^2+6720\,A\,a^2\,b^3\,d^3\,e^3+7680\,B\,a\,b^4\,d^5\,e-5376\,A\,a\,b^4\,d^4\,e^2-2048\,B\,b^5\,d^6+1536\,A\,b^5\,d^5\,e}{63\,b\,e^8}+\frac {x^3\,\left (420\,B\,a^3\,b^2\,e^6-672\,B\,a^2\,b^3\,d\,e^5+420\,A\,a^2\,b^3\,e^6+480\,B\,a\,b^4\,d^2\,e^4-336\,A\,a\,b^4\,d\,e^5-128\,B\,b^5\,d^3\,e^3+96\,A\,b^5\,d^2\,e^4\right )}{63\,b\,e^8}+\frac {2\,b^3\,x^5\,\left (3\,A\,b\,e+15\,B\,a\,e-4\,B\,b\,d\right )}{21\,e^3}-\frac {x\,\left (126\,B\,a^5\,e^6-2520\,B\,a^4\,b\,d\,e^5+630\,A\,a^4\,b\,e^6+10080\,B\,a^3\,b^2\,d^2\,e^4-5040\,A\,a^3\,b^2\,d\,e^5-16128\,B\,a^2\,b^3\,d^3\,e^3+10080\,A\,a^2\,b^3\,d^2\,e^4+11520\,B\,a\,b^4\,d^4\,e^2-8064\,A\,a\,b^4\,d^3\,e^3-3072\,B\,b^5\,d^5\,e+2304\,A\,b^5\,d^4\,e^2\right )}{63\,b\,e^8}+\frac {2\,b^2\,x^4\,\left (42\,B\,a^2\,e^2-30\,B\,a\,b\,d\,e+21\,A\,a\,b\,e^2+8\,B\,b^2\,d^2-6\,A\,b^2\,d\,e\right )}{21\,e^4}+\frac {2\,B\,b^4\,x^6}{9\,e^2}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (63\,a\,e^8+63\,b\,d\,e^7\right )\,\sqrt {d+e\,x}}{63\,b\,e^8}} \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)^(5/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((x^2*(630*B*a^4*b*e^6 + 1260*A*a^3*b^2*e^6 - 576*A*b^5*d^3*e^3 + 768*B*b^5*d
^4*e^2 + 2016*A*a*b^4*d^2*e^4 - 2520*A*a^2*b^3*d*e^5 - 2880*B*a*b^4*d^3*e^3 - 2520*B*a^3*b^2*d*e^5 + 4032*B*a^
2*b^3*d^2*e^4))/(63*b*e^8) - (42*A*a^5*e^6 - 2048*B*b^5*d^6 + 1536*A*b^5*d^5*e + 84*B*a^5*d*e^5 - 5376*A*a*b^4
*d^4*e^2 - 1680*B*a^4*b*d^2*e^4 + 6720*A*a^2*b^3*d^3*e^3 - 3360*A*a^3*b^2*d^2*e^4 - 10752*B*a^2*b^3*d^4*e^2 +
6720*B*a^3*b^2*d^3*e^3 + 420*A*a^4*b*d*e^5 + 7680*B*a*b^4*d^5*e)/(63*b*e^8) + (x^3*(420*A*a^2*b^3*e^6 + 420*B*
a^3*b^2*e^6 + 96*A*b^5*d^2*e^4 - 128*B*b^5*d^3*e^3 + 480*B*a*b^4*d^2*e^4 - 672*B*a^2*b^3*d*e^5 - 336*A*a*b^4*d
*e^5))/(63*b*e^8) + (2*b^3*x^5*(3*A*b*e + 15*B*a*e - 4*B*b*d))/(21*e^3) - (x*(126*B*a^5*e^6 + 630*A*a^4*b*e^6
- 3072*B*b^5*d^5*e + 2304*A*b^5*d^4*e^2 - 8064*A*a*b^4*d^3*e^3 - 5040*A*a^3*b^2*d*e^5 + 11520*B*a*b^4*d^4*e^2
+ 10080*A*a^2*b^3*d^2*e^4 - 16128*B*a^2*b^3*d^3*e^3 + 10080*B*a^3*b^2*d^2*e^4 - 2520*B*a^4*b*d*e^5))/(63*b*e^8
) + (2*b^2*x^4*(42*B*a^2*e^2 + 8*B*b^2*d^2 + 21*A*a*b*e^2 - 6*A*b^2*d*e - 30*B*a*b*d*e))/(21*e^4) + (2*B*b^4*x
^6)/(9*e^2)))/(x^2*(d + e*x)^(1/2) + (a*d*(d + e*x)^(1/2))/(b*e) + (x*(63*a*e^8 + 63*b*d*e^7)*(d + e*x)^(1/2))
/(63*b*e^8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(5/2),x)

[Out]

Timed out

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