3.17.36 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=446 \[ -\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-5 a B e-A b e+6 b B d)}{9 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)} \]

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Rubi [A]  time = 0.22, 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)^{9/2} (-5 a B e-A b e+6 b B d)}{9 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x)}-\frac {4 b^2 \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+2 b B d)}{e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4 (-a B e-5 A b e+6 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)^5 (B d-A e)}{e^7 (a+b x) \sqrt {d+e x}}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 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)^(3/2),x]

[Out]

(-2*(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)*Sqrt[d + e*x]) - (2*(b*d - a*e)^4*
(6*b*B*d - 5*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)) + (10*b*(b*d - a*e)^3
*(3*b*B*d - 2*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)) - (4*b^2*(b*d -
a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (10*b^3*(b*d
- a*e)*(3*b*B*d - A*b*e - 2*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^4*
(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) + (2*b^5*B*(d + e
*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*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)^{3/2}} \, 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)^{3/2}} \, 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)^{3/2}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 \sqrt {d+e x}}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) \sqrt {d+e x}}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{5/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{7/2}}{e^6}+\frac {b^{10} B (d+e x)^{9/2}}{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}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 (b d-a e)^4 (6 b B d-5 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 {10 b (b d-a e)^3 (3 b B d-2 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 {4 b^2 (b d-a e)^2 (2 b B d-A b e-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 {10 b^3 (b d-a e) (3 b B d-A b e-2 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^4 (6 b B d-A b e-5 a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 239, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-77 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+495 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-1386 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+1155 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-693 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)-693 (b d-a e)^5 (B d-A e)+63 b^5 B (d+e x)^6\right )}{693 e^7 (a+b x) \sqrt {d+e x}} \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)^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-693*(b*d - a*e)^5*(B*d - A*e) - 693*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)
+ 1155*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 1386*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*
B*e)*(d + e*x)^3 + 495*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 77*b^4*(6*b*B*d - A*b*e - 5*a
*B*e)*(d + e*x)^5 + 63*b^5*B*(d + e*x)^6))/(693*e^7*(a + b*x)*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 38.16, size = 812, normalized size = 1.82 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (-693 b^5 B d^6+693 A b^5 e d^5+3465 a b^4 B e d^5-4158 b^5 B (d+e x) d^5-3465 a A b^4 e^2 d^4-6930 a^2 b^3 B e^2 d^4+3465 b^5 B (d+e x)^2 d^4+3465 A b^5 e (d+e x) d^4+17325 a b^4 B e (d+e x) d^4+6930 a^2 A b^3 e^3 d^3+6930 a^3 b^2 B e^3 d^3-2772 b^5 B (d+e x)^3 d^3-2310 A b^5 e (d+e x)^2 d^3-11550 a b^4 B e (d+e x)^2 d^3-13860 a A b^4 e^2 (d+e x) d^3-27720 a^2 b^3 B e^2 (d+e x) d^3-6930 a^3 A b^2 e^4 d^2-3465 a^4 b B e^4 d^2+1485 b^5 B (d+e x)^4 d^2+1386 A b^5 e (d+e x)^3 d^2+6930 a b^4 B e (d+e x)^3 d^2+6930 a A b^4 e^2 (d+e x)^2 d^2+13860 a^2 b^3 B e^2 (d+e x)^2 d^2+20790 a^2 A b^3 e^3 (d+e x) d^2+20790 a^3 b^2 B e^3 (d+e x) d^2+3465 a^4 A b e^5 d+693 a^5 B e^5 d-462 b^5 B (d+e x)^5 d-495 A b^5 e (d+e x)^4 d-2475 a b^4 B e (d+e x)^4 d-2772 a A b^4 e^2 (d+e x)^3 d-5544 a^2 b^3 B e^2 (d+e x)^3 d-6930 a^2 A b^3 e^3 (d+e x)^2 d-6930 a^3 b^2 B e^3 (d+e x)^2 d-13860 a^3 A b^2 e^4 (d+e x) d-6930 a^4 b B e^4 (d+e x) d-693 a^5 A e^6+63 b^5 B (d+e x)^6+77 A b^5 e (d+e x)^5+385 a b^4 B e (d+e x)^5+495 a A b^4 e^2 (d+e x)^4+990 a^2 b^3 B e^2 (d+e x)^4+1386 a^2 A b^3 e^3 (d+e x)^3+1386 a^3 b^2 B e^3 (d+e x)^3+2310 a^3 A b^2 e^4 (d+e x)^2+1155 a^4 b B e^4 (d+e x)^2+3465 a^4 A b e^5 (d+e x)+693 a^5 B e^5 (d+e x)\right )}{693 e^6 \sqrt {d+e x} (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)^(3/2),x]

