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

Optimal. Leaf size=452 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{7 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{5 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-5 a B e-A b e+6 b B d)}{13 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)} \]

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Rubi [A]  time = 0.21, antiderivative size = 452, 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)^{13/2} (-5 a B e-A b e+6 b B d)}{13 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{7 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{5 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^5*(B*d - A*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*d - a*e)
^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*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)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) + (1
0*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x
)) - (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (
2*b^5*B*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*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 (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (A+B x) \sqrt {d+e x} \, 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) \sqrt {d+e x}}{e^6}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{3/2}}{e^6}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^{5/2}}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{9/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{11/2}}{e^6}+\frac {b^{10} B (d+e x)^{13/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^5 (B d-A e) (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 d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 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)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 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)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 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)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 239, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (-3465 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+20475 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-50050 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+32175 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-9009 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+15015 (b d-a e)^5 (B d-A e)+3003 b^5 B (d+e x)^6\right )}{45045 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(15015*(b*d - a*e)^5*(B*d - A*e) - 9009*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e
- a*B*e)*(d + e*x) + 32175*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 50050*b^2*(b*d - a*e)^2*(
2*b*B*d - A*b*e - a*B*e)*(d + e*x)^3 + 20475*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 3465*b^
4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5 + 3003*b^5*B*(d + e*x)^6))/(45045*e^7*(a + b*x))

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IntegrateAlgebraic [A]  time = 53.72, size = 812, normalized size = 1.80 \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (15015 b^5 B d^6-15015 A b^5 e d^5-75075 a b^4 B e d^5-54054 b^5 B (d+e x) d^5+75075 a A b^4 e^2 d^4+150150 a^2 b^3 B e^2 d^4+96525 b^5 B (d+e x)^2 d^4+45045 A b^5 e (d+e x) d^4+225225 a b^4 B e (d+e x) d^4-150150 a^2 A b^3 e^3 d^3-150150 a^3 b^2 B e^3 d^3-100100 b^5 B (d+e x)^3 d^3-64350 A b^5 e (d+e x)^2 d^3-321750 a b^4 B e (d+e x)^2 d^3-180180 a A b^4 e^2 (d+e x) d^3-360360 a^2 b^3 B e^2 (d+e x) d^3+150150 a^3 A b^2 e^4 d^2+75075 a^4 b B e^4 d^2+61425 b^5 B (d+e x)^4 d^2+50050 A b^5 e (d+e x)^3 d^2+250250 a b^4 B e (d+e x)^3 d^2+193050 a A b^4 e^2 (d+e x)^2 d^2+386100 a^2 b^3 B e^2 (d+e x)^2 d^2+270270 a^2 A b^3 e^3 (d+e x) d^2+270270 a^3 b^2 B e^3 (d+e x) d^2-75075 a^4 A b e^5 d-15015 a^5 B e^5 d-20790 b^5 B (d+e x)^5 d-20475 A b^5 e (d+e x)^4 d-102375 a b^4 B e (d+e x)^4 d-100100 a A b^4 e^2 (d+e x)^3 d-200200 a^2 b^3 B e^2 (d+e x)^3 d-193050 a^2 A b^3 e^3 (d+e x)^2 d-193050 a^3 b^2 B e^3 (d+e x)^2 d-180180 a^3 A b^2 e^4 (d+e x) d-90090 a^4 b B e^4 (d+e x) d+15015 a^5 A e^6+3003 b^5 B (d+e x)^6+3465 A b^5 e (d+e x)^5+17325 a b^4 B e (d+e x)^5+20475 a A b^4 e^2 (d+e x)^4+40950 a^2 b^3 B e^2 (d+e x)^4+50050 a^2 A b^3 e^3 (d+e x)^3+50050 a^3 b^2 B e^3 (d+e x)^3+64350 a^3 A b^2 e^4 (d+e x)^2+32175 a^4 b B e^4 (d+e x)^2+45045 a^4 A b e^5 (d+e x)+9009 a^5 B e^5 (d+e x)\right )}{45045 e^6 (a e+b x e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(15015*b^5*B*d^6 - 15015*A*b^5*d^5*e - 75075*a*b^4*B*d^5*e + 7507
5*a*A*b^4*d^4*e^2 + 150150*a^2*b^3*B*d^4*e^2 - 150150*a^2*A*b^3*d^3*e^3 - 150150*a^3*b^2*B*d^3*e^3 + 150150*a^
3*A*b^2*d^2*e^4 + 75075*a^4*b*B*d^2*e^4 - 75075*a^4*A*b*d*e^5 - 15015*a^5*B*d*e^5 + 15015*a^5*A*e^6 - 54054*b^
5*B*d^5*(d + e*x) + 45045*A*b^5*d^4*e*(d + e*x) + 225225*a*b^4*B*d^4*e*(d + e*x) - 180180*a*A*b^4*d^3*e^2*(d +
e*x) - 360360*a^2*b^3*B*d^3*e^2*(d + e*x) + 270270*a^2*A*b^3*d^2*e^3*(d + e*x) + 270270*a^3*b^2*B*d^2*e^3*(d
+ e*x) - 180180*a^3*A*b^2*d*e^4*(d + e*x) - 90090*a^4*b*B*d*e^4*(d + e*x) + 45045*a^4*A*b*e^5*(d + e*x) + 9009
*a^5*B*e^5*(d + e*x) + 96525*b^5*B*d^4*(d + e*x)^2 - 64350*A*b^5*d^3*e*(d + e*x)^2 - 321750*a*b^4*B*d^3*e*(d +
e*x)^2 + 193050*a*A*b^4*d^2*e^2*(d + e*x)^2 + 386100*a^2*b^3*B*d^2*e^2*(d + e*x)^2 - 193050*a^2*A*b^3*d*e^3*(
d + e*x)^2 - 193050*a^3*b^2*B*d*e^3*(d + e*x)^2 + 64350*a^3*A*b^2*e^4*(d + e*x)^2 + 32175*a^4*b*B*e^4*(d + e*x
)^2 - 100100*b^5*B*d^3*(d + e*x)^3 + 50050*A*b^5*d^2*e*(d + e*x)^3 + 250250*a*b^4*B*d^2*e*(d + e*x)^3 - 100100
*a*A*b^4*d*e^2*(d + e*x)^3 - 200200*a^2*b^3*B*d*e^2*(d + e*x)^3 + 50050*a^2*A*b^3*e^3*(d + e*x)^3 + 50050*a^3*
b^2*B*e^3*(d + e*x)^3 + 61425*b^5*B*d^2*(d + e*x)^4 - 20475*A*b^5*d*e*(d + e*x)^4 - 102375*a*b^4*B*d*e*(d + e*
x)^4 + 20475*a*A*b^4*e^2*(d + e*x)^4 + 40950*a^2*b^3*B*e^2*(d + e*x)^4 - 20790*b^5*B*d*(d + e*x)^5 + 3465*A*b^
5*e*(d + e*x)^5 + 17325*a*b^4*B*e*(d + e*x)^5 + 3003*b^5*B*(d + e*x)^6))/(45045*e^6*(a*e + b*e*x))

