3.18.86 \(\int (A+B x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2) \, dx\) [1786]

Optimal. Leaf size=128 \[ -\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{5/2}}{5 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{7/2}}{7 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{9/2}}{9 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4} \]

[Out]

-2/5*(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^(5/2)/e^4+2/7*(-a*e+b*d)*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(7/2)/e^4-2/9*b
*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(9/2)/e^4+2/11*b^2*B*(e*x+d)^(11/2)/e^4

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Rubi [A]
time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 78} \begin {gather*} -\frac {2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac {2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac {2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(7/2))/(7*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(9/2))/(9*e^4) + (2*b^2*B*(d + e*x)^(11/2))/(11*e^
4)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 78

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]))))

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 139, normalized size = 1.09 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (99 a^2 e^2 (-2 B d+7 A e+5 B e x)+22 a b e \left (9 A e (-2 d+5 e x)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+b^2 \left (11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(d + e*x)^(5/2)*(99*a^2*e^2*(-2*B*d + 7*A*e + 5*B*e*x) + 22*a*b*e*(9*A*e*(-2*d + 5*e*x) + B*(8*d^2 - 20*d*e
*x + 35*e^2*x^2)) + b^2*(11*A*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) - 3*B*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 10
5*e^3*x^3))))/(3465*e^4)

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Maple [A]
time = 0.65, size = 148, normalized size = 1.16

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -B d \right ) b^{2}+B \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) \left (2 a b e -2 b^{2} d \right )+B \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (A e -B d \right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\)	\(148\)
default	\(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -B d \right ) b^{2}+B \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) \left (2 a b e -2 b^{2} d \right )+B \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (A e -B d \right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\)	\(148\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (315 b^{2} B \,x^{3} e^{3}+385 A \,b^{2} e^{3} x^{2}+770 B a b \,e^{3} x^{2}-210 B \,b^{2} d \,e^{2} x^{2}+990 A a b \,e^{3} x -220 A \,b^{2} d \,e^{2} x +495 B \,a^{2} e^{3} x -440 B a b d \,e^{2} x +120 B \,b^{2} d^{2} e x +693 a^{2} A \,e^{3}-396 A a b d \,e^{2}+88 A \,b^{2} d^{2} e -198 B \,a^{2} d \,e^{2}+176 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{3465 e^{4}}\)	\(169\)
trager	\(\frac {2 \left (315 b^{2} B \,e^{5} x^{5}+385 A \,b^{2} e^{5} x^{4}+770 B a b \,e^{5} x^{4}+420 b^{2} B d \,e^{4} x^{4}+990 A a b \,e^{5} x^{3}+550 A \,b^{2} d \,e^{4} x^{3}+495 B \,e^{5} a^{2} x^{3}+1100 B a b d \,e^{4} x^{3}+15 b^{2} B \,d^{2} e^{3} x^{3}+693 A \,a^{2} e^{5} x^{2}+1584 A a b d \,e^{4} x^{2}+33 A \,b^{2} d^{2} e^{3} x^{2}+792 B \,a^{2} d \,e^{4} x^{2}+66 B a b \,d^{2} e^{3} x^{2}-18 b^{2} B \,d^{3} e^{2} x^{2}+1386 A \,a^{2} d \,e^{4} x +198 A a b \,d^{2} e^{3} x -44 A \,b^{2} d^{3} e^{2} x +99 B \,a^{2} d^{2} e^{3} x -88 B a b \,d^{3} e^{2} x +24 B \,b^{2} d^{4} e x +693 a^{2} A \,d^{2} e^{3}-396 A a b \,d^{3} e^{2}+88 A \,b^{2} d^{4} e -198 B \,a^{2} d^{3} e^{2}+176 B a b \,d^{4} e -48 b^{2} B \,d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\)	\(341\)
risch	\(\frac {2 \left (315 b^{2} B \,e^{5} x^{5}+385 A \,b^{2} e^{5} x^{4}+770 B a b \,e^{5} x^{4}+420 b^{2} B d \,e^{4} x^{4}+990 A a b \,e^{5} x^{3}+550 A \,b^{2} d \,e^{4} x^{3}+495 B \,e^{5} a^{2} x^{3}+1100 B a b d \,e^{4} x^{3}+15 b^{2} B \,d^{2} e^{3} x^{3}+693 A \,a^{2} e^{5} x^{2}+1584 A a b d \,e^{4} x^{2}+33 A \,b^{2} d^{2} e^{3} x^{2}+792 B \,a^{2} d \,e^{4} x^{2}+66 B a b \,d^{2} e^{3} x^{2}-18 b^{2} B \,d^{3} e^{2} x^{2}+1386 A \,a^{2} d \,e^{4} x +198 A a b \,d^{2} e^{3} x -44 A \,b^{2} d^{3} e^{2} x +99 B \,a^{2} d^{2} e^{3} x -88 B a b \,d^{3} e^{2} x +24 B \,b^{2} d^{4} e x +693 a^{2} A \,d^{2} e^{3}-396 A a b \,d^{3} e^{2}+88 A \,b^{2} d^{4} e -198 B \,a^{2} d^{3} e^{2}+176 B a b \,d^{4} e -48 b^{2} B \,d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\)	\(341\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

