3.1736 \(\int (a+b x)^3 (A+B x) (d+e x)^{3/2} \, dx\)

Optimal. Leaf size=173 \[ -\frac{2 b^2 (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5}+\frac{2 b (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5}-\frac{2 (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5}+\frac{2 b^3 B (d+e x)^{13/2}}{13 e^5} \]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(7/2))/(7*e^5) + (2*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2))/(3*e^5) - (2*b^2*(4*b*B*d - A*b*
e - 3*a*B*e)*(d + e*x)^(11/2))/(11*e^5) + (2*b^3*B*(d + e*x)^(13/2))/(13*e^5)

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Rubi [A]  time = 0.0693775, antiderivative size = 173, normalized size of antiderivative = 1., 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} \[ -\frac{2 b^2 (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5}+\frac{2 b (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5}-\frac{2 (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5}+\frac{2 b^3 B (d+e x)^{13/2}}{13 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(7/2))/(7*e^5) + (2*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2))/(3*e^5) - (2*b^2*(4*b*B*d - A*b*
e - 3*a*B*e)*(d + e*x)^(11/2))/(11*e^5) + (2*b^3*B*(d + e*x)^(13/2))/(13*e^5)

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)^3 (A+B x) (d+e x)^{3/2} \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e) (d+e x)^{3/2}}{e^4}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{5/2}}{e^4}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^4}+\frac{b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{9/2}}{e^4}+\frac{b^3 B (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac{2 (b d-a e)^3 (B d-A e) (d+e x)^{5/2}}{5 e^5}-\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{7/2}}{7 e^5}+\frac{2 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{9/2}}{3 e^5}-\frac{2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{11/2}}{11 e^5}+\frac{2 b^3 B (d+e x)^{13/2}}{13 e^5}\\ \end{align*}

Mathematica [A]  time = 0.161379, size = 145, normalized size = 0.84 \[ \frac{2 (d+e x)^{5/2} \left (-1365 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+5005 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-2145 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+3003 (b d-a e)^3 (B d-A e)+1155 b^3 B (d+e x)^4\right )}{15015 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(3003*(b*d - a*e)^3*(B*d - A*e) - 2145*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
+ 5005*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 1365*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^3
+ 1155*b^3*B*(d + e*x)^4))/(15015*e^5)

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Maple [A]  time = 0.004, size = 301, normalized size = 1.7 \begin{align*}{\frac{2310\,{b}^{3}B{x}^{4}{e}^{4}+2730\,A{b}^{3}{e}^{4}{x}^{3}+8190\,Ba{b}^{2}{e}^{4}{x}^{3}-1680\,B{b}^{3}d{e}^{3}{x}^{3}+10010\,Aa{b}^{2}{e}^{4}{x}^{2}-1820\,A{b}^{3}d{e}^{3}{x}^{2}+10010\,B{a}^{2}b{e}^{4}{x}^{2}-5460\,Ba{b}^{2}d{e}^{3}{x}^{2}+1120\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+12870\,A{a}^{2}b{e}^{4}x-5720\,Aa{b}^{2}d{e}^{3}x+1040\,A{b}^{3}{d}^{2}{e}^{2}x+4290\,B{a}^{3}{e}^{4}x-5720\,B{a}^{2}bd{e}^{3}x+3120\,Ba{b}^{2}{d}^{2}{e}^{2}x-640\,B{b}^{3}{d}^{3}ex+6006\,{a}^{3}A{e}^{4}-5148\,A{a}^{2}bd{e}^{3}+2288\,Aa{b}^{2}{d}^{2}{e}^{2}-416\,A{b}^{3}{d}^{3}e-1716\,B{a}^{3}d{e}^{3}+2288\,B{a}^{2}b{d}^{2}{e}^{2}-1248\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{15015\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(e*x+d)^(5/2)*(1155*B*b^3*e^4*x^4+1365*A*b^3*e^4*x^3+4095*B*a*b^2*e^4*x^3-840*B*b^3*d*e^3*x^3+5005*A*a
*b^2*e^4*x^2-910*A*b^3*d*e^3*x^2+5005*B*a^2*b*e^4*x^2-2730*B*a*b^2*d*e^3*x^2+560*B*b^3*d^2*e^2*x^2+6435*A*a^2*
b*e^4*x-2860*A*a*b^2*d*e^3*x+520*A*b^3*d^2*e^2*x+2145*B*a^3*e^4*x-2860*B*a^2*b*d*e^3*x+1560*B*a*b^2*d^2*e^2*x-
320*B*b^3*d^3*e*x+3003*A*a^3*e^4-2574*A*a^2*b*d*e^3+1144*A*a*b^2*d^2*e^2-208*A*b^3*d^3*e-858*B*a^3*d*e^3+1144*
B*a^2*b*d^2*e^2-624*B*a*b^2*d^3*e+128*B*b^3*d^4)/e^5

