3.501 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{256 b^4 (a+b x)^{7/2} (10 A b-17 a B)}{765765 a^6 x^{7/2}}-\frac{128 b^3 (a+b x)^{7/2} (10 A b-17 a B)}{109395 a^5 x^{9/2}}+\frac{32 b^2 (a+b x)^{7/2} (10 A b-17 a B)}{12155 a^4 x^{11/2}}-\frac{16 b (a+b x)^{7/2} (10 A b-17 a B)}{3315 a^3 x^{13/2}}+\frac{2 (a+b x)^{7/2} (10 A b-17 a B)}{255 a^2 x^{15/2}}-\frac{2 A (a+b x)^{7/2}}{17 a x^{17/2}} \]

[Out]

(-2*A*(a + b*x)^(7/2))/(17*a*x^(17/2)) + (2*(10*A*b - 17*a*B)*(a + b*x)^(7/2))/(
255*a^2*x^(15/2)) - (16*b*(10*A*b - 17*a*B)*(a + b*x)^(7/2))/(3315*a^3*x^(13/2))
+ (32*b^2*(10*A*b - 17*a*B)*(a + b*x)^(7/2))/(12155*a^4*x^(11/2)) - (128*b^3*(1
0*A*b - 17*a*B)*(a + b*x)^(7/2))/(109395*a^5*x^(9/2)) + (256*b^4*(10*A*b - 17*a*
B)*(a + b*x)^(7/2))/(765765*a^6*x^(7/2))

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Rubi [A]  time = 0.228787, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{256 b^4 (a+b x)^{7/2} (10 A b-17 a B)}{765765 a^6 x^{7/2}}-\frac{128 b^3 (a+b x)^{7/2} (10 A b-17 a B)}{109395 a^5 x^{9/2}}+\frac{32 b^2 (a+b x)^{7/2} (10 A b-17 a B)}{12155 a^4 x^{11/2}}-\frac{16 b (a+b x)^{7/2} (10 A b-17 a B)}{3315 a^3 x^{13/2}}+\frac{2 (a+b x)^{7/2} (10 A b-17 a B)}{255 a^2 x^{15/2}}-\frac{2 A (a+b x)^{7/2}}{17 a x^{17/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^(19/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(17*a*x^(17/2)) + (2*(10*A*b - 17*a*B)*(a + b*x)^(7/2))/(
255*a^2*x^(15/2)) - (16*b*(10*A*b - 17*a*B)*(a + b*x)^(7/2))/(3315*a^3*x^(13/2))
+ (32*b^2*(10*A*b - 17*a*B)*(a + b*x)^(7/2))/(12155*a^4*x^(11/2)) - (128*b^3*(1
0*A*b - 17*a*B)*(a + b*x)^(7/2))/(109395*a^5*x^(9/2)) + (256*b^4*(10*A*b - 17*a*
B)*(a + b*x)^(7/2))/(765765*a^6*x^(7/2))

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Rubi in Sympy [A]  time = 19.6989, size = 184, normalized size = 1.01 \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{17 a x^{\frac{17}{2}}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (10 A b - 17 B a\right )}{255 a^{2} x^{\frac{15}{2}}} - \frac{16 b \left (a + b x\right )^{\frac{7}{2}} \left (10 A b - 17 B a\right )}{3315 a^{3} x^{\frac{13}{2}}} + \frac{32 b^{2} \left (a + b x\right )^{\frac{7}{2}} \left (10 A b - 17 B a\right )}{12155 a^{4} x^{\frac{11}{2}}} - \frac{128 b^{3} \left (a + b x\right )^{\frac{7}{2}} \left (10 A b - 17 B a\right )}{109395 a^{5} x^{\frac{9}{2}}} + \frac{256 b^{4} \left (a + b x\right )^{\frac{7}{2}} \left (10 A b - 17 B a\right )}{765765 a^{6} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**(19/2),x)

[Out]

-2*A*(a + b*x)**(7/2)/(17*a*x**(17/2)) + 2*(a + b*x)**(7/2)*(10*A*b - 17*B*a)/(2
55*a**2*x**(15/2)) - 16*b*(a + b*x)**(7/2)*(10*A*b - 17*B*a)/(3315*a**3*x**(13/2
)) + 32*b**2*(a + b*x)**(7/2)*(10*A*b - 17*B*a)/(12155*a**4*x**(11/2)) - 128*b**
3*(a + b*x)**(7/2)*(10*A*b - 17*B*a)/(109395*a**5*x**(9/2)) + 256*b**4*(a + b*x)
**(7/2)*(10*A*b - 17*B*a)/(765765*a**6*x**(7/2))

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Mathematica [A]  time = 0.16638, size = 114, normalized size = 0.62 \[ -\frac{2 (a+b x)^{7/2} \left (3003 a^5 (15 A+17 B x)-462 a^4 b x (65 A+68 B x)+336 a^3 b^2 x^2 (55 A+51 B x)-224 a^2 b^3 x^3 (45 A+34 B x)+128 a b^4 x^4 (35 A+17 B x)-1280 A b^5 x^5\right )}{765765 a^6 x^{17/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(19/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(-1280*A*b^5*x^5 + 3003*a^5*(15*A + 17*B*x) + 128*a*b^4*x^4*
(35*A + 17*B*x) - 224*a^2*b^3*x^3*(45*A + 34*B*x) + 336*a^3*b^2*x^2*(55*A + 51*B
*x) - 462*a^4*b*x*(65*A + 68*B*x)))/(765765*a^6*x^(17/2))