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fricas [B]  time = 0.43, size = 702, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (3003 \, B b^{5} e^{7} x^{7} + 1024 \, B b^{5} d^{7} + 15015 \, A a^{5} d e^{6} - 1280 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{6} e + 8320 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} e^{2} - 22880 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} e^{3} + 17160 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e^{4} - 6006 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{5} + 231 \, {\left (B b^{5} d e^{6} + 15 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{7}\right )} x^{6} - 63 \, {\left (4 \, B b^{5} d^{2} e^{5} - 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{6} - 325 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{7}\right )} x^{5} + 35 \, {\left (8 \, B b^{5} d^{3} e^{4} - 10 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{5} + 65 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{6} + 1430 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{7}\right )} x^{4} - 5 \, {\left (64 \, B b^{5} d^{4} e^{3} - 80 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{4} + 520 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{5} - 1430 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{6} - 6435 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{7}\right )} x^{3} + 3 \, {\left (128 \, B b^{5} d^{5} e^{2} - 160 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{3} + 1040 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{4} - 2860 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{5} + 2145 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{6} + 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{7}\right )} x^{2} - {\left (512 \, B b^{5} d^{6} e - 15015 \, A a^{5} e^{7} - 640 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e^{2} + 4160 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{3} - 11440 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{4} + 8580 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{5} - 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, 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)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^5*e^7*x^7 + 1024*B*b^5*d^7 + 15015*A*a^5*d*e^6 - 1280*(5*B*a*b^4 + A*b^5)*d^6*e + 8320*(2*B*
a^2*b^3 + A*a*b^4)*d^5*e^2 - 22880*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^3 + 17160*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^4 - 6
006*(B*a^5 + 5*A*a^4*b)*d^2*e^5 + 231*(B*b^5*d*e^6 + 15*(5*B*a*b^4 + A*b^5)*e^7)*x^6 - 63*(4*B*b^5*d^2*e^5 - 5
*(5*B*a*b^4 + A*b^5)*d*e^6 - 325*(2*B*a^2*b^3 + A*a*b^4)*e^7)*x^5 + 35*(8*B*b^5*d^3*e^4 - 10*(5*B*a*b^4 + A*b^
5)*d^2*e^5 + 65*(2*B*a^2*b^3 + A*a*b^4)*d*e^6 + 1430*(B*a^3*b^2 + A*a^2*b^3)*e^7)*x^4 - 5*(64*B*b^5*d^4*e^3 -
80*(5*B*a*b^4 + A*b^5)*d^3*e^4 + 520*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^5 - 1430*(B*a^3*b^2 + A*a^2*b^3)*d*e^6 - 64
35*(B*a^4*b + 2*A*a^3*b^2)*e^7)*x^3 + 3*(128*B*b^5*d^5*e^2 - 160*(5*B*a*b^4 + A*b^5)*d^4*e^3 + 1040*(2*B*a^2*b
^3 + A*a*b^4)*d^3*e^4 - 2860*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^5 + 2145*(B*a^4*b + 2*A*a^3*b^2)*d*e^6 + 3003*(B*a^
5 + 5*A*a^4*b)*e^7)*x^2 - (512*B*b^5*d^6*e - 15015*A*a^5*e^7 - 640*(5*B*a*b^4 + A*b^5)*d^5*e^2 + 4160*(2*B*a^2
*b^3 + A*a*b^4)*d^4*e^3 - 11440*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^4 + 8580*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^5 - 3003*
(B*a^5 + 5*A*a^4*b)*d*e^6)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.39, size = 1682, normalized size = 3.72

