3.1718 \(\int (a+b x)^3 (A+B x) (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=173 \[ -\frac{2 b^2 (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5}+\frac{6 b (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5}+\frac{2 (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5}+\frac{2 b^3 B (d+e x)^{15/2}}{15 e^5} \]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^5) - (2*(b*d - a*e)^2*(4*b*B*
d - 3*A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*
e - a*B*e)*(d + e*x)^(11/2))/(11*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d +
e*x)^(13/2))/(13*e^5) + (2*b^3*B*(d + e*x)^(15/2))/(15*e^5)

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Rubi [A]  time = 0.218867, 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 \[ -\frac{2 b^2 (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5}+\frac{6 b (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5}+\frac{2 (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5}+\frac{2 b^3 B (d+e x)^{15/2}}{15 e^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^5) - (2*(b*d - a*e)^2*(4*b*B*
d - 3*A*b*e - a*B*e)*(d + e*x)^(9/2))/(9*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*
e - a*B*e)*(d + e*x)^(11/2))/(11*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d +
e*x)^(13/2))/(13*e^5) + (2*b^3*B*(d + e*x)^(15/2))/(15*e^5)

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Rubi in Sympy [A]  time = 46.5546, size = 170, normalized size = 0.98 \[ \frac{2 B b^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{5}} + \frac{2 b^{2} \left (d + e x\right )^{\frac{13}{2}} \left (A b e + 3 B a e - 4 B b d\right )}{13 e^{5}} + \frac{6 b \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{11 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{9 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{3}}{7 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

2*B*b**3*(d + e*x)**(15/2)/(15*e**5) + 2*b**2*(d + e*x)**(13/2)*(A*b*e + 3*B*a*e
- 4*B*b*d)/(13*e**5) + 6*b*(d + e*x)**(11/2)*(a*e - b*d)*(A*b*e + B*a*e - 2*B*b
*d)/(11*e**5) + 2*(d + e*x)**(9/2)*(a*e - b*d)**2*(3*A*b*e + B*a*e - 4*B*b*d)/(9
*e**5) + 2*(d + e*x)**(7/2)*(A*e - B*d)*(a*e - b*d)**3/(7*e**5)

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Mathematica [A]  time = 0.338188, size = 228, normalized size = 1.32 \[ \frac{2 (d+e x)^{7/2} \left (715 a^3 e^3 (9 A e-2 B d+7 B e x)+195 a^2 b e^2 \left (11 A e (7 e x-2 d)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-15 a b^2 e \left (3 B \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )-13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+b^3 \left (15 A e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+B \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )\right )}{45045 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(2*(d + e*x)^(7/2)*(715*a^3*e^3*(-2*B*d + 9*A*e + 7*B*e*x) + 195*a^2*b*e^2*(11*A
*e*(-2*d + 7*e*x) + B*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 15*a*b^2*e*(-13*A*e*(8*
d^2 - 28*d*e*x + 63*e^2*x^2) + 3*B*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^
3*x^3)) + b^3*(15*A*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + B*(
128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4))))/(45
045*e^5)

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Maple [A]  time = 0.012, size = 301, normalized size = 1.7 \[{\frac{6006\,B{b}^{3}{x}^{4}{e}^{4}+6930\,A{b}^{3}{e}^{4}{x}^{3}+20790\,Ba{b}^{2}{e}^{4}{x}^{3}-3696\,B{b}^{3}d{e}^{3}{x}^{3}+24570\,Aa{b}^{2}{e}^{4}{x}^{2}-3780\,A{b}^{3}d{e}^{3}{x}^{2}+24570\,B{a}^{2}b{e}^{4}{x}^{2}-11340\,Ba{b}^{2}d{e}^{3}{x}^{2}+2016\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+30030\,A{a}^{2}b{e}^{4}x-10920\,Aa{b}^{2}d{e}^{3}x+1680\,A{b}^{3}{d}^{2}{e}^{2}x+10010\,B{a}^{3}{e}^{4}x-10920\,B{a}^{2}bd{e}^{3}x+5040\,Ba{b}^{2}{d}^{2}{e}^{2}x-896\,B{b}^{3}{d}^{3}ex+12870\,{a}^{3}A{e}^{4}-8580\,A{a}^{2}bd{e}^{3}+3120\,Aa{b}^{2}{d}^{2}{e}^{2}-480\,A{b}^{3}{d}^{3}e-2860\,B{a}^{3}d{e}^{3}+3120\,B{a}^{2}b{d}^{2}{e}^{2}-1440\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)*(e*x+d)^(5/2),x)

[Out]

