3.2216 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=201 \[ \frac{16 b^2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^4}+\frac{8 b (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(7*b*B
*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2))
+ (8*b*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(1287*e*(b*d - a*e)^3*(d
+ e*x)^(9/2)) + (16*b^2*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(9009*e
*(b*d - a*e)^4*(d + e*x)^(7/2))

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Rubi [A]  time = 0.364228, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{16 b^2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^4}+\frac{8 b (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(7*b*B
*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2))
+ (8*b*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(1287*e*(b*d - a*e)^3*(d
+ e*x)^(9/2)) + (16*b^2*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(a + b*x)^(7/2))/(9009*e
*(b*d - a*e)^4*(d + e*x)^(7/2))

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Rubi in Sympy [A]  time = 36.1463, size = 192, normalized size = 0.96 \[ \frac{16 b^{2} \left (a + b x\right )^{\frac{7}{2}} \left (6 A b e - 13 B a e + 7 B b d\right )}{9009 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4}} - \frac{8 b \left (a + b x\right )^{\frac{7}{2}} \left (6 A b e - 13 B a e + 7 B b d\right )}{1287 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (6 A b e - 13 B a e + 7 B b d\right )}{143 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{13 e \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(15/2),x)

[Out]

16*b**2*(a + b*x)**(7/2)*(6*A*b*e - 13*B*a*e + 7*B*b*d)/(9009*e*(d + e*x)**(7/2)
*(a*e - b*d)**4) - 8*b*(a + b*x)**(7/2)*(6*A*b*e - 13*B*a*e + 7*B*b*d)/(1287*e*(
d + e*x)**(9/2)*(a*e - b*d)**3) + 2*(a + b*x)**(7/2)*(6*A*b*e - 13*B*a*e + 7*B*b
*d)/(143*e*(d + e*x)**(11/2)*(a*e - b*d)**2) - 2*(a + b*x)**(7/2)*(A*e - B*d)/(1
3*e*(d + e*x)**(13/2)*(a*e - b*d))

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Mathematica [A]  time = 0.62586, size = 255, normalized size = 1.27 \[ \frac{2 \sqrt{a+b x} \left (\frac{8 b^5 (d+e x)^6 (-13 a B e+6 A b e+7 b B d)}{(b d-a e)^4}+\frac{4 b^4 (d+e x)^5 (-13 a B e+6 A b e+7 b B d)}{(b d-a e)^3}+\frac{3 b^3 (d+e x)^4 (-13 a B e+6 A b e+7 b B d)}{(b d-a e)^2}-\frac{b^2 (d+e x)^3 (1469 a B e+15 A b e-1484 b B d)}{a e-b d}+7 b (d+e x)^2 (-299 a B e-159 A b e+458 b B d)-63 (d+e x) (a e-b d) (13 a B e+27 A b e-40 b B d)+693 (b d-a e)^2 (B d-A e)\right )}{9009 e^4 (d+e x)^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(2*Sqrt[a + b*x]*(693*(b*d - a*e)^2*(B*d - A*e) - 63*(-(b*d) + a*e)*(-40*b*B*d +
27*A*b*e + 13*a*B*e)*(d + e*x) + 7*b*(458*b*B*d - 159*A*b*e - 299*a*B*e)*(d + e
*x)^2 - (b^2*(-1484*b*B*d + 15*A*b*e + 1469*a*B*e)*(d + e*x)^3)/(-(b*d) + a*e) +
(3*b^3*(7*b*B*d + 6*A*b*e - 13*a*B*e)*(d + e*x)^4)/(b*d - a*e)^2 + (4*b^4*(7*b*
B*d + 6*A*b*e - 13*a*B*e)*(d + e*x)^5)/(b*d - a*e)^3 + (8*b^5*(7*b*B*d + 6*A*b*e
- 13*a*B*e)*(d + e*x)^6)/(b*d - a*e)^4))/(9009*e^4*(d + e*x)^(13/2))

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Maple [A]  time = 0.013, size = 322, normalized size = 1.6 \[ -{\frac{-96\,A{b}^{3}{e}^{3}{x}^{3}+208\,Ba{b}^{2}{e}^{3}{x}^{3}-112\,B{b}^{3}d{e}^{2}{x}^{3}+336\,Aa{b}^{2}{e}^{3}{x}^{2}-624\,A{b}^{3}d{e}^{2}{x}^{2}-728\,B{a}^{2}b{e}^{3}{x}^{2}+1744\,Ba{b}^{2}d{e}^{2}{x}^{2}-728\,B{b}^{3}{d}^{2}e{x}^{2}-756\,A{a}^{2}b{e}^{3}x+2184\,Aa{b}^{2}d{e}^{2}x-1716\,A{b}^{3}{d}^{2}ex+1638\,B{a}^{3}{e}^{3}x-5614\,B{a}^{2}bd{e}^{2}x+6266\,Ba{b}^{2}{d}^{2}ex-2002\,B{b}^{3}{d}^{3}x+1386\,A{a}^{3}{e}^{3}-4914\,A{a}^{2}bd{e}^{2}+6006\,Aa{b}^{2}{d}^{2}e-2574\,A{b}^{3}{d}^{3}+252\,B{a}^{3}d{e}^{2}-728\,B{a}^{2}b{d}^{2}e+572\,Ba{b}^{2}{d}^{3}}{9009\,{e}^{4}{a}^{4}-36036\,b{e}^{3}d{a}^{3}+54054\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-36036\,a{b}^{3}{d}^{3}e+9009\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(15/2),x)