2/e^4*(1/11*b^2*B*(e*x+d)^(11/2)+1/9*((A*e-B*d)*b^2+B*(2*a*b*e-2*b^2*d))*(e*x+d)^(9/2)+1/7*((A*e-B*d)*(2*a*b*e
-2*b^2*d)+B*(a^2*e^2-2*a*b*d*e+b^2*d^2))*(e*x+d)^(7/2)+1/5*(A*e-B*d)*(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(5/2)
)

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Maxima [A]
time = 0.27, size = 168, normalized size = 1.31 \begin {gather*} \frac {2}{3465} \, {\left (315 \, {\left (x e + d\right )}^{\frac {11}{2}} B b^{2} - 385 \, {\left (3 \, B b^{2} d - 2 \, B a b e - A b^{2} e\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 495 \, {\left (3 \, B b^{2} d^{2} + B a^{2} e^{2} + 2 \, A a b e^{2} - 2 \, {\left (2 \, B a b e + A b^{2} e\right )} d\right )} {\left (x e + d\right )}^{\frac {7}{2}} - 693 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b e + A b^{2} e\right )} d^{2} + {\left (B a^{2} e^{2} + 2 \, A a b e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

2/3465*(315*(x*e + d)^(11/2)*B*b^2 - 385*(3*B*b^2*d - 2*B*a*b*e - A*b^2*e)*(x*e + d)^(9/2) + 495*(3*B*b^2*d^2
+ B*a^2*e^2 + 2*A*a*b*e^2 - 2*(2*B*a*b*e + A*b^2*e)*d)*(x*e + d)^(7/2) - 693*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b
*e + A*b^2*e)*d^2 + (B*a^2*e^2 + 2*A*a*b*e^2)*d)*(x*e + d)^(5/2))*e^(-4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (119) = 238\).
time = 0.96, size = 277, normalized size = 2.16 \begin {gather*} -\frac {2}{3465} \, {\left (48 \, B b^{2} d^{5} - {\left (315 \, B b^{2} x^{5} + 693 \, A a^{2} x^{2} + 385 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 495 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3}\right )} e^{5} - 2 \, {\left (210 \, B b^{2} d x^{4} + 693 \, A a^{2} d x + 275 \, {\left (2 \, B a b + A b^{2}\right )} d x^{3} + 396 \, {\left (B a^{2} + 2 \, A a b\right )} d x^{2}\right )} e^{4} - 3 \, {\left (5 \, B b^{2} d^{2} x^{3} + 231 \, A a^{2} d^{2} + 11 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} x^{2} + 33 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} x\right )} e^{3} + 2 \, {\left (9 \, B b^{2} d^{3} x^{2} + 22 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} x + 99 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3}\right )} e^{2} - 8 \, {\left (3 \, B b^{2} d^{4} x + 11 \, {\left (2 \, B a b + A b^{2}\right )} d^{4}\right )} e\right )} \sqrt {x e + d} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