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)^(1/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^5*d*e^(-1)*sgn(b*x + a) + 75075*((x*e + d)^(3/2) - 3*
sqrt(x*e + d)*d)*A*a^4*b*d*e^(-1)*sgn(b*x + a) + 15015*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e
+ d)*d^2)*B*a^4*b*d*e^(-2)*sgn(b*x + a) + 30030*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*
d^2)*A*a^3*b^2*d*e^(-2)*sgn(b*x + a) + 12870*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^
2 - 35*sqrt(x*e + d)*d^3)*B*a^3*b^2*d*e^(-3)*sgn(b*x + a) + 12870*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d +
35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^2*b^3*d*e^(-3)*sgn(b*x + a) + 1430*(35*(x*e + d)^(9/2) - 18
0*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^2*b^3*d*e
^(-4)*sgn(b*x + a) + 715*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)
^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a*b^4*d*e^(-4)*sgn(b*x + a) + 325*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(
9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5
)*B*a*b^4*d*e^(-5)*sgn(b*x + a) + 65*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 -
1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*b^5*d*e^(-5)*sgn(b*x + a) + 15*
(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(
x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*b^5*d*e^(-6)*sgn(b*x + a) + 3003*(3*
(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^5*e^(-1)*sgn(b*x + a) + 15015*(3*(x*e + d)^
(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^4*b*e^(-1)*sgn(b*x + a) + 6435*(5*(x*e + d)^(7/2) - 2
1*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^4*b*e^(-2)*sgn(b*x + a) + 12870*(5*(x
*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^3*b^2*e^(-2)*sgn(b*x
+ a) + 1430*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 +
315*sqrt(x*e + d)*d^4)*B*a^3*b^2*e^(-3)*sgn(b*x + a) + 1430*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378
*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^2*b^3*e^(-3)*sgn(b*x + a) + 650*(6
3*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e +
d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^2*b^3*e^(-4)*sgn(b*x + a) + 325*(63*(x*e + d)^(11/2) - 385*(x*e + d)
^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d
^5)*A*a*b^4*e^(-4)*sgn(b*x + a) + 75*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^
2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B
*a*b^4*e^(-5)*sgn(b*x + a) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8
580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*b^5*
e^(-5)*sgn(b*x + a) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(
x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*
sqrt(x*e + d)*d^7)*B*b^5*e^(-6)*sgn(b*x + a) + 45045*sqrt(x*e + d)*A*a^5*d*sgn(b*x + a) + 15015*((x*e + d)^(3/
2) - 3*sqrt(x*e + d)*d)*A*a^5*sgn(b*x + a))*e^(-1)

