3.800 \(\int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=320 \[ \frac{4 a^2 b^2 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 b^4 x^{19/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{19 (a+b x)}+\frac{10 a b^3 x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{17 (a+b x)}+\frac{2 b^5 B x^{21/2} \sqrt{a^2+2 a b x+b^2 x^2}}{21 (a+b x)}+\frac{2 a^5 A x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{2 a^4 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{11 (a+b x)}+\frac{10 a^3 b x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{13 (a+b x)} \]

[Out]

(2*a^5*A*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*a^4*(5*A*b +
a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (10*a^3*b*(2*A*b +
a*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (4*a^2*b^2*(A*b +
a*B)*x^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (10*a*b^3*(A*b + 2
*a*B)*x^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x)) + (2*b^4*(A*b + 5*a
*B)*x^(19/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(19*(a + b*x)) + (2*b^5*B*x^(21/2)*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(21*(a + b*x))

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Rubi [A]  time = 0.360384, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{4 a^2 b^2 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 b^4 x^{19/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{19 (a+b x)}+\frac{10 a b^3 x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{17 (a+b x)}+\frac{2 b^5 B x^{21/2} \sqrt{a^2+2 a b x+b^2 x^2}}{21 (a+b x)}+\frac{2 a^5 A x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{2 a^4 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{11 (a+b x)}+\frac{10 a^3 b x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{13 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*a^5*A*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*a^4*(5*A*b +
a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (10*a^3*b*(2*A*b +
a*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (4*a^2*b^2*(A*b +
a*B)*x^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (10*a*b^3*(A*b + 2
*a*B)*x^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x)) + (2*b^4*(A*b + 5*a
*B)*x^(19/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(19*(a + b*x)) + (2*b^5*B*x^(21/2)*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(21*(a + b*x))

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Rubi in Sympy [A]  time = 35.4571, size = 320, normalized size = 1. \[ \frac{B x^{\frac{9}{2}} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{21 b} + \frac{512 a^{5} x^{\frac{9}{2}} \left (7 A b - 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2909907 b \left (a + b x\right )} + \frac{256 a^{4} x^{\frac{9}{2}} \left (7 A b - 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{323323 b} + \frac{64 a^{3} x^{\frac{9}{2}} \left (3 a + 3 b x\right ) \left (7 A b - 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{88179 b} + \frac{32 a^{2} x^{\frac{9}{2}} \left (7 A b - 3 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{6783 b} + \frac{4 a x^{\frac{9}{2}} \left (5 a + 5 b x\right ) \left (7 A b - 3 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{2261 b} + \frac{2 x^{\frac{9}{2}} \left (7 A b - 3 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{133 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

B*x**(9/2)*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(21*b) + 512*a**5*x
**(9/2)*(7*A*b - 3*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2909907*b*(a + b*x)) +
256*a**4*x**(9/2)*(7*A*b - 3*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(323323*b) +
64*a**3*x**(9/2)*(3*a + 3*b*x)*(7*A*b - 3*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)
/(88179*b) + 32*a**2*x**(9/2)*(7*A*b - 3*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2
)/(6783*b) + 4*a*x**(9/2)*(5*a + 5*b*x)*(7*A*b - 3*B*a)*(a**2 + 2*a*b*x + b**2*x
**2)**(3/2)/(2261*b) + 2*x**(9/2)*(7*A*b - 3*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**
(5/2)/(133*b)

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Mathematica [A]  time = 0.107041, size = 127, normalized size = 0.4 \[ \frac{2 x^{9/2} \sqrt{(a+b x)^2} \left (29393 a^5 (11 A+9 B x)+101745 a^4 b x (13 A+11 B x)+149226 a^3 b^2 x^2 (15 A+13 B x)+114114 a^2 b^3 x^3 (17 A+15 B x)+45045 a b^4 x^4 (19 A+17 B x)+7293 b^5 x^5 (21 A+19 B x)\right )}{2909907 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*x^(9/2)*Sqrt[(a + b*x)^2]*(29393*a^5*(11*A + 9*B*x) + 101745*a^4*b*x*(13*A +
11*B*x) + 149226*a^3*b^2*x^2*(15*A + 13*B*x) + 114114*a^2*b^3*x^3*(17*A + 15*B*x
) + 45045*a*b^4*x^4*(19*A + 17*B*x) + 7293*b^5*x^5*(21*A + 19*B*x)))/(2909907*(a
+ b*x))

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Maple [A]  time = 0.011, size = 140, normalized size = 0.4 \[{\frac{277134\,B{b}^{5}{x}^{6}+306306\,A{x}^{5}{b}^{5}+1531530\,B{x}^{5}a{b}^{4}+1711710\,A{x}^{4}a{b}^{4}+3423420\,B{x}^{4}{a}^{2}{b}^{3}+3879876\,A{x}^{3}{a}^{2}{b}^{3}+3879876\,B{x}^{3}{a}^{3}{b}^{2}+4476780\,A{x}^{2}{a}^{3}{b}^{2}+2238390\,B{x}^{2}{a}^{4}b+2645370\,Ax{a}^{4}b+529074\,Bx{a}^{5}+646646\,A{a}^{5}}{2909907\, \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