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(-693*b^5*B*d^6 + 693*A*b^5*d^5*e + 3465*a*b^4*B*d^5*e - 3465*a*A*b^4*d^4*e^2 - 6
930*a^2*b^3*B*d^4*e^2 + 6930*a^2*A*b^3*d^3*e^3 + 6930*a^3*b^2*B*d^3*e^3 - 6930*a^3*A*b^2*d^2*e^4 - 3465*a^4*b*
B*d^2*e^4 + 3465*a^4*A*b*d*e^5 + 693*a^5*B*d*e^5 - 693*a^5*A*e^6 - 4158*b^5*B*d^5*(d + e*x) + 3465*A*b^5*d^4*e
*(d + e*x) + 17325*a*b^4*B*d^4*e*(d + e*x) - 13860*a*A*b^4*d^3*e^2*(d + e*x) - 27720*a^2*b^3*B*d^3*e^2*(d + e*
x) + 20790*a^2*A*b^3*d^2*e^3*(d + e*x) + 20790*a^3*b^2*B*d^2*e^3*(d + e*x) - 13860*a^3*A*b^2*d*e^4*(d + e*x) -
6930*a^4*b*B*d*e^4*(d + e*x) + 3465*a^4*A*b*e^5*(d + e*x) + 693*a^5*B*e^5*(d + e*x) + 3465*b^5*B*d^4*(d + e*x
)^2 - 2310*A*b^5*d^3*e*(d + e*x)^2 - 11550*a*b^4*B*d^3*e*(d + e*x)^2 + 6930*a*A*b^4*d^2*e^2*(d + e*x)^2 + 1386
0*a^2*b^3*B*d^2*e^2*(d + e*x)^2 - 6930*a^2*A*b^3*d*e^3*(d + e*x)^2 - 6930*a^3*b^2*B*d*e^3*(d + e*x)^2 + 2310*a
^3*A*b^2*e^4*(d + e*x)^2 + 1155*a^4*b*B*e^4*(d + e*x)^2 - 2772*b^5*B*d^3*(d + e*x)^3 + 1386*A*b^5*d^2*e*(d + e
*x)^3 + 6930*a*b^4*B*d^2*e*(d + e*x)^3 - 2772*a*A*b^4*d*e^2*(d + e*x)^3 - 5544*a^2*b^3*B*d*e^2*(d + e*x)^3 + 1
386*a^2*A*b^3*e^3*(d + e*x)^3 + 1386*a^3*b^2*B*e^3*(d + e*x)^3 + 1485*b^5*B*d^2*(d + e*x)^4 - 495*A*b^5*d*e*(d
+ e*x)^4 - 2475*a*b^4*B*d*e*(d + e*x)^4 + 495*a*A*b^4*e^2*(d + e*x)^4 + 990*a^2*b^3*B*e^2*(d + e*x)^4 - 462*b
^5*B*d*(d + e*x)^5 + 77*A*b^5*e*(d + e*x)^5 + 385*a*b^4*B*e*(d + e*x)^5 + 63*b^5*B*(d + e*x)^6))/(693*e^6*Sqrt
[d + e*x]*(a*e + b*e*x))