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maple [A]  time = 0.05, size = 689, normalized size = 1.52 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 B \,b^{5} e^{6} x^{6}+3465 A \,b^{5} e^{6} x^{5}+17325 B a \,b^{4} e^{6} x^{5}-2772 B \,b^{5} d \,e^{5} x^{5}+20475 A a \,b^{4} e^{6} x^{4}-3150 A \,b^{5} d \,e^{5} x^{4}+40950 B \,a^{2} b^{3} e^{6} x^{4}-15750 B a \,b^{4} d \,e^{5} x^{4}+2520 B \,b^{5} d^{2} e^{4} x^{4}+50050 A \,a^{2} b^{3} e^{6} x^{3}-18200 A a \,b^{4} d \,e^{5} x^{3}+2800 A \,b^{5} d^{2} e^{4} x^{3}+50050 B \,a^{3} b^{2} e^{6} x^{3}-36400 B \,a^{2} b^{3} d \,e^{5} x^{3}+14000 B a \,b^{4} d^{2} e^{4} x^{3}-2240 B \,b^{5} d^{3} e^{3} x^{3}+64350 A \,a^{3} b^{2} e^{6} x^{2}-42900 A \,a^{2} b^{3} d \,e^{5} x^{2}+15600 A a \,b^{4} d^{2} e^{4} x^{2}-2400 A \,b^{5} d^{3} e^{3} x^{2}+32175 B \,a^{4} b \,e^{6} x^{2}-42900 B \,a^{3} b^{2} d \,e^{5} x^{2}+31200 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-12000 B a \,b^{4} d^{3} e^{3} x^{2}+1920 B \,b^{5} d^{4} e^{2} x^{2}+45045 A \,a^{4} b \,e^{6} x -51480 A \,a^{3} b^{2} d \,e^{5} x +34320 A \,a^{2} b^{3} d^{2} e^{4} x -12480 A a \,b^{4} d^{3} e^{3} x +1920 A \,b^{5} d^{4} e^{2} x +9009 B \,a^{5} e^{6} x -25740 B \,a^{4} b d \,e^{5} x +34320 B \,a^{3} b^{2} d^{2} e^{4} x -24960 B \,a^{2} b^{3} d^{3} e^{3} x +9600 B a \,b^{4} d^{4} e^{2} x -1536 B \,b^{5} d^{5} e x +15015 A \,a^{5} e^{6}-30030 A \,a^{4} b d \,e^{5}+34320 A \,a^{3} b^{2} d^{2} e^{4}-22880 A \,a^{2} b^{3} d^{3} e^{3}+8320 A a \,b^{4} d^{4} e^{2}-1280 A \,b^{5} d^{5} e -6006 B \,a^{5} d \,e^{5}+17160 B \,a^{4} b \,d^{2} e^{4}-22880 B \,a^{3} b^{2} d^{3} e^{3}+16640 B \,a^{2} b^{3} d^{4} e^{2}-6400 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}}}{45045 \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)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*B*b^5*e^6*x^6+3465*A*b^5*e^6*x^5+17325*B*a*b^4*e^6*x^5-2772*B*b^5*d*e^5*x^5+20475*
A*a*b^4*e^6*x^4-3150*A*b^5*d*e^5*x^4+40950*B*a^2*b^3*e^6*x^4-15750*B*a*b^4*d*e^5*x^4+2520*B*b^5*d^2*e^4*x^4+50
050*A*a^2*b^3*e^6*x^3-18200*A*a*b^4*d*e^5*x^3+2800*A*b^5*d^2*e^4*x^3+50050*B*a^3*b^2*e^6*x^3-36400*B*a^2*b^3*d
*e^5*x^3+14000*B*a*b^4*d^2*e^4*x^3-2240*B*b^5*d^3*e^3*x^3+64350*A*a^3*b^2*e^6*x^2-42900*A*a^2*b^3*d*e^5*x^2+15
600*A*a*b^4*d^2*e^4*x^2-2400*A*b^5*d^3*e^3*x^2+32175*B*a^4*b*e^6*x^2-42900*B*a^3*b^2*d*e^5*x^2+31200*B*a^2*b^3
*d^2*e^4*x^2-12000*B*a*b^4*d^3*e^3*x^2+1920*B*b^5*d^4*e^2*x^2+45045*A*a^4*b*e^6*x-51480*A*a^3*b^2*d*e^5*x+3432
0*A*a^2*b^3*d^2*e^4*x-12480*A*a*b^4*d^3*e^3*x+1920*A*b^5*d^4*e^2*x+9009*B*a^5*e^6*x-25740*B*a^4*b*d*e^5*x+3432
0*B*a^3*b^2*d^2*e^4*x-24960*B*a^2*b^3*d^3*e^3*x+9600*B*a*b^4*d^4*e^2*x-1536*B*b^5*d^5*e*x+15015*A*a^5*e^6-3003
0*A*a^4*b*d*e^5+34320*A*a^3*b^2*d^2*e^4-22880*A*a^2*b^3*d^3*e^3+8320*A*a*b^4*d^4*e^2-1280*A*b^5*d^5*e-6006*B*a
^5*d*e^5+17160*B*a^4*b*d^2*e^4-22880*B*a^3*b^2*d^3*e^3+16640*B*a^2*b^3*d^4*e^2-6400*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 [B]  time = 0.81, size = 760, normalized size = 1.68 \begin {gather*} \frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d} A}{9009 \, e^{6}} + \frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} + 1024 \, b^{5} d^{7} - 6400 \, a b^{4} d^{6} e + 16640 \, a^{2} b^{3} d^{5} e^{2} - 22880 \, a^{3} b^{2} d^{4} e^{3} + 17160 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} + 231 \, {\left (b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{5} d^{2} e^{5} - 25 \, a b^{4} d e^{6} - 650 \, a^{2} b^{3} e^{7}\right )} x^{5} + 70 \, {\left (4 \, b^{5} d^{3} e^{4} - 25 \, a b^{4} d^{2} e^{5} + 65 \, a^{2} b^{3} d e^{6} + 715 \, a^{3} b^{2} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{5} d^{4} e^{3} - 400 \, a b^{4} d^{3} e^{4} + 1040 \, a^{2} b^{3} d^{2} e^{5} - 1430 \, a^{3} b^{2} d e^{6} - 6435 \, a^{4} b e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{5} e^{2} - 800 \, a b^{4} d^{4} e^{3} + 2080 \, a^{2} b^{3} d^{3} e^{4} - 2860 \, a^{3} b^{2} d^{2} e^{5} + 2145 \, a^{4} b d e^{6} + 3003 \, a^{5} e^{7}\right )} x^{2} - {\left (512 \, b^{5} d^{6} e - 3200 \, a b^{4} d^{5} e^{2} + 8320 \, a^{2} b^{3} d^{4} e^{3} - 11440 \, a^{3} b^{2} d^{3} e^{4} + 8580 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d} B}{45045 \, 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)^(1/2),x, algorithm="maxima")