2/2909907*x^(9/2)*(138567*B*b^5*x^6+153153*A*b^5*x^5+765765*B*a*b^4*x^5+855855*A
*a*b^4*x^4+1711710*B*a^2*b^3*x^4+1939938*A*a^2*b^3*x^3+1939938*B*a^3*b^2*x^3+223
8390*A*a^3*b^2*x^2+1119195*B*a^4*b*x^2+1322685*A*a^4*b*x+264537*B*a^5*x+323323*A
*a^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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Maxima [A]  time = 0.70358, size = 325, normalized size = 1.02 \[ \frac{2}{2078505} \,{\left (6435 \,{\left (17 \, b^{5} x^{2} + 19 \, a b^{4} x\right )} x^{\frac{15}{2}} + 32604 \,{\left (15 \, a b^{4} x^{2} + 17 \, a^{2} b^{3} x\right )} x^{\frac{13}{2}} + 63954 \,{\left (13 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x\right )} x^{\frac{11}{2}} + 58140 \,{\left (11 \, a^{3} b^{2} x^{2} + 13 \, a^{4} b x\right )} x^{\frac{9}{2}} + 20995 \,{\left (9 \, a^{4} b x^{2} + 11 \, a^{5} x\right )} x^{\frac{7}{2}}\right )} A + \frac{2}{4849845} \,{\left (12155 \,{\left (19 \, b^{5} x^{2} + 21 \, a b^{4} x\right )} x^{\frac{17}{2}} + 60060 \,{\left (17 \, a b^{4} x^{2} + 19 \, a^{2} b^{3} x\right )} x^{\frac{15}{2}} + 114114 \,{\left (15 \, a^{2} b^{3} x^{2} + 17 \, a^{3} b^{2} x\right )} x^{\frac{13}{2}} + 99484 \,{\left (13 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x\right )} x^{\frac{11}{2}} + 33915 \,{\left (11 \, a^{4} b x^{2} + 13 \, a^{5} x\right )} x^{\frac{9}{2}}\right )} B \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^(7/2),x, algorithm="maxima")

[Out]

2/2078505*(6435*(17*b^5*x^2 + 19*a*b^4*x)*x^(15/2) + 32604*(15*a*b^4*x^2 + 17*a^
2*b^3*x)*x^(13/2) + 63954*(13*a^2*b^3*x^2 + 15*a^3*b^2*x)*x^(11/2) + 58140*(11*a
^3*b^2*x^2 + 13*a^4*b*x)*x^(9/2) + 20995*(9*a^4*b*x^2 + 11*a^5*x)*x^(7/2))*A + 2
/4849845*(12155*(19*b^5*x^2 + 21*a*b^4*x)*x^(17/2) + 60060*(17*a*b^4*x^2 + 19*a^
2*b^3*x)*x^(15/2) + 114114*(15*a^2*b^3*x^2 + 17*a^3*b^2*x)*x^(13/2) + 99484*(13*
a^3*b^2*x^2 + 15*a^4*b*x)*x^(11/2) + 33915*(11*a^4*b*x^2 + 13*a^5*x)*x^(9/2))*B

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Fricas [A]  time = 0.269524, size = 167, normalized size = 0.52 \[ \frac{2}{2909907} \,{\left (138567 \, B b^{5} x^{10} + 323323 \, A a^{5} x^{4} + 153153 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + 855855 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1939938 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + 1119195 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 264537 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^(7/2),x, algorithm="fricas")

[Out]

2/2909907*(138567*B*b^5*x^10 + 323323*A*a^5*x^4 + 153153*(5*B*a*b^4 + A*b^5)*x^9
+ 855855*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 1939938*(B*a^3*b^2 + A*a^2*b^3)*x^7 + 11
19195*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 264537*(B*a^5 + 5*A*a^4*b)*x^5)*sqrt(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(x**(7/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274231, size = 266, normalized size = 0.83 \[ \frac{2}{21} \, B b^{5} x^{\frac{21}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{19} \, B a b^{4} x^{\frac{19}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{19} \, A b^{5} x^{\frac{19}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{17} \, B a^{2} b^{3} x^{\frac{17}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{17} \, A a b^{4} x^{\frac{17}{2}}{\rm sign}\left (b x + a\right ) + \frac{4}{3} \, B a^{3} b^{2} x^{\frac{15}{2}}{\rm sign}\left (b x + a\right ) + \frac{4}{3} \, A a^{2} b^{3} x^{\frac{15}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{13} \, B a^{4} b x^{\frac{13}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{13} \, A a^{3} b^{2} x^{\frac{13}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{11} \, B a^{5} x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{11} \, A a^{4} b x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{9} \, A a^{5} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^(7/2),x, algorithm="giac")

[Out]

2/21*B*b^5*x^(21/2)*sign(b*x + a) + 10/19*B*a*b^4*x^(19/2)*sign(b*x + a) + 2/19*
A*b^5*x^(19/2)*sign(b*x + a) + 20/17*B*a^2*b^3*x^(17/2)*sign(b*x + a) + 10/17*A*
a*b^4*x^(17/2)*sign(b*x + a) + 4/3*B*a^3*b^2*x^(15/2)*sign(b*x + a) + 4/3*A*a^2*
b^3*x^(15/2)*sign(b*x + a) + 10/13*B*a^4*b*x^(13/2)*sign(b*x + a) + 20/13*A*a^3*
b^2*x^(13/2)*sign(b*x + a) + 2/11*B*a^5*x^(11/2)*sign(b*x + a) + 10/11*A*a^4*b*x
^(11/2)*sign(b*x + a) + 2/9*A*a^5*x^(9/2)*sign(b*x + a)