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fricas [A]  time = 0.43, size = 569, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left (63 \, B b^{5} e^{6} x^{6} - 3072 \, B b^{5} d^{6} - 693 \, A a^{5} e^{6} + 2816 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 12672 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 22176 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 9240 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 7 \, {\left (12 \, B b^{5} d e^{5} - 11 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \, {\left (24 \, B b^{5} d^{2} e^{4} - 22 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 99 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 2 \, {\left (96 \, B b^{5} d^{3} e^{3} - 88 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 396 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 693 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + {\left (384 \, B b^{5} d^{4} e^{2} - 352 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 1584 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 2772 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 1155 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - {\left (1536 \, B b^{5} d^{5} e - 1408 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 6336 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 11088 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 4620 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 693 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt {e x + d}}{693 \, {\left (e^{8} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/693*(63*B*b^5*e^6*x^6 - 3072*B*b^5*d^6 - 693*A*a^5*e^6 + 2816*(5*B*a*b^4 + A*b^5)*d^5*e - 12672*(2*B*a^2*b^3
+ A*a*b^4)*d^4*e^2 + 22176*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 9240*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 1386*(B*a
^5 + 5*A*a^4*b)*d*e^5 - 7*(12*B*b^5*d*e^5 - 11*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(24*B*b^5*d^2*e^4 - 22*(5*B*a*
b^4 + A*b^5)*d*e^5 + 99*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 2*(96*B*b^5*d^3*e^3 - 88*(5*B*a*b^4 + A*b^5)*d^2*e^
4 + 396*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - 693*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + (384*B*b^5*d^4*e^2 - 352*(5*B*a
*b^4 + A*b^5)*d^3*e^3 + 1584*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 2772*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 1155*(B*a^
4*b + 2*A*a^3*b^2)*e^6)*x^2 - (1536*B*b^5*d^5*e - 1408*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 6336*(2*B*a^2*b^3 + A*a*b
^4)*d^3*e^3 - 11088*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 4620*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - 693*(B*a^5 + 5*A*a^
4*b)*e^6)*x)*sqrt(e*x + d)/(e^8*x + d*e^7)

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giac [B]  time = 0.33, size = 1125, normalized size = 2.52

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)^(3/2),x, algorithm="giac")

[Out]

