3.2215 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac{4 b (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(7*b*B
*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^(7/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) +
(4*b*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^(7/2))/(693*e*(b*d - a*e)^3*(d +
e*x)^(7/2))

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Rubi [A]  time = 0.268184, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{4 b (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-11 a B e+4 A b e+7 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(7*b*B
*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^(7/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) +
(4*b*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^(7/2))/(693*e*(b*d - a*e)^3*(d +
e*x)^(7/2))

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Rubi in Sympy [A]  time = 25.0861, size = 138, normalized size = 0.94 \[ - \frac{4 b \left (a + b x\right )^{\frac{7}{2}} \left (4 A b e - 11 B a e + 7 B b d\right )}{693 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (4 A b e - 11 B a e + 7 B b d\right )}{99 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{11 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(13/2),x)

[Out]

-4*b*(a + b*x)**(7/2)*(4*A*b*e - 11*B*a*e + 7*B*b*d)/(693*e*(d + e*x)**(7/2)*(a*
e - b*d)**3) + 2*(a + b*x)**(7/2)*(4*A*b*e - 11*B*a*e + 7*B*b*d)/(99*e*(d + e*x)
**(9/2)*(a*e - b*d)**2) - 2*(a + b*x)**(7/2)*(A*e - B*d)/(11*e*(d + e*x)**(11/2)
*(a*e - b*d))

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Mathematica [A]  time = 0.417297, size = 135, normalized size = 0.92 \[ \frac{2 (a+b x)^{7/2} \left (A \left (63 a^2 e^2-14 a b e (11 d+2 e x)+b^2 \left (99 d^2+44 d e x+8 e^2 x^2\right )\right )+B \left (7 a^2 e (2 d+11 e x)-2 a b \left (11 d^2+85 d e x+11 e^2 x^2\right )+7 b^2 d x (11 d+2 e x)\right )\right )}{693 (d+e x)^{11/2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(2*(a + b*x)^(7/2)*(A*(63*a^2*e^2 - 14*a*b*e*(11*d + 2*e*x) + b^2*(99*d^2 + 44*d
*e*x + 8*e^2*x^2)) + B*(7*b^2*d*x*(11*d + 2*e*x) + 7*a^2*e*(2*d + 11*e*x) - 2*a*
b*(11*d^2 + 85*d*e*x + 11*e^2*x^2))))/(693*(b*d - a*e)^3*(d + e*x)^(11/2))

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Maple [A]  time = 0.011, size = 177, normalized size = 1.2 \[ -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-44\,Bab{e}^{2}{x}^{2}+28\,B{b}^{2}de{x}^{2}-56\,Aab{e}^{2}x+88\,A{b}^{2}dex+154\,B{a}^{2}{e}^{2}x-340\,Babdex+154\,B{b}^{2}{d}^{2}x+126\,A{a}^{2}{e}^{2}-308\,Aabde+198\,A{b}^{2}{d}^{2}+28\,B{a}^{2}de-44\,Bab{d}^{2}}{693\,{a}^{3}{e}^{3}-2079\,{a}^{2}bd{e}^{2}+2079\,a{b}^{2}{d}^{2}e-693\,{b}^{3}{d}^{3}} \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(13/2),x)

[Out]

