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

Optimal. Leaf size=316 \[ \frac{20 a^2 b^2 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{2 b^4 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac{10 a b^3 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac{2 b^5 B x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac{2 a^5 A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 a^4 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 (a+b x)}+\frac{2 a^3 b x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]

[Out]

(2*a^5*A*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*a^4*(5*A*b + a*B)
*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*a^3*b*(2*A*b + a*B)*x
^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (20*a^2*b^2*(A*b + a*B)*x^(7/2
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (10*a*b^3*(A*b + 2*a*B)*x^(9/2)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(11/2)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (2*b^5*B*x^(13/2)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(13*(a + b*x))

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Rubi [A]  time = 0.335562, antiderivative size = 316, 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 \[ \frac{20 a^2 b^2 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{2 b^4 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac{10 a b^3 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac{2 b^5 B x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac{2 a^5 A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 a^4 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 (a+b x)}+\frac{2 a^3 b x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^5*A*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*a^4*(5*A*b + a*B)
*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*a^3*b*(2*A*b + a*B)*x
^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (20*a^2*b^2*(A*b + a*B)*x^(7/2
)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (10*a*b^3*(A*b + 2*a*B)*x^(9/2)
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(11/2)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (2*b^5*B*x^(13/2)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(13*(a + b*x))

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Rubi in Sympy [A]  time = 35.5564, size = 309, normalized size = 0.98 \[ \frac{B \sqrt{x} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{13 b} + \frac{512 a^{5} \sqrt{x} \left (13 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 b \left (a + b x\right )} + \frac{256 a^{4} \sqrt{x} \left (13 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 b} + \frac{64 a^{3} \sqrt{x} \left (3 a + 3 b x\right ) \left (13 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 b} + \frac{160 a^{2} \sqrt{x} \left (13 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9009 b} + \frac{4 a \sqrt{x} \left (5 a + 5 b x\right ) \left (13 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{1287 b} + \frac{2 \sqrt{x} \left (13 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{143 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/x**(1/2),x)

[Out]

B*sqrt(x)*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(13*b) + 512*a**5*sq
rt(x)*(13*A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*b*(a + b*x)) + 256*a
**4*sqrt(x)*(13*A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*b) + 64*a**3*s
qrt(x)*(3*a + 3*b*x)*(13*A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*b) +
160*a**2*sqrt(x)*(13*A*b - B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(9009*b) + 4
*a*sqrt(x)*(5*a + 5*b*x)*(13*A*b - B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(128
7*b) + 2*sqrt(x)*(13*A*b - B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(143*b)

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Mathematica [A]  time = 0.126822, size = 126, normalized size = 0.4 \[ \frac{2 \sqrt{x} \sqrt{(a+b x)^2} \left (3003 a^5 (3 A+B x)+3003 a^4 b x (5 A+3 B x)+2574 a^3 b^2 x^2 (7 A+5 B x)+1430 a^2 b^3 x^3 (9 A+7 B x)+455 a b^4 x^4 (11 A+9 B x)+63 b^5 x^5 (13 A+11 B x)\right )}{9009 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*Sqrt[(a + b*x)^2]*(3003*a^5*(3*A + B*x) + 3003*a^4*b*x*(5*A + 3*B*x)
+ 2574*a^3*b^2*x^2*(7*A + 5*B*x) + 1430*a^2*b^3*x^3*(9*A + 7*B*x) + 455*a*b^4*x^
4*(11*A + 9*B*x) + 63*b^5*x^5*(13*A + 11*B*x)))/(9009*(a + b*x))

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Maple [A]  time = 0.011, size = 140, normalized size = 0.4 \[{\frac{1386\,B{b}^{5}{x}^{6}+1638\,A{x}^{5}{b}^{5}+8190\,B{x}^{5}a{b}^{4}+10010\,A{x}^{4}a{b}^{4}+20020\,B{x}^{4}{a}^{2}{b}^{3}+25740\,A{x}^{3}{a}^{2}{b}^{3}+25740\,B{x}^{3}{a}^{3}{b}^{2}+36036\,A{x}^{2}{a}^{3}{b}^{2}+18018\,B{x}^{2}{a}^{4}b+30030\,Ax{a}^{4}b+6006\,Bx{a}^{5}+18018\,A{a}^{5}}{9009\, \left ( bx+a \right ) ^{5}}\sqrt{x} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^(1/2),x)