2/45045*(e*x+d)^(7/2)*(3003*B*b^3*e^4*x^4+3465*A*b^3*e^4*x^3+10395*B*a*b^2*e^4*x
^3-1848*B*b^3*d*e^3*x^3+12285*A*a*b^2*e^4*x^2-1890*A*b^3*d*e^3*x^2+12285*B*a^2*b
*e^4*x^2-5670*B*a*b^2*d*e^3*x^2+1008*B*b^3*d^2*e^2*x^2+15015*A*a^2*b*e^4*x-5460*
A*a*b^2*d*e^3*x+840*A*b^3*d^2*e^2*x+5005*B*a^3*e^4*x-5460*B*a^2*b*d*e^3*x+2520*B
*a*b^2*d^2*e^2*x-448*B*b^3*d^3*e*x+6435*A*a^3*e^4-4290*A*a^2*b*d*e^3+1560*A*a*b^
2*d^2*e^2-240*A*b^3*d^3*e-1430*B*a^3*d*e^3+1560*B*a^2*b*d^2*e^2-720*B*a*b^2*d^3*
e+128*B*b^3*d^4)/e^5

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Maxima [A]  time = 1.35407, size = 358, normalized size = 2.07 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} B b^{3} - 3465 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 12285 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 5005 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*B*b^3 - 3465*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*
(e*x + d)^(13/2) + 12285*(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)^(11/2) - 5005*(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)^(9/2) + 6435*(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)^(7/2))/e^5

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Fricas [A]  time = 0.228234, size = 728, normalized size = 4.21 \[ \frac{2 \,{\left (3003 \, B b^{3} e^{7} x^{7} + 128 \, B b^{3} d^{7} + 6435 \, A a^{3} d^{3} e^{4} - 240 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e + 1560 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{2} - 1430 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{3} + 231 \,{\left (31 \, B b^{3} d e^{6} + 15 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{7}\right )} x^{6} + 63 \,{\left (71 \, B b^{3} d^{2} e^{5} + 135 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{6} + 195 \,{\left (B a^{2} b + A a b^{2}\right )} e^{7}\right )} x^{5} + 35 \,{\left (B b^{3} d^{3} e^{4} + 159 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{5} + 897 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{6} + 143 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{4} e^{3} - 1287 \, A a^{3} e^{7} - 15 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{4} - 4407 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{5} - 2717 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{5} e^{2} + 6435 \, A a^{3} d e^{6} - 30 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{3} + 195 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{4} + 3575 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{6} e - 19305 \, A a^{3} d^{2} e^{5} - 120 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{2} + 780 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{3} - 715 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^3*e^7*x^7 + 128*B*b^3*d^7 + 6435*A*a^3*d^3*e^4 - 240*(3*B*a*b^
2 + A*b^3)*d^6*e + 1560*(B*a^2*b + A*a*b^2)*d^5*e^2 - 1430*(B*a^3 + 3*A*a^2*b)*d
^4*e^3 + 231*(31*B*b^3*d*e^6 + 15*(3*B*a*b^2 + A*b^3)*e^7)*x^6 + 63*(71*B*b^3*d^
2*e^5 + 135*(3*B*a*b^2 + A*b^3)*d*e^6 + 195*(B*a^2*b + A*a*b^2)*e^7)*x^5 + 35*(B
*b^3*d^3*e^4 + 159*(3*B*a*b^2 + A*b^3)*d^2*e^5 + 897*(B*a^2*b + A*a*b^2)*d*e^6 +
143*(B*a^3 + 3*A*a^2*b)*e^7)*x^4 - 5*(8*B*b^3*d^4*e^3 - 1287*A*a^3*e^7 - 15*(3*
B*a*b^2 + A*b^3)*d^3*e^4 - 4407*(B*a^2*b + A*a*b^2)*d^2*e^5 - 2717*(B*a^3 + 3*A*
a^2*b)*d*e^6)*x^3 + 3*(16*B*b^3*d^5*e^2 + 6435*A*a^3*d*e^6 - 30*(3*B*a*b^2 + A*b
^3)*d^4*e^3 + 195*(B*a^2*b + A*a*b^2)*d^3*e^4 + 3575*(B*a^3 + 3*A*a^2*b)*d^2*e^5
)*x^2 - (64*B*b^3*d^6*e - 19305*A*a^3*d^2*e^5 - 120*(3*B*a*b^2 + A*b^3)*d^5*e^2
+ 780*(B*a^2*b + A*a*b^2)*d^4*e^3 - 715*(B*a^3 + 3*A*a^2*b)*d^3*e^4)*x)*sqrt(e*x
+ d)/e^5

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Sympy [A]  time = 19.223, size = 1564, normalized size = 9.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

A*a**3*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) +
4*A*a**3*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 2*A*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 + 6*A*a**2*b*d**
2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*A*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**2 + 6*A*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**2 + 6*A*a*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*
x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*A*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**3
+ 6*A*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**3 + 2*A*b**3*
d**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**4 + 4*A*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**4 + 2*A*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**4 + 2*B*a**3*d**2*(-d*(d + e*x)**(3/2)/3 + (d +
e*x)**(5/2)/5)/e**2 + 4*B*a**3*d*(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*B*a**3*(-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**2 + 6*B*a**2*b*
d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**
3 + 12*B*a**2*b*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**3 + 6*B*a**2*b*(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**3 + 6*B*a*b**2*d**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**4 + 12
*B*a*b**2*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**4 + 6*B*a*b**2*
(-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
**4 + 2*B*b**3*d**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**5 + 4*B
*b**3*d*(-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 + 2*B*b**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15
*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2
)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5

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GIAC/XCAS [A]  time = 0.238391, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done