[Out]

-2/9009*(b*x+a)^(7/2)*(-48*A*b^3*e^3*x^3+104*B*a*b^2*e^3*x^3-56*B*b^3*d*e^2*x^3+
168*A*a*b^2*e^3*x^2-312*A*b^3*d*e^2*x^2-364*B*a^2*b*e^3*x^2+872*B*a*b^2*d*e^2*x^
2-364*B*b^3*d^2*e*x^2-378*A*a^2*b*e^3*x+1092*A*a*b^2*d*e^2*x-858*A*b^3*d^2*e*x+8
19*B*a^3*e^3*x-2807*B*a^2*b*d*e^2*x+3133*B*a*b^2*d^2*e*x-1001*B*b^3*d^3*x+693*A*
a^3*e^3-2457*A*a^2*b*d*e^2+3003*A*a*b^2*d^2*e-1287*A*b^3*d^3+126*B*a^3*d*e^2-364
*B*a^2*b*d^2*e+286*B*a*b^2*d^3)/(e*x+d)^(13/2)/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*
d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)

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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)^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 14.6447, size = 1413, normalized size = 7.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(15/2),x, algorithm="fricas")

[Out]

-2/9009*(693*A*a^6*e^3 - 8*(7*B*b^6*d*e^2 - (13*B*a*b^5 - 6*A*b^6)*e^3)*x^6 - 4*
(91*B*b^6*d^2*e - 2*(88*B*a*b^5 - 39*A*b^6)*d*e^2 + (13*B*a^2*b^4 - 6*A*a*b^5)*e
^3)*x^5 - (1001*B*b^6*d^3 - 13*(157*B*a*b^5 - 66*A*b^6)*d^2*e + (359*B*a^2*b^4 -
156*A*a*b^5)*d*e^2 - 3*(13*B*a^3*b^3 - 6*A*a^2*b^4)*e^3)*x^4 + 143*(2*B*a^4*b^2
- 9*A*a^3*b^3)*d^3 - 91*(4*B*a^5*b - 33*A*a^4*b^2)*d^2*e + 63*(2*B*a^6 - 39*A*a
^5*b)*d*e^2 - (143*(19*B*a*b^5 + 9*A*b^6)*d^3 - 13*(611*B*a^2*b^4 + 33*A*a*b^5)*
d^2*e + (5735*B*a^3*b^3 + 117*A*a^2*b^4)*d*e^2 - (1469*B*a^4*b^2 + 15*A*a^3*b^3)
*e^3)*x^3 - (429*(5*B*a^2*b^4 + 9*A*a*b^5)*d^3 - 13*(611*B*a^3*b^3 + 495*A*a^2*b
^4)*d^2*e + (7171*B*a^4*b^2 + 4407*A*a^3*b^3)*d*e^2 - 7*(299*B*a^5*b + 159*A*a^4
*b^2)*e^3)*x^2 - (143*(B*a^3*b^3 + 27*A*a^2*b^4)*d^3 - 13*(157*B*a^4*b^2 + 627*A
*a^3*b^3)*d^2*e + 7*(347*B*a^5*b + 897*A*a^4*b^2)*d*e^2 - 63*(13*B*a^6 + 27*A*a^
5*b)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^4*d^11 - 4*a*b^3*d^10*e + 6*a^2*b^2*
d^9*e^2 - 4*a^3*b*d^8*e^3 + a^4*d^7*e^4 + (b^4*d^4*e^7 - 4*a*b^3*d^3*e^8 + 6*a^2
*b^2*d^2*e^9 - 4*a^3*b*d*e^10 + a^4*e^11)*x^7 + 7*(b^4*d^5*e^6 - 4*a*b^3*d^4*e^7
+ 6*a^2*b^2*d^3*e^8 - 4*a^3*b*d^2*e^9 + a^4*d*e^10)*x^6 + 21*(b^4*d^6*e^5 - 4*a
*b^3*d^5*e^6 + 6*a^2*b^2*d^4*e^7 - 4*a^3*b*d^3*e^8 + a^4*d^2*e^9)*x^5 + 35*(b^4*
d^7*e^4 - 4*a*b^3*d^6*e^5 + 6*a^2*b^2*d^5*e^6 - 4*a^3*b*d^4*e^7 + a^4*d^3*e^8)*x
^4 + 35*(b^4*d^8*e^3 - 4*a*b^3*d^7*e^4 + 6*a^2*b^2*d^6*e^5 - 4*a^3*b*d^5*e^6 + a
^4*d^4*e^7)*x^3 + 21*(b^4*d^9*e^2 - 4*a*b^3*d^8*e^3 + 6*a^2*b^2*d^7*e^4 - 4*a^3*
b*d^6*e^5 + a^4*d^5*e^6)*x^2 + 7*(b^4*d^10*e - 4*a*b^3*d^9*e^2 + 6*a^2*b^2*d^8*e
^3 - 4*a^3*b*d^7*e^4 + a^4*d^6*e^5)*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)**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.572231, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(15/2),x, algorithm="giac")

[Out]

Done