2/693*(63*(x*e + d)^(11/2)*B*b^5*e^70*sgn(b*x + a) - 462*(x*e + d)^(9/2)*B*b^5*d*e^70*sgn(b*x + a) + 1485*(x*e
+ d)^(7/2)*B*b^5*d^2*e^70*sgn(b*x + a) - 2772*(x*e + d)^(5/2)*B*b^5*d^3*e^70*sgn(b*x + a) + 3465*(x*e + d)^(3
/2)*B*b^5*d^4*e^70*sgn(b*x + a) - 4158*sqrt(x*e + d)*B*b^5*d^5*e^70*sgn(b*x + a) + 385*(x*e + d)^(9/2)*B*a*b^4
*e^71*sgn(b*x + a) + 77*(x*e + d)^(9/2)*A*b^5*e^71*sgn(b*x + a) - 2475*(x*e + d)^(7/2)*B*a*b^4*d*e^71*sgn(b*x
+ a) - 495*(x*e + d)^(7/2)*A*b^5*d*e^71*sgn(b*x + a) + 6930*(x*e + d)^(5/2)*B*a*b^4*d^2*e^71*sgn(b*x + a) + 13
86*(x*e + d)^(5/2)*A*b^5*d^2*e^71*sgn(b*x + a) - 11550*(x*e + d)^(3/2)*B*a*b^4*d^3*e^71*sgn(b*x + a) - 2310*(x
*e + d)^(3/2)*A*b^5*d^3*e^71*sgn(b*x + a) + 17325*sqrt(x*e + d)*B*a*b^4*d^4*e^71*sgn(b*x + a) + 3465*sqrt(x*e
+ d)*A*b^5*d^4*e^71*sgn(b*x + a) + 990*(x*e + d)^(7/2)*B*a^2*b^3*e^72*sgn(b*x + a) + 495*(x*e + d)^(7/2)*A*a*b
^4*e^72*sgn(b*x + a) - 5544*(x*e + d)^(5/2)*B*a^2*b^3*d*e^72*sgn(b*x + a) - 2772*(x*e + d)^(5/2)*A*a*b^4*d*e^7
2*sgn(b*x + a) + 13860*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^72*sgn(b*x + a) + 6930*(x*e + d)^(3/2)*A*a*b^4*d^2*e^72
*sgn(b*x + a) - 27720*sqrt(x*e + d)*B*a^2*b^3*d^3*e^72*sgn(b*x + a) - 13860*sqrt(x*e + d)*A*a*b^4*d^3*e^72*sgn
(b*x + a) + 1386*(x*e + d)^(5/2)*B*a^3*b^2*e^73*sgn(b*x + a) + 1386*(x*e + d)^(5/2)*A*a^2*b^3*e^73*sgn(b*x + a
) - 6930*(x*e + d)^(3/2)*B*a^3*b^2*d*e^73*sgn(b*x + a) - 6930*(x*e + d)^(3/2)*A*a^2*b^3*d*e^73*sgn(b*x + a) +
20790*sqrt(x*e + d)*B*a^3*b^2*d^2*e^73*sgn(b*x + a) + 20790*sqrt(x*e + d)*A*a^2*b^3*d^2*e^73*sgn(b*x + a) + 11
55*(x*e + d)^(3/2)*B*a^4*b*e^74*sgn(b*x + a) + 2310*(x*e + d)^(3/2)*A*a^3*b^2*e^74*sgn(b*x + a) - 6930*sqrt(x*
e + d)*B*a^4*b*d*e^74*sgn(b*x + a) - 13860*sqrt(x*e + d)*A*a^3*b^2*d*e^74*sgn(b*x + a) + 693*sqrt(x*e + d)*B*a
^5*e^75*sgn(b*x + a) + 3465*sqrt(x*e + d)*A*a^4*b*e^75*sgn(b*x + a))*e^(-77) - 2*(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*s