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Maxima [A]  time = 1.36497, size = 358, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (1155 \,{\left (e x + d\right )}^{\frac{13}{2}} B b^{3} - 1365 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 5005 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 2145 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 3003 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{15015 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*(e*x + d)^(13/2)*B*b^3 - 1365*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x + d)^(11/2) + 5005*(2*B*b
^3*d^2 - (3*B*a*b^2 + A*b^3)*d*e + (B*a^2*b + A*a*b^2)*e^2)*(e*x + d)^(9/2) - 2145*(4*B*b^3*d^3 - 3*(3*B*a*b^2
+ A*b^3)*d^2*e + 6*(B*a^2*b + A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*(e*x + d)^(7/2) + 3003*(B*b^3*d^4 + A
*a^3*e^4 - (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3)*(e*x + d)^(5
/2))/e^5

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Fricas [B]  time = 1.37393, size = 994, normalized size = 5.75 \begin{align*} \frac{2 \,{\left (1155 \, B b^{3} e^{6} x^{6} + 128 \, B b^{3} d^{6} + 3003 \, A a^{3} d^{2} e^{4} - 208 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e + 1144 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{2} - 858 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{3} + 105 \,{\left (14 \, B b^{3} d e^{5} + 13 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{6}\right )} x^{5} + 35 \,{\left (B b^{3} d^{2} e^{4} + 52 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{5} + 143 \,{\left (B a^{2} b + A a b^{2}\right )} e^{6}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{3} e^{3} - 13 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{4} - 1430 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{5} - 429 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{4} e^{2} + 1001 \, A a^{3} e^{6} - 26 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{3} + 143 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{4} + 1144 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{5} e - 6006 \, A a^{3} d e^{5} - 104 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{2} + 572 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{3} - 429 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*B*b^3*e^6*x^6 + 128*B*b^3*d^6 + 3003*A*a^3*d^2*e^4 - 208*(3*B*a*b^2 + A*b^3)*d^5*e + 1144*(B*a^2
*b + A*a*b^2)*d^4*e^2 - 858*(B*a^3 + 3*A*a^2*b)*d^3*e^3 + 105*(14*B*b^3*d*e^5 + 13*(3*B*a*b^2 + A*b^3)*e^6)*x^
5 + 35*(B*b^3*d^2*e^4 + 52*(3*B*a*b^2 + A*b^3)*d*e^5 + 143*(B*a^2*b + A*a*b^2)*e^6)*x^4 - 5*(8*B*b^3*d^3*e^3 -
13*(3*B*a*b^2 + A*b^3)*d^2*e^4 - 1430*(B*a^2*b + A*a*b^2)*d*e^5 - 429*(B*a^3 + 3*A*a^2*b)*e^6)*x^3 + 3*(16*B*
b^3*d^4*e^2 + 1001*A*a^3*e^6 - 26*(3*B*a*b^2 + A*b^3)*d^3*e^3 + 143*(B*a^2*b + A*a*b^2)*d^2*e^4 + 1144*(B*a^3
+ 3*A*a^2*b)*d*e^5)*x^2 - (64*B*b^3*d^5*e - 6006*A*a^3*d*e^5 - 104*(3*B*a*b^2 + A*b^3)*d^4*e^2 + 572*(B*a^2*b
+ A*a*b^2)*d^3*e^3 - 429*(B*a^3 + 3*A*a^2*b)*d^2*e^4)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 24.5762, size = 913, normalized size = 5.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**3*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a**3*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 6*A*a**2*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 6*A*a**2*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 + 6*A*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 + 6*A*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*A*b**3*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*A*b**3*(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 + 2*B
*a**3*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 2*B*a**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)*
*(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*B*a**2*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 + 6*B*a**2*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 + 6*B*a*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 + 6*B*a*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 + 2*B*b**3*d*(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
**5 + 2*B*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e
*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5

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Giac [B]  time = 1.73428, size = 1091, normalized size = 6.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^3*d*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 5*(x*e + d)
^(3/2)*d)*A*a^2*b*d*e^(-1) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^2*b
*d*e^(-2) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a*b^2*d*e^(-2) + 429*(
35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a*b^2*d*e^(-
3) + 143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*b^
3*d*e^(-3) + 13*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/
2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*b^3*d*e^(-4) + 15015*(x*e + d)^(3/2)*A*a^3*d + 429*(15*(x*e + d)^(7/2) -
42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^3*e^(-1) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d
+ 35*(x*e + d)^(3/2)*d^2)*A*a^2*b*e^(-1) + 429*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/
2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a^2*b*e^(-2) + 429*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e
+ d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a*b^2*e^(-2) + 39*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d +
2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a*b^2*e^(-3) + 13*(315*(x*e
+ d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(
3/2)*d^4)*A*b^3*e^(-3) + 5*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870
*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*b^3*e^(-4) + 3003*(3*(x*e + d)^(
5/2) - 5*(x*e + d)^(3/2)*d)*A*a^3)*e^(-1)