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Maple [A]  time = 0.009, size = 125, normalized size = 0.7 \[ -{\frac{-2560\,A{b}^{5}{x}^{5}+4352\,B{x}^{5}a{b}^{4}+8960\,aA{b}^{4}{x}^{4}-15232\,B{x}^{4}{a}^{2}{b}^{3}-20160\,{a}^{2}A{b}^{3}{x}^{3}+34272\,B{x}^{3}{a}^{3}{b}^{2}+36960\,{a}^{3}A{b}^{2}{x}^{2}-62832\,B{x}^{2}{a}^{4}b-60060\,{a}^{4}Abx+102102\,{a}^{5}Bx+90090\,A{a}^{5}}{765765\,{a}^{6}} \left ( bx+a \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^(19/2),x)

[Out]

-2/765765*(b*x+a)^(7/2)*(-1280*A*b^5*x^5+2176*B*a*b^4*x^5+4480*A*a*b^4*x^4-7616*
B*a^2*b^3*x^4-10080*A*a^2*b^3*x^3+17136*B*a^3*b^2*x^3+18480*A*a^3*b^2*x^2-31416*
B*a^4*b*x^2-30030*A*a^4*b*x+51051*B*a^5*x+45045*A*a^5)/x^(17/2)/a^6

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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)/x^(19/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233411, size = 267, normalized size = 1.46 \[ -\frac{2 \,{\left (45045 \, A a^{8} + 128 \,{\left (17 \, B a b^{7} - 10 \, A b^{8}\right )} x^{8} - 64 \,{\left (17 \, B a^{2} b^{6} - 10 \, A a b^{7}\right )} x^{7} + 48 \,{\left (17 \, B a^{3} b^{5} - 10 \, A a^{2} b^{6}\right )} x^{6} - 40 \,{\left (17 \, B a^{4} b^{4} - 10 \, A a^{3} b^{5}\right )} x^{5} + 35 \,{\left (17 \, B a^{5} b^{3} - 10 \, A a^{4} b^{4}\right )} x^{4} + 63 \,{\left (1207 \, B a^{6} b^{2} + 5 \, A a^{5} b^{3}\right )} x^{3} + 231 \,{\left (527 \, B a^{7} b + 275 \, A a^{6} b^{2}\right )} x^{2} + 3003 \,{\left (17 \, B a^{8} + 35 \, A a^{7} b\right )} x\right )} \sqrt{b x + a}}{765765 \, a^{6} x^{\frac{17}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(19/2),x, algorithm="fricas")

[Out]

-2/765765*(45045*A*a^8 + 128*(17*B*a*b^7 - 10*A*b^8)*x^8 - 64*(17*B*a^2*b^6 - 10
*A*a*b^7)*x^7 + 48*(17*B*a^3*b^5 - 10*A*a^2*b^6)*x^6 - 40*(17*B*a^4*b^4 - 10*A*a
^3*b^5)*x^5 + 35*(17*B*a^5*b^3 - 10*A*a^4*b^4)*x^4 + 63*(1207*B*a^6*b^2 + 5*A*a^
5*b^3)*x^3 + 231*(527*B*a^7*b + 275*A*a^6*b^2)*x^2 + 3003*(17*B*a^8 + 35*A*a^7*b
)*x)*sqrt(b*x + a)/(a^6*x^(17/2))

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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)/x**(19/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.251594, size = 305, normalized size = 1.67 \[ \frac{{\left ({\left (8 \,{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (17 \, B a^{3} b^{16} - 10 \, A a^{2} b^{17}\right )}{\left (b x + a\right )}}{a^{9} b^{27}} - \frac{17 \,{\left (17 \, B a^{4} b^{16} - 10 \, A a^{3} b^{17}\right )}}{a^{9} b^{27}}\right )} + \frac{255 \,{\left (17 \, B a^{5} b^{16} - 10 \, A a^{4} b^{17}\right )}}{a^{9} b^{27}}\right )} - \frac{1105 \,{\left (17 \, B a^{6} b^{16} - 10 \, A a^{5} b^{17}\right )}}{a^{9} b^{27}}\right )}{\left (b x + a\right )} + \frac{12155 \,{\left (17 \, B a^{7} b^{16} - 10 \, A a^{6} b^{17}\right )}}{a^{9} b^{27}}\right )}{\left (b x + a\right )} - \frac{109395 \,{\left (B a^{8} b^{16} - A a^{7} b^{17}\right )}}{a^{9} b^{27}}\right )}{\left (b x + a\right )}^{\frac{7}{2}} b}{200740700160 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{17}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(19/2),x, algorithm="giac")

[Out]

1/200740700160*((8*(2*(b*x + a)*(4*(b*x + a)*(2*(17*B*a^3*b^16 - 10*A*a^2*b^17)*
(b*x + a)/(a^9*b^27) - 17*(17*B*a^4*b^16 - 10*A*a^3*b^17)/(a^9*b^27)) + 255*(17*
B*a^5*b^16 - 10*A*a^4*b^17)/(a^9*b^27)) - 1105*(17*B*a^6*b^16 - 10*A*a^5*b^17)/(
a^9*b^27))*(b*x + a) + 12155*(17*B*a^7*b^16 - 10*A*a^6*b^17)/(a^9*b^27))*(b*x +
a) - 109395*(B*a^8*b^16 - A*a^7*b^17)/(a^9*b^27))*(b*x + a)^(7/2)*b/(((b*x + a)*
b - a*b)^(17/2)*abs(b))