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

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

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Rubi [A]  time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \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)^2*(A + B*x)*(d + e*x)^(3/2),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 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]))))

Rubi steps

\begin {align*} \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 = 0.14, size = 107, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (-385 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+495 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-693 (b d-a e)^2 (B d-A e)+315 b^2 B (d+e x)^3\right )}{3465 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.10, size = 193, normalized size = 1.51 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (693 a^2 A e^3+495 a^2 B e^2 (d+e x)-693 a^2 B d e^2+990 a A b e^2 (d+e x)-1386 a A b d e^2+1386 a b B d^2 e-1980 a b B d e (d+e x)+770 a b B e (d+e x)^2+693 A b^2 d^2 e-990 A b^2 d e (d+e x)+385 A b^2 e (d+e x)^2-693 b^2 B d^3+1485 b^2 B d^2 (d+e x)-1155 b^2 B d (d+e x)^2+315 b^2 B (d+e x)^3\right )}{3465 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(-693*b^2*B*d^3 + 693*A*b^2*d^2*e + 1386*a*b*B*d^2*e - 1386*a*A*b*d*e^2 - 693*a^2*B*d*e^2 +
693*a^2*A*e^3 + 1485*b^2*B*d^2*(d + e*x) - 990*A*b^2*d*e*(d + e*x) - 1980*a*b*B*d*e*(d + e*x) + 990*a*A*b*e^2
*(d + e*x) + 495*a^2*B*e^2*(d + e*x) - 1155*b^2*B*d*(d + e*x)^2 + 385*A*b^2*e*(d + e*x)^2 + 770*a*b*B*e*(d + e
*x)^2 + 315*b^2*B*(d + e*x)^3))/(3465*e^4)

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fricas [B]  time = 1.44, size = 289, normalized size = 2.26 \begin {gather*} \frac {2 \, {\left (315 \, B b^{2} e^{5} x^{5} - 48 \, B b^{2} d^{5} + 693 \, A a^{2} d^{2} e^{3} + 88 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e - 198 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2} + 35 \, {\left (12 \, B b^{2} d e^{4} + 11 \, {\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{4} + 5 \, {\left (3 \, B b^{2} d^{2} e^{3} + 110 \, {\left (2 \, B a b + A b^{2}\right )} d e^{4} + 99 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{3} - 3 \, {\left (6 \, B b^{2} d^{3} e^{2} - 231 \, A a^{2} e^{5} - 11 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} - 264 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{2} + {\left (24 \, B b^{2} d^{4} e + 1386 \, A a^{2} d e^{4} - 44 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 99 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*B*b^2*e^5*x^5 - 48*B*b^2*d^5 + 693*A*a^2*d^2*e^3 + 88*(2*B*a*b + A*b^2)*d^4*e - 198*(B*a^2 + 2*A*a
*b)*d^3*e^2 + 35*(12*B*b^2*d*e^4 + 11*(2*B*a*b + A*b^2)*e^5)*x^4 + 5*(3*B*b^2*d^2*e^3 + 110*(2*B*a*b + A*b^2)*
d*e^4 + 99*(B*a^2 + 2*A*a*b)*e^5)*x^3 - 3*(6*B*b^2*d^3*e^2 - 231*A*a^2*e^5 - 11*(2*B*a*b + A*b^2)*d^2*e^3 - 26
4*(B*a^2 + 2*A*a*b)*d*e^4)*x^2 + (24*B*b^2*d^4*e + 1386*A*a^2*d*e^4 - 44*(2*B*a*b + A*b^2)*d^3*e^2 + 99*(B*a^2
+ 2*A*a*b)*d^2*e^3)*x)*sqrt(e*x + d)/e^4

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giac [B]  time = 1.64, size = 901, normalized size = 7.04

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(3/2),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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maple [A]  time = 0.01, size = 169, normalized size = 1.32 \begin {gather*} \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 \,b^{2} d^{3}\right )}{3465 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(e*x+d)^(5/2)*(315*B*b^2*e^3*x^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*a^2*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*b^2*d^3)/e^4

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maxima [A]  time = 0.64, size = 159, normalized size = 1.24 \begin {gather*} \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B b^{2} - 385 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{3465 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)*(a + b*x)^2*(d + e*x)^(3/2),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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sympy [A]  time = 22.59, 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)**2*(B*x+A)*(e*x+d)**(3/2),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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