-2/693*(b*x+a)^(7/2)*(8*A*b^2*e^2*x^2-22*B*a*b*e^2*x^2+14*B*b^2*d*e*x^2-28*A*a*b
*e^2*x+44*A*b^2*d*e*x+77*B*a^2*e^2*x-170*B*a*b*d*e*x+77*B*b^2*d^2*x+63*A*a^2*e^2
-154*A*a*b*d*e+99*A*b^2*d^2+14*B*a^2*d*e-22*B*a*b*d^2)/(e*x+d)^(11/2)/(a^3*e^3-3
*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.90702, size = 936, normalized size = 6.37 \[ \frac{2 \,{\left (63 \, A a^{5} e^{2} + 2 \,{\left (7 \, B b^{5} d e -{\left (11 \, B a b^{4} - 4 \, A b^{5}\right )} e^{2}\right )} x^{5} +{\left (77 \, B b^{5} d^{2} - 4 \,{\left (32 \, B a b^{4} - 11 \, A b^{5}\right )} d e +{\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} e^{2}\right )} x^{4} +{\left (11 \,{\left (19 \, B a b^{4} + 9 \, A b^{5}\right )} d^{2} - 2 \,{\left (227 \, B a^{2} b^{3} + 11 \, A a b^{4}\right )} d e + 3 \,{\left (55 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{2}\right )} x^{3} - 11 \,{\left (2 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} d^{2} + 14 \,{\left (B a^{5} - 11 \, A a^{4} b\right )} d e +{\left (33 \,{\left (5 \, B a^{2} b^{3} + 9 \, A a b^{4}\right )} d^{2} - 2 \,{\left (227 \, B a^{3} b^{2} + 165 \, A a^{2} b^{3}\right )} d e +{\left (209 \, B a^{4} b + 113 \, A a^{3} b^{2}\right )} e^{2}\right )} x^{2} +{\left (11 \,{\left (B a^{3} b^{2} + 27 \, A a^{2} b^{3}\right )} d^{2} - 2 \,{\left (64 \, B a^{4} b + 209 \, A a^{3} b^{2}\right )} d e + 7 \,{\left (11 \, B a^{5} + 23 \, A a^{4} b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{693 \,{\left (b^{3} d^{9} - 3 \, a b^{2} d^{8} e + 3 \, a^{2} b d^{7} e^{2} - a^{3} d^{6} e^{3} +{\left (b^{3} d^{3} e^{6} - 3 \, a b^{2} d^{2} e^{7} + 3 \, a^{2} b d e^{8} - a^{3} e^{9}\right )} x^{6} + 6 \,{\left (b^{3} d^{4} e^{5} - 3 \, a b^{2} d^{3} e^{6} + 3 \, a^{2} b d^{2} e^{7} - a^{3} d e^{8}\right )} x^{5} + 15 \,{\left (b^{3} d^{5} e^{4} - 3 \, a b^{2} d^{4} e^{5} + 3 \, a^{2} b d^{3} e^{6} - a^{3} d^{2} e^{7}\right )} x^{4} + 20 \,{\left (b^{3} d^{6} e^{3} - 3 \, a b^{2} d^{5} e^{4} + 3 \, a^{2} b d^{4} e^{5} - a^{3} d^{3} e^{6}\right )} x^{3} + 15 \,{\left (b^{3} d^{7} e^{2} - 3 \, a b^{2} d^{6} e^{3} + 3 \, a^{2} b d^{5} e^{4} - a^{3} d^{4} e^{5}\right )} x^{2} + 6 \,{\left (b^{3} d^{8} e - 3 \, a b^{2} d^{7} e^{2} + 3 \, a^{2} b d^{6} e^{3} - a^{3} d^{5} e^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(13/2),x, algorithm="fricas")

[Out]

