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

Optimal. Leaf size=308 \[ -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)} \]

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Rubi [A]  time = 0.14, antiderivative size = 308, 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^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (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)^2 (-a B e-3 A b e+4 b B d)}{5 e^5 (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)^3 (B d-A e)}{3 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (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)^(3/2),x]

[Out]

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

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Mathematica [A]  time = 0.17, size = 163, normalized size = 0.53 \begin {gather*} \frac {2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{3/2} \left (-385 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+1485 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-693 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+1155 (b d-a e)^3 (B d-A e)+315 b^3 B (d+e x)^4\right )}{3465 e^5 (a+b x)^3} \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)^(3/2),x]

[Out]

(2*((a + b*x)^2)^(3/2)*(d + e*x)^(3/2)*(1155*(b*d - a*e)^3*(B*d - A*e) - 693*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e
- a*B*e)*(d + e*x) + 1485*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 385*b^2*(4*b*B*d - A*b*e - 3*a
*B*e)*(d + e*x)^3 + 315*b^3*B*(d + e*x)^4))/(3465*e^5*(a + b*x)^3)

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IntegrateAlgebraic [A]  time = 52.82, size = 374, normalized size = 1.21 \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (1155 a^3 A e^4+693 a^3 B e^3 (d+e x)-1155 a^3 B d e^3+2079 a^2 A b e^3 (d+e x)-3465 a^2 A b d e^3+3465 a^2 b B d^2 e^2-4158 a^2 b B d e^2 (d+e x)+1485 a^2 b B e^2 (d+e x)^2+3465 a A b^2 d^2 e^2-4158 a A b^2 d e^2 (d+e x)+1485 a A b^2 e^2 (d+e x)^2-3465 a b^2 B d^3 e+6237 a b^2 B d^2 e (d+e x)-4455 a b^2 B d e (d+e x)^2+1155 a b^2 B e (d+e x)^3-1155 A b^3 d^3 e+2079 A b^3 d^2 e (d+e x)-1485 A b^3 d e (d+e x)^2+385 A b^3 e (d+e x)^3+1155 b^3 B d^4-2772 b^3 B d^3 (d+e x)+2970 b^3 B d^2 (d+e x)^2-1540 b^3 B d (d+e x)^3+315 b^3 B (d+e x)^4\right )}{3465 e^4 (a e+b e x)} \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)^(3/2),x]

[Out]

(2*(d + e*x)^(3/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(1155*b^3*B*d^4 - 1155*A*b^3*d^3*e - 3465*a*b^2*B*d^3*e + 3465*a*
A*b^2*d^2*e^2 + 3465*a^2*b*B*d^2*e^2 - 3465*a^2*A*b*d*e^3 - 1155*a^3*B*d*e^3 + 1155*a^3*A*e^4 - 2772*b^3*B*d^3
*(d + e*x) + 2079*A*b^3*d^2*e*(d + e*x) + 6237*a*b^2*B*d^2*e*(d + e*x) - 4158*a*A*b^2*d*e^2*(d + e*x) - 4158*a
^2*b*B*d*e^2*(d + e*x) + 2079*a^2*A*b*e^3*(d + e*x) + 693*a^3*B*e^3*(d + e*x) + 2970*b^3*B*d^2*(d + e*x)^2 - 1
485*A*b^3*d*e*(d + e*x)^2 - 4455*a*b^2*B*d*e*(d + e*x)^2 + 1485*a*A*b^2*e^2*(d + e*x)^2 + 1485*a^2*b*B*e^2*(d
+ e*x)^2 - 1540*b^3*B*d*(d + e*x)^3 + 385*A*b^3*e*(d + e*x)^3 + 1155*a*b^2*B*e*(d + e*x)^3 + 315*b^3*B*(d + e*
x)^4))/(3465*e^4*(a*e + b*e*x))