gn(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*sgn(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)/sqrt(x*e + d)

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maple [A]  time = 0.06, size = 689, normalized size = 1.54 \begin {gather*} -\frac {2 \left (-63 B \,b^{5} e^{6} x^{6}-77 A \,b^{5} e^{6} x^{5}-385 B a \,b^{4} e^{6} x^{5}+84 B \,b^{5} d \,e^{5} x^{5}-495 A a \,b^{4} e^{6} x^{4}+110 A \,b^{5} d \,e^{5} x^{4}-990 B \,a^{2} b^{3} e^{6} x^{4}+550 B a \,b^{4} d \,e^{5} x^{4}-120 B \,b^{5} d^{2} e^{4} x^{4}-1386 A \,a^{2} b^{3} e^{6} x^{3}+792 A a \,b^{4} d \,e^{5} x^{3}-176 A \,b^{5} d^{2} e^{4} x^{3}-1386 B \,a^{3} b^{2} e^{6} x^{3}+1584 B \,a^{2} b^{3} d \,e^{5} x^{3}-880 B a \,b^{4} d^{2} e^{4} x^{3}+192 B \,b^{5} d^{3} e^{3} x^{3}-2310 A \,a^{3} b^{2} e^{6} x^{2}+2772 A \,a^{2} b^{3} d \,e^{5} x^{2}-1584 A a \,b^{4} d^{2} e^{4} x^{2}+352 A \,b^{5} d^{3} e^{3} x^{2}-1155 B \,a^{4} b \,e^{6} x^{2}+2772 B \,a^{3} b^{2} d \,e^{5} x^{2}-3168 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1760 B a \,b^{4} d^{3} e^{3} x^{2}-384 B \,b^{5} d^{4} e^{2} x^{2}-3465 A \,a^{4} b \,e^{6} x +9240 A \,a^{3} b^{2} d \,e^{5} x -11088 A \,a^{2} b^{3} d^{2} e^{4} x +6336 A a \,b^{4} d^{3} e^{3} x -1408 A \,b^{5} d^{4} e^{2} x -693 B \,a^{5} e^{6} x +4620 B \,a^{4} b d \,e^{5} x -11088 B \,a^{3} b^{2} d^{2} e^{4} x +12672 B \,a^{2} b^{3} d^{3} e^{3} x -7040 B a \,b^{4} d^{4} e^{2} x +1536 B \,b^{5} d^{5} e x +693 A \,a^{5} e^{6}-6930 A \,a^{4} b d \,e^{5}+18480 A \,a^{3} b^{2} d^{2} e^{4}-22176 A \,a^{2} b^{3} d^{3} e^{3}+12672 A a \,b^{4} d^{4} e^{2}-2816 A \,b^{5} d^{5} e -1386 B \,a^{5} d \,e^{5}+9240 B \,a^{4} b \,d^{2} e^{4}-22176 B \,a^{3} b^{2} d^{3} e^{3}+25344 B \,a^{2} b^{3} d^{4} e^{2}-14080 B a \,b^{4} d^{5} e +3072 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{693 \sqrt {e x +d}\, \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)^(3/2),x)