-2/3465*(48*B*b^2*d^5 - (315*B*b^2*x^5 + 693*A*a^2*x^2 + 385*(2*B*a*b + A*b^2)*x^4 + 495*(B*a^2 + 2*A*a*b)*x^3
)*e^5 - 2*(210*B*b^2*d*x^4 + 693*A*a^2*d*x + 275*(2*B*a*b + A*b^2)*d*x^3 + 396*(B*a^2 + 2*A*a*b)*d*x^2)*e^4 -
3*(5*B*b^2*d^2*x^3 + 231*A*a^2*d^2 + 11*(2*B*a*b + A*b^2)*d^2*x^2 + 33*(B*a^2 + 2*A*a*b)*d^2*x)*e^3 + 2*(9*B*b
^2*d^3*x^2 + 22*(2*B*a*b + A*b^2)*d^3*x + 99*(B*a^2 + 2*A*a*b)*d^3)*e^2 - 8*(3*B*b^2*d^4*x + 11*(2*B*a*b + A*b
^2)*d^4)*e)*sqrt(x*e + d)*e^(-4)

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Sympy [A]
time = 11.16, size = 586, normalized size = 4.58 \begin {gather*} A a^{2} d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 A a^{2} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {4 A a b d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {4 A a b \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {2 A b^{2} d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 A b^{2} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {2 B a^{2} d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {2 B a^{2} \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {4 B a b d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {4 B a b \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {2 B b^{2} d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {2 B b^{2} \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a**2*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 4*A*a*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*A*a*b*(d**2*(d + e*x
)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*A*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d
+ e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 2*A*b**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 -
3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 2*B*a**2*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e*
*2 + 2*B*a**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 4*B*a*b*d*(d**2*(
d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 4*B*a*b*(-d**3*(d + e*x)**(3/2)/3 + 3*
d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 2*B*b**2*d*(-d**3*(d + e*x)**(3/
2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 2*B*b**2*(d**4*(d + e*x
)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2
)/11)/e**4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (119) = 238\).
time = 1.28, size = 901, normalized size = 7.04 \begin {gather*} \frac {2}{3465} \, {\left (1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{2} d^{2} e^{\left (-1\right )} + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a b d^{2} e^{\left (-1\right )} + 462 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a b d^{2} e^{\left (-2\right )} + 231 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A b^{2} d^{2} e^{\left (-2\right )} + 99 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B b^{2} d^{2} e^{\left (-3\right )} + 462 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{2} d e^{\left (-1\right )} + 924 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a b d e^{\left (-1\right )} + 396 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a b d e^{\left (-2\right )} + 198 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A b^{2} d e^{\left (-2\right )} + 22 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B b^{2} d e^{\left (-3\right )} + 3465 \, \sqrt {x e + d} A a^{2} d^{2} + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{2} d + 99 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a^{2} e^{\left (-1\right )} + 198 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A a b e^{\left (-1\right )} + 22 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B a b e^{\left (-2\right )} + 11 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} A b^{2} e^{\left (-2\right )} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} B b^{2} e^{\left (-3\right )} + 231 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^2*d^2*e^(-1) + 2310*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*
d)*A*a*b*d^2*e^(-1) + 462*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a*b*d^2*e^(-2) +
231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*d^2*e^(-2) + 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)*B*b^2*d^2*e^(-3) + 462*(3*(x*e + d)^
(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^2*d*e^(-1) + 924*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3
/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*b*d*e^(-1) + 396*(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*d*e^(-2) + 198*(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^2*d*e^(-2) + 22*(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^2*d*e^(-3) + 3465*sqrt(x*e + d)*A*a^2*d^2
+ 2310*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^2*d + 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)*B*a^2*e^(-1) + 198*(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*e^(-1) + 22*(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*b*e^(-2) + 11*(35*(x*e + d)^(9/2) - 1
80*(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^2*e^(-2)
+ 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 - 693*sqrt(x*e + d)*d^5)*B*b^2*e^(-3) + 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15
*sqrt(x*e + d)*d^2)*A*a^2)*e^(-1)

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Mupad [B]
time = 0.07, size = 115, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{9\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{7\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(9/2)*(2*A*b^2*e - 6*B*b^2*d + 4*B*a*b*e))/(9*e^4) + (2*B*b^2*(d + e*x)^(11/2))/(11*e^4) + (2*(a*e
- b*d)*(d + e*x)^(7/2)*(2*A*b*e + B*a*e - 3*B*b*d))/(7*e^4) + (2*(A*e - B*d)*(a*e - b*d)^2*(d + e*x)^(5/2))/(5
*e^4)

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