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fricas [A]  time = 0.44, size = 353, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (315 \, B b^{3} e^{5} x^{5} + 128 \, B b^{3} d^{5} + 1155 \, A a^{3} d e^{4} - 176 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 792 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} - 462 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3} + 35 \, {\left (B b^{3} d e^{4} + 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{2} e^{3} - 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} - 297 \, {\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{3} e^{2} - 22 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 99 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{4} + 231 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{4} e - 1155 \, A a^{3} e^{5} - 88 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 396 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} - 231 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \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)^(3/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^3*e^5*x^5 + 128*B*b^3*d^5 + 1155*A*a^3*d*e^4 - 176*(3*B*a*b^2 + A*b^3)*d^4*e + 792*(B*a^2*b +
A*a*b^2)*d^3*e^2 - 462*(B*a^3 + 3*A*a^2*b)*d^2*e^3 + 35*(B*b^3*d*e^4 + 11*(3*B*a*b^2 + A*b^3)*e^5)*x^4 - 5*(8*
B*b^3*d^2*e^3 - 11*(3*B*a*b^2 + A*b^3)*d*e^4 - 297*(B*a^2*b + A*a*b^2)*e^5)*x^3 + 3*(16*B*b^3*d^3*e^2 - 22*(3*
B*a*b^2 + A*b^3)*d^2*e^3 + 99*(B*a^2*b + A*a*b^2)*d*e^4 + 231*(B*a^3 + 3*A*a^2*b)*e^5)*x^2 - (64*B*b^3*d^4*e -
1155*A*a^3*e^5 - 88*(3*B*a*b^2 + A*b^3)*d^3*e^2 + 396*(B*a^2*b + A*a*b^2)*d^2*e^3 - 231*(B*a^3 + 3*A*a^2*b)*d
*e^4)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.27, size = 898, normalized size = 2.92

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)^(3/2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*d*e^(-1)*sgn(b*x + a) + 3465*((x*e + d)^(3/2) - 3*sqr
t(x*e + d)*d)*A*a^2*b*d*e^(-1)*sgn(b*x + a) + 693*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)
*d^2)*B*a^2*b*d*e^(-2)*sgn(b*x + a) + 693*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*
a*b^2*d*e^(-2)*sgn(b*x + a) + 297*(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*b^2*d*e^(-3)*sgn(b*x + a) + 99*(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*b^3*d*e^(-3)*sgn(b*x + a) + 11*(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*b^3*d*e^(-4)*sgn(b*x + a) + 231*(
3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^3*e^(-1)*sgn(b*x + a) + 693*(3*(x*e + d)^
(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*b*e^(-1)*sgn(b*x + a) + 297*(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^2*b*e^(-2)*sgn(b*x + a) + 297*(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*b^2*e^(-2)*sgn(b*x + a)
+ 33*(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*sq
rt(x*e + d)*d^4)*B*a*b^2*e^(-3)*sgn(b*x + a) + 11*(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*b^3*e^(-3)*sgn(b*x + a) + 5*(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 - 69
3*sqrt(x*e + d)*d^5)*B*b^3*e^(-4)*sgn(b*x + a) + 3465*sqrt(x*e + d)*A*a^3*d*sgn(b*x + a) + 1155*((x*e + d)^(3/
2) - 3*sqrt(x*e + d)*d)*A*a^3*sgn(b*x + a))*e^(-1)

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maple [A]  time = 0.06, size = 317, normalized size = 1.03 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 b^{3} B \,x^{4} e^{4}+385 A \,b^{3} e^{4} x^{3}+1155 B a \,b^{2} e^{4} x^{3}-280 B \,b^{3} d \,e^{3} x^{3}+1485 A a \,b^{2} e^{4} x^{2}-330 A \,b^{3} d \,e^{3} x^{2}+1485 B \,a^{2} b \,e^{4} x^{2}-990 B a \,b^{2} d \,e^{3} x^{2}+240 B \,b^{3} d^{2} e^{2} x^{2}+2079 A \,a^{2} b \,e^{4} x -1188 A a \,b^{2} d \,e^{3} x +264 A \,b^{3} d^{2} e^{2} x +693 B \,a^{3} e^{4} x -1188 B \,a^{2} b d \,e^{3} x +792 B a \,b^{2} d^{2} e^{2} x -192 B \,b^{3} d^{3} e x +1155 A \,a^{3} e^{4}-1386 A \,a^{2} b d \,e^{3}+792 A a \,b^{2} d^{2} e^{2}-176 A \,b^{3} d^{3} e -462 B \,a^{3} d \,e^{3}+792 B \,a^{2} b \,d^{2} e^{2}-528 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3465 \left (b x +a \right )^{3} e^{5}} \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)^(3/2)*(e*x+d)^(1/2),x)