[Out]

-2/693/(e*x+d)^(1/2)*(-63*B*b^5*e^6*x^6-77*A*b^5*e^6*x^5-385*B*a*b^4*e^6*x^5+84*B*b^5*d*e^5*x^5-495*A*a*b^4*e^
6*x^4+110*A*b^5*d*e^5*x^4-990*B*a^2*b^3*e^6*x^4+550*B*a*b^4*d*e^5*x^4-120*B*b^5*d^2*e^4*x^4-1386*A*a^2*b^3*e^6
*x^3+792*A*a*b^4*d*e^5*x^3-176*A*b^5*d^2*e^4*x^3-1386*B*a^3*b^2*e^6*x^3+1584*B*a^2*b^3*d*e^5*x^3-880*B*a*b^4*d
^2*e^4*x^3+192*B*b^5*d^3*e^3*x^3-2310*A*a^3*b^2*e^6*x^2+2772*A*a^2*b^3*d*e^5*x^2-1584*A*a*b^4*d^2*e^4*x^2+352*
A*b^5*d^3*e^3*x^2-1155*B*a^4*b*e^6*x^2+2772*B*a^3*b^2*d*e^5*x^2-3168*B*a^2*b^3*d^2*e^4*x^2+1760*B*a*b^4*d^3*e^
3*x^2-384*B*b^5*d^4*e^2*x^2-3465*A*a^4*b*e^6*x+9240*A*a^3*b^2*d*e^5*x-11088*A*a^2*b^3*d^2*e^4*x+6336*A*a*b^4*d
^3*e^3*x-1408*A*b^5*d^4*e^2*x-693*B*a^5*e^6*x+4620*B*a^4*b*d*e^5*x-11088*B*a^3*b^2*d^2*e^4*x+12672*B*a^2*b^3*d
^3*e^3*x-7040*B*a*b^4*d^4*e^2*x+1536*B*b^5*d^5*e*x+693*A*a^5*e^6-6930*A*a^4*b*d*e^5+18480*A*a^3*b^2*d^2*e^4-22
176*A*a^2*b^3*d^3*e^3+12672*A*a*b^4*d^4*e^2-2816*A*b^5*d^5*e-1386*B*a^5*d*e^5+9240*B*a^4*b*d^2*e^4-22176*B*a^3
*b^2*d^3*e^3+25344*B*a^2*b^3*d^4*e^2-14080*B*a*b^4*d^5*e+3072*B*b^5*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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maxima [A]  time = 0.66, size = 603, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} A}{63 \, \sqrt {e x + d} e^{6}} + \frac {2 \, {\left (63 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 14080 \, a b^{4} d^{5} e - 25344 \, a^{2} b^{3} d^{4} e^{2} + 22176 \, a^{3} b^{2} d^{3} e^{3} - 9240 \, a^{4} b d^{2} e^{4} + 1386 \, a^{5} d e^{5} - 7 \, {\left (12 \, b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 10 \, {\left (12 \, b^{5} d^{2} e^{4} - 55 \, a b^{4} d e^{5} + 99 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (96 \, b^{5} d^{3} e^{3} - 440 \, a b^{4} d^{2} e^{4} + 792 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} + {\left (384 \, b^{5} d^{4} e^{2} - 1760 \, a b^{4} d^{3} e^{3} + 3168 \, a^{2} b^{3} d^{2} e^{4} - 2772 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} - {\left (1536 \, b^{5} d^{5} e - 7040 \, a b^{4} d^{4} e^{2} + 12672 \, a^{2} b^{3} d^{3} e^{3} - 11088 \, a^{3} b^{2} d^{2} e^{4} + 4620 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} B}{693 \, \sqrt {e x + d} e^{7}} \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)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b
*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*
x^3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e - 576*a
*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*A/(sqrt(e*x + d)*e^6) + 2/693*(63*
b^5*e^6*x^6 - 3072*b^5*d^6 + 14080*a*b^4*d^5*e - 25344*a^2*b^3*d^4*e^2 + 22176*a^3*b^2*d^3*e^3 - 9240*a^4*b*d^
2*e^4 + 1386*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 55*a*b^4*d*e^5 + 99*a^2*b^
3*e^6)*x^4 - 2*(96*b^5*d^3*e^3 - 440*a*b^4*d^2*e^4 + 792*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (384*b^5*d^4*e
^2 - 1760*a*b^4*d^3*e^3 + 3168*a^2*b^3*d^2*e^4 - 2772*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (1536*b^5*d^5*e -
7040*a*b^4*d^4*e^2 + 12672*a^2*b^3*d^3*e^3 - 11088*a^3*b^2*d^2*e^4 + 4620*a^4*b*d*e^5 - 693*a^5*e^6)*x)*B/(sqr
t(e*x + d)*e^7)

