3.791 \(\int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=220 \[ \frac{2 b^2 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{15 (a+b x)}+\frac{6 a b x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{13 (a+b x)}+\frac{2 a^2 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{11 (a+b x)}+\frac{2 b^3 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)}+\frac{2 a^3 A x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]

[Out]

(2*a^3*A*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*a^2*(3*A*b +
a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (6*a*b*(A*b + a*B)
*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*b^2*(A*b + 3*a*B)*x
^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*(a + b*x)) + (2*b^3*B*x^(17/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x))

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Rubi [A]  time = 0.261258, antiderivative size = 220, 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 \[ \frac{2 b^2 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{15 (a+b x)}+\frac{6 a b x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{13 (a+b x)}+\frac{2 a^2 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{11 (a+b x)}+\frac{2 b^3 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)}+\frac{2 a^3 A x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*a^3*A*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*a^2*(3*A*b +
a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (6*a*b*(A*b + a*B)
*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*b^2*(A*b + 3*a*B)*x
^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*(a + b*x)) + (2*b^3*B*x^(17/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x))

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Rubi in Sympy [A]  time = 25.0546, size = 223, normalized size = 1.01 \[ \frac{B x^{\frac{9}{2}} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{17 b} + \frac{32 a^{3} x^{\frac{9}{2}} \left (17 A b - 9 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{109395 b \left (a + b x\right )} + \frac{16 a^{2} x^{\frac{9}{2}} \left (17 A b - 9 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12155 b} + \frac{4 a x^{\frac{9}{2}} \left (3 a + 3 b x\right ) \left (17 A b - 9 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3315 b} + \frac{2 x^{\frac{9}{2}} \left (17 A b - 9 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{255 b} \]

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)**(3/2),x)

[Out]

B*x**(9/2)*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(17*b) + 32*a**3*x*
*(9/2)*(17*A*b - 9*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(109395*b*(a + b*x)) +
16*a**2*x**(9/2)*(17*A*b - 9*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(12155*b) + 4
*a*x**(9/2)*(3*a + 3*b*x)*(17*A*b - 9*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(331
5*b) + 2*x**(9/2)*(17*A*b - 9*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(255*b)

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Mathematica [A]  time = 0.0797375, size = 89, normalized size = 0.4 \[ \frac{2 x^{9/2} \sqrt{(a+b x)^2} \left (1105 a^3 (11 A+9 B x)+2295 a^2 b x (13 A+11 B x)+1683 a b^2 x^2 (15 A+13 B x)+429 b^3 x^3 (17 A+15 B x)\right )}{109395 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*x^(9/2)*Sqrt[(a + b*x)^2]*(1105*a^3*(11*A + 9*B*x) + 2295*a^2*b*x*(13*A + 11*
B*x) + 1683*a*b^2*x^2*(15*A + 13*B*x) + 429*b^3*x^3*(17*A + 15*B*x)))/(109395*(a
+ b*x))

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Maple [A]  time = 0.01, size = 92, normalized size = 0.4 \[{\frac{12870\,B{x}^{4}{b}^{3}+14586\,A{b}^{3}{x}^{3}+43758\,B{x}^{3}a{b}^{2}+50490\,A{x}^{2}a{b}^{2}+50490\,B{x}^{2}{a}^{2}b+59670\,A{a}^{2}bx+19890\,{a}^{3}Bx+24310\,A{a}^{3}}{109395\, \left ( bx+a \right ) ^{3}}{x}^{{\frac{9}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

2/109395*x^(9/2)*(6435*B*b^3*x^4+7293*A*b^3*x^3+21879*B*a*b^2*x^3+25245*A*a*b^2*
x^2+25245*B*a^2*b*x^2+29835*A*a^2*b*x+9945*B*a^3*x+12155*A*a^3)*((b*x+a)^2)^(3/2
)/(b*x+a)^3

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Maxima [A]  time = 0.696098, size = 185, normalized size = 0.84 \[ \frac{2}{6435} \,{\left (33 \,{\left (13 \, b^{3} x^{2} + 15 \, a b^{2} x\right )} x^{\frac{11}{2}} + 90 \,{\left (11 \, a b^{2} x^{2} + 13 \, a^{2} b x\right )} x^{\frac{9}{2}} + 65 \,{\left (9 \, a^{2} b x^{2} + 11 \, a^{3} x\right )} x^{\frac{7}{2}}\right )} A + \frac{2}{36465} \,{\left (143 \,{\left (15 \, b^{3} x^{2} + 17 \, a b^{2} x\right )} x^{\frac{13}{2}} + 374 \,{\left (13 \, a b^{2} x^{2} + 15 \, a^{2} b x\right )} x^{\frac{11}{2}} + 255 \,{\left (11 \, a^{2} b x^{2} + 13 \, a^{3} x\right )} x^{\frac{9}{2}}\right )} B \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*x^(7/2),x, algorithm="maxima")

[Out]

2/6435*(33*(13*b^3*x^2 + 15*a*b^2*x)*x^(11/2) + 90*(11*a*b^2*x^2 + 13*a^2*b*x)*x
^(9/2) + 65*(9*a^2*b*x^2 + 11*a^3*x)*x^(7/2))*A + 2/36465*(143*(15*b^3*x^2 + 17*
a*b^2*x)*x^(13/2) + 374*(13*a*b^2*x^2 + 15*a^2*b*x)*x^(11/2) + 255*(11*a^2*b*x^2
+ 13*a^3*x)*x^(9/2))*B

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Fricas [A]  time = 0.306903, size = 105, normalized size = 0.48 \[ \frac{2}{109395} \,{\left (6435 \, B b^{3} x^{8} + 12155 \, A a^{3} x^{4} + 7293 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + 25245 \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + 9945 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*x^(7/2),x, algorithm="fricas")

[Out]

2/109395*(6435*B*b^3*x^8 + 12155*A*a^3*x^4 + 7293*(3*B*a*b^2 + A*b^3)*x^7 + 2524
5*(B*a^2*b + A*a*b^2)*x^6 + 9945*(B*a^3 + 3*A*a^2*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275502, size = 169, normalized size = 0.77 \[ \frac{2}{17} \, B b^{3} x^{\frac{17}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{5} \, B a b^{2} x^{\frac{15}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{15} \, A b^{3} x^{\frac{15}{2}}{\rm sign}\left (b x + a\right ) + \frac{6}{13} \, B a^{2} b x^{\frac{13}{2}}{\rm sign}\left (b x + a\right ) + \frac{6}{13} \, A a b^{2} x^{\frac{13}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{11} \, B a^{3} x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{6}{11} \, A a^{2} b x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*x^(7/2),x, algorithm="giac")

[Out]

2/17*B*b^3*x^(17/2)*sign(b*x + a) + 2/5*B*a*b^2*x^(15/2)*sign(b*x + a) + 2/15*A*
b^3*x^(15/2)*sign(b*x + a) + 6/13*B*a^2*b*x^(13/2)*sign(b*x + a) + 6/13*A*a*b^2*
x^(13/2)*sign(b*x + a) + 2/11*B*a^3*x^(11/2)*sign(b*x + a) + 6/11*A*a^2*b*x^(11/
2)*sign(b*x + a) + 2/9*A*a^3*x^(9/2)*sign(b*x + a)