[Out]

2/3465*(e*x+d)^(3/2)*(315*B*b^3*e^4*x^4+385*A*b^3*e^4*x^3+1155*B*a*b^2*e^4*x^3-280*B*b^3*d*e^3*x^3+1485*A*a*b^
2*e^4*x^2-330*A*b^3*d*e^3*x^2+1485*B*a^2*b*e^4*x^2-990*B*a*b^2*d*e^3*x^2+240*B*b^3*d^2*e^2*x^2+2079*A*a^2*b*e^
4*x-1188*A*a*b^2*d*e^3*x+264*A*b^3*d^2*e^2*x+693*B*a^3*e^4*x-1188*B*a^2*b*d*e^3*x+792*B*a*b^2*d^2*e^2*x-192*B*
b^3*d^3*e*x+1155*A*a^3*e^4-1386*A*a^2*b*d*e^3+792*A*a*b^2*d^2*e^2-176*A*b^3*d^3*e-462*B*a^3*d*e^3+792*B*a^2*b*
d^2*e^2-528*B*a*b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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maxima [A]  time = 0.61, size = 384, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \, {\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} + {\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt {e x + d} A}{315 \, e^{4}} + \frac {2 \, {\left (315 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 528 \, a b^{2} d^{4} e + 792 \, a^{2} b d^{3} e^{2} - 462 \, a^{3} d^{2} e^{3} + 35 \, {\left (b^{3} d e^{4} + 33 \, a b^{2} e^{5}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{2} e^{3} - 33 \, a b^{2} d e^{4} - 297 \, a^{2} b e^{5}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{3} e^{2} - 66 \, a b^{2} d^{2} e^{3} + 99 \, a^{2} b d e^{4} + 231 \, a^{3} e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{4} e - 264 \, a b^{2} d^{3} e^{2} + 396 \, a^{2} b d^{2} e^{3} - 231 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d} B}{3465 \, e^{5}} \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)^(3/2)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*b^3*e^4*x^4 - 16*b^3*d^4 + 72*a*b^2*d^3*e - 126*a^2*b*d^2*e^2 + 105*a^3*d*e^3 + 5*(b^3*d*e^3 + 27*a*
b^2*e^4)*x^3 - 3*(2*b^3*d^2*e^2 - 9*a*b^2*d*e^3 - 63*a^2*b*e^4)*x^2 + (8*b^3*d^3*e - 36*a*b^2*d^2*e^2 + 63*a^2
*b*d*e^3 + 105*a^3*e^4)*x)*sqrt(e*x + d)*A/e^4 + 2/3465*(315*b^3*e^5*x^5 + 128*b^3*d^5 - 528*a*b^2*d^4*e + 792
*a^2*b*d^3*e^2 - 462*a^3*d^2*e^3 + 35*(b^3*d*e^4 + 33*a*b^2*e^5)*x^4 - 5*(8*b^3*d^2*e^3 - 33*a*b^2*d*e^4 - 297
*a^2*b*e^5)*x^3 + 3*(16*b^3*d^3*e^2 - 66*a*b^2*d^2*e^3 + 99*a^2*b*d*e^4 + 231*a^3*e^5)*x^2 - (64*b^3*d^4*e - 2
64*a*b^2*d^3*e^2 + 396*a^2*b*d^2*e^3 - 231*a^3*d*e^4)*x)*sqrt(e*x + d)*B/e^5

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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 )}^{3/2} \,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)^(3/2),x)

[Out]

int((A + B*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), 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 {3}{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)**(3/2)*(e*x+d)**(1/2),x)

[Out]

Integral((A + B*x)*sqrt(d + e*x)*((a + b*x)**2)**(3/2), x)

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