2/693*(63*A*a^5*e^2 + 2*(7*B*b^5*d*e - (11*B*a*b^4 - 4*A*b^5)*e^2)*x^5 + (77*B*b
^5*d^2 - 4*(32*B*a*b^4 - 11*A*b^5)*d*e + (11*B*a^2*b^3 - 4*A*a*b^4)*e^2)*x^4 + (
11*(19*B*a*b^4 + 9*A*b^5)*d^2 - 2*(227*B*a^2*b^3 + 11*A*a*b^4)*d*e + 3*(55*B*a^3
*b^2 + A*a^2*b^3)*e^2)*x^3 - 11*(2*B*a^4*b - 9*A*a^3*b^2)*d^2 + 14*(B*a^5 - 11*A
*a^4*b)*d*e + (33*(5*B*a^2*b^3 + 9*A*a*b^4)*d^2 - 2*(227*B*a^3*b^2 + 165*A*a^2*b
^3)*d*e + (209*B*a^4*b + 113*A*a^3*b^2)*e^2)*x^2 + (11*(B*a^3*b^2 + 27*A*a^2*b^3
)*d^2 - 2*(64*B*a^4*b + 209*A*a^3*b^2)*d*e + 7*(11*B*a^5 + 23*A*a^4*b)*e^2)*x)*s
qrt(b*x + a)*sqrt(e*x + d)/(b^3*d^9 - 3*a*b^2*d^8*e + 3*a^2*b*d^7*e^2 - a^3*d^6*
e^3 + (b^3*d^3*e^6 - 3*a*b^2*d^2*e^7 + 3*a^2*b*d*e^8 - a^3*e^9)*x^6 + 6*(b^3*d^4
*e^5 - 3*a*b^2*d^3*e^6 + 3*a^2*b*d^2*e^7 - a^3*d*e^8)*x^5 + 15*(b^3*d^5*e^4 - 3*
a*b^2*d^4*e^5 + 3*a^2*b*d^3*e^6 - a^3*d^2*e^7)*x^4 + 20*(b^3*d^6*e^3 - 3*a*b^2*d
^5*e^4 + 3*a^2*b*d^4*e^5 - a^3*d^3*e^6)*x^3 + 15*(b^3*d^7*e^2 - 3*a*b^2*d^6*e^3
+ 3*a^2*b*d^5*e^4 - a^3*d^4*e^5)*x^2 + 6*(b^3*d^8*e - 3*a*b^2*d^7*e^2 + 3*a^2*b*
d^6*e^3 - a^3*d^5*e^4)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.441619, size = 887, normalized size = 6.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(13/2),x, algorithm="giac")

[Out]

-1/2838528*((b*x + a)*(2*(7*B*b^14*d^3*abs(b)*e^6 - 25*B*a*b^13*d^2*abs(b)*e^7 +
4*A*b^14*d^2*abs(b)*e^7 + 29*B*a^2*b^12*d*abs(b)*e^8 - 8*A*a*b^13*d*abs(b)*e^8
- 11*B*a^3*b^11*abs(b)*e^9 + 4*A*a^2*b^12*abs(b)*e^9)*(b*x + a)/(b^24*d^6*e^12 -
6*a*b^23*d^5*e^13 + 15*a^2*b^22*d^4*e^14 - 20*a^3*b^21*d^3*e^15 + 15*a^4*b^20*d
^2*e^16 - 6*a^5*b^19*d*e^17 + a^6*b^18*e^18) + 11*(7*B*b^15*d^4*abs(b)*e^5 - 32*
B*a*b^14*d^3*abs(b)*e^6 + 4*A*b^15*d^3*abs(b)*e^6 + 54*B*a^2*b^13*d^2*abs(b)*e^7
- 12*A*a*b^14*d^2*abs(b)*e^7 - 40*B*a^3*b^12*d*abs(b)*e^8 + 12*A*a^2*b^13*d*abs
(b)*e^8 + 11*B*a^4*b^11*abs(b)*e^9 - 4*A*a^3*b^12*abs(b)*e^9)/(b^24*d^6*e^12 - 6
*a*b^23*d^5*e^13 + 15*a^2*b^22*d^4*e^14 - 20*a^3*b^21*d^3*e^15 + 15*a^4*b^20*d^2
*e^16 - 6*a^5*b^19*d*e^17 + a^6*b^18*e^18)) - 99*(B*a*b^15*d^4*abs(b)*e^5 - A*b^
16*d^4*abs(b)*e^5 - 4*B*a^2*b^14*d^3*abs(b)*e^6 + 4*A*a*b^15*d^3*abs(b)*e^6 + 6*
B*a^3*b^13*d^2*abs(b)*e^7 - 6*A*a^2*b^14*d^2*abs(b)*e^7 - 4*B*a^4*b^12*d*abs(b)*
e^8 + 4*A*a^3*b^13*d*abs(b)*e^8 + B*a^5*b^11*abs(b)*e^9 - A*a^4*b^12*abs(b)*e^9)
/(b^24*d^6*e^12 - 6*a*b^23*d^5*e^13 + 15*a^2*b^22*d^4*e^14 - 20*a^3*b^21*d^3*e^1
5 + 15*a^4*b^20*d^2*e^16 - 6*a^5*b^19*d*e^17 + a^6*b^18*e^18))*(b*x + a)^(7/2)/(
b^2*d + (b*x + a)*b*e - a*b*e)^(11/2)