[Out]

2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*
a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*
a^2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^
5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e
- 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(
e*x + d)*A/e^6 + 2/45045*(3003*b^5*e^7*x^7 + 1024*b^5*d^7 - 6400*a*b^4*d^6*e + 16640*a^2*b^3*d^5*e^2 - 22880*a
^3*b^2*d^4*e^3 + 17160*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 + 231*(b^5*d*e^6 + 75*a*b^4*e^7)*x^6 - 63*(4*b^5*d^2*e
^5 - 25*a*b^4*d*e^6 - 650*a^2*b^3*e^7)*x^5 + 70*(4*b^5*d^3*e^4 - 25*a*b^4*d^2*e^5 + 65*a^2*b^3*d*e^6 + 715*a^3
*b^2*e^7)*x^4 - 5*(64*b^5*d^4*e^3 - 400*a*b^4*d^3*e^4 + 1040*a^2*b^3*d^2*e^5 - 1430*a^3*b^2*d*e^6 - 6435*a^4*b
*e^7)*x^3 + 3*(128*b^5*d^5*e^2 - 800*a*b^4*d^4*e^3 + 2080*a^2*b^3*d^3*e^4 - 2860*a^3*b^2*d^2*e^5 + 2145*a^4*b*
d*e^6 + 3003*a^5*e^7)*x^2 - (512*b^5*d^6*e - 3200*a*b^4*d^5*e^2 + 8320*a^2*b^3*d^4*e^3 - 11440*a^3*b^2*d^3*e^4
+ 8580*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*sqrt(e*x + d)*B/e^7

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((A + B*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B x\right ) \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, 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)**(1/2),x)

[Out]

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

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