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mupad [B]  time = 3.74, size = 659, normalized size = 1.48 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {x^2\,\left (2310\,B\,a^4\,b\,e^6-5544\,B\,a^3\,b^2\,d\,e^5+4620\,A\,a^3\,b^2\,e^6+6336\,B\,a^2\,b^3\,d^2\,e^4-5544\,A\,a^2\,b^3\,d\,e^5-3520\,B\,a\,b^4\,d^3\,e^3+3168\,A\,a\,b^4\,d^2\,e^4+768\,B\,b^5\,d^4\,e^2-704\,A\,b^5\,d^3\,e^3\right )}{693\,b\,e^7}-\frac {-2772\,B\,a^5\,d\,e^5+1386\,A\,a^5\,e^6+18480\,B\,a^4\,b\,d^2\,e^4-13860\,A\,a^4\,b\,d\,e^5-44352\,B\,a^3\,b^2\,d^3\,e^3+36960\,A\,a^3\,b^2\,d^2\,e^4+50688\,B\,a^2\,b^3\,d^4\,e^2-44352\,A\,a^2\,b^3\,d^3\,e^3-28160\,B\,a\,b^4\,d^5\,e+25344\,A\,a\,b^4\,d^4\,e^2+6144\,B\,b^5\,d^6-5632\,A\,b^5\,d^5\,e}{693\,b\,e^7}+\frac {x^3\,\left (2772\,B\,a^3\,b^2\,e^6-3168\,B\,a^2\,b^3\,d\,e^5+2772\,A\,a^2\,b^3\,e^6+1760\,B\,a\,b^4\,d^2\,e^4-1584\,A\,a\,b^4\,d\,e^5-384\,B\,b^5\,d^3\,e^3+352\,A\,b^5\,d^2\,e^4\right )}{693\,b\,e^7}+\frac {2\,b^3\,x^5\,\left (11\,A\,b\,e+55\,B\,a\,e-12\,B\,b\,d\right )}{99\,e^2}+\frac {x\,\left (1386\,B\,a^5\,e^6-9240\,B\,a^4\,b\,d\,e^5+6930\,A\,a^4\,b\,e^6+22176\,B\,a^3\,b^2\,d^2\,e^4-18480\,A\,a^3\,b^2\,d\,e^5-25344\,B\,a^2\,b^3\,d^3\,e^3+22176\,A\,a^2\,b^3\,d^2\,e^4+14080\,B\,a\,b^4\,d^4\,e^2-12672\,A\,a\,b^4\,d^3\,e^3-3072\,B\,b^5\,d^5\,e+2816\,A\,b^5\,d^4\,e^2\right )}{693\,b\,e^7}+\frac {10\,b^2\,x^4\,\left (198\,B\,a^2\,e^2-110\,B\,a\,b\,d\,e+99\,A\,a\,b\,e^2+24\,B\,b^2\,d^2-22\,A\,b^2\,d\,e\right )}{693\,e^3}+\frac {2\,B\,b^4\,x^6}{11\,e}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \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)^(3/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((x^2*(2310*B*a^4*b*e^6 + 4620*A*a^3*b^2*e^6 - 704*A*b^5*d^3*e^3 + 768*B*b^5*
d^4*e^2 + 3168*A*a*b^4*d^2*e^4 - 5544*A*a^2*b^3*d*e^5 - 3520*B*a*b^4*d^3*e^3 - 5544*B*a^3*b^2*d*e^5 + 6336*B*a
^2*b^3*d^2*e^4))/(693*b*e^7) - (1386*A*a^5*e^6 + 6144*B*b^5*d^6 - 5632*A*b^5*d^5*e - 2772*B*a^5*d*e^5 + 25344*
A*a*b^4*d^4*e^2 + 18480*B*a^4*b*d^2*e^4 - 44352*A*a^2*b^3*d^3*e^3 + 36960*A*a^3*b^2*d^2*e^4 + 50688*B*a^2*b^3*
d^4*e^2 - 44352*B*a^3*b^2*d^3*e^3 - 13860*A*a^4*b*d*e^5 - 28160*B*a*b^4*d^5*e)/(693*b*e^7) + (x^3*(2772*A*a^2*
b^3*e^6 + 2772*B*a^3*b^2*e^6 + 352*A*b^5*d^2*e^4 - 384*B*b^5*d^3*e^3 + 1760*B*a*b^4*d^2*e^4 - 3168*B*a^2*b^3*d
*e^5 - 1584*A*a*b^4*d*e^5))/(693*b*e^7) + (2*b^3*x^5*(11*A*b*e + 55*B*a*e - 12*B*b*d))/(99*e^2) + (x*(1386*B*a
^5*e^6 + 6930*A*a^4*b*e^6 - 3072*B*b^5*d^5*e + 2816*A*b^5*d^4*e^2 - 12672*A*a*b^4*d^3*e^3 - 18480*A*a^3*b^2*d*
e^5 + 14080*B*a*b^4*d^4*e^2 + 22176*A*a^2*b^3*d^2*e^4 - 25344*B*a^2*b^3*d^3*e^3 + 22176*B*a^3*b^2*d^2*e^4 - 92
40*B*a^4*b*d*e^5))/(693*b*e^7) + (10*b^2*x^4*(198*B*a^2*e^2 + 24*B*b^2*d^2 + 99*A*a*b*e^2 - 22*A*b^2*d*e - 110
*B*a*b*d*e))/(693*e^3) + (2*B*b^4*x^6)/(11*e)))/(x*(d + e*x)^(1/2) + (a*(d + e*x)^(1/2))/b)

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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)**(3/2),x)

[Out]

Timed out

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