[Out]

2/9009*x^(1/2)*(693*B*b^5*x^6+819*A*b^5*x^5+4095*B*a*b^4*x^5+5005*A*a*b^4*x^4+10
010*B*a^2*b^3*x^4+12870*A*a^2*b^3*x^3+12870*B*a^3*b^2*x^3+18018*A*a^3*b^2*x^2+90
09*B*a^4*b*x^2+15015*A*a^4*b*x+3003*B*a^5*x+9009*A*a^5)*((b*x+a)^2)^(5/2)/(b*x+a
)^5

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Maxima [A]  time = 0.709382, size = 324, normalized size = 1.03 \[ \frac{2}{3465} \,{\left (35 \,{\left (9 \, b^{5} x^{2} + 11 \, a b^{4} x\right )} x^{\frac{7}{2}} + 220 \,{\left (7 \, a b^{4} x^{2} + 9 \, a^{2} b^{3} x\right )} x^{\frac{5}{2}} + 594 \,{\left (5 \, a^{2} b^{3} x^{2} + 7 \, a^{3} b^{2} x\right )} x^{\frac{3}{2}} + 924 \,{\left (3 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x\right )} \sqrt{x} + \frac{1155 \,{\left (a^{4} b x^{2} + 3 \, a^{5} x\right )}}{\sqrt{x}}\right )} A + \frac{2}{45045} \,{\left (315 \,{\left (11 \, b^{5} x^{2} + 13 \, a b^{4} x\right )} x^{\frac{9}{2}} + 1820 \,{\left (9 \, a b^{4} x^{2} + 11 \, a^{2} b^{3} x\right )} x^{\frac{7}{2}} + 4290 \,{\left (7 \, a^{2} b^{3} x^{2} + 9 \, a^{3} b^{2} x\right )} x^{\frac{5}{2}} + 5148 \,{\left (5 \, a^{3} b^{2} x^{2} + 7 \, a^{4} b x\right )} x^{\frac{3}{2}} + 3003 \,{\left (3 \, a^{4} b x^{2} + 5 \, a^{5} x\right )} \sqrt{x}\right )} B \]

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)/sqrt(x),x, algorithm="maxima")

[Out]

2/3465*(35*(9*b^5*x^2 + 11*a*b^4*x)*x^(7/2) + 220*(7*a*b^4*x^2 + 9*a^2*b^3*x)*x^
(5/2) + 594*(5*a^2*b^3*x^2 + 7*a^3*b^2*x)*x^(3/2) + 924*(3*a^3*b^2*x^2 + 5*a^4*b
*x)*sqrt(x) + 1155*(a^4*b*x^2 + 3*a^5*x)/sqrt(x))*A + 2/45045*(315*(11*b^5*x^2 +
13*a*b^4*x)*x^(9/2) + 1820*(9*a*b^4*x^2 + 11*a^2*b^3*x)*x^(7/2) + 4290*(7*a^2*b
^3*x^2 + 9*a^3*b^2*x)*x^(5/2) + 5148*(5*a^3*b^2*x^2 + 7*a^4*b*x)*x^(3/2) + 3003*
(3*a^4*b*x^2 + 5*a^5*x)*sqrt(x))*B

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Fricas [A]  time = 0.277672, size = 161, normalized size = 0.51 \[ \frac{2}{9009} \,{\left (693 \, B b^{5} x^{6} + 9009 \, A a^{5} + 819 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 5005 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 12870 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 9009 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 3003 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )} \sqrt{x} \]

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)/sqrt(x),x, algorithm="fricas")

[Out]

2/9009*(693*B*b^5*x^6 + 9009*A*a^5 + 819*(5*B*a*b^4 + A*b^5)*x^5 + 5005*(2*B*a^2
*b^3 + A*a*b^4)*x^4 + 12870*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 9009*(B*a^4*b + 2*A*a^
3*b^2)*x^2 + 3003*(B*a^5 + 5*A*a^4*b)*x)*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((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275035, size = 266, normalized size = 0.84 \[ \frac{2}{13} \, B b^{5} x^{\frac{13}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{11} \, B a b^{4} x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{11} \, A b^{5} x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{9} \, B a^{2} b^{3} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{9} \, A a b^{4} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{7} \, B a^{3} b^{2} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{7} \, A a^{2} b^{3} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + 2 \, B a^{4} b x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + 4 \, A a^{3} b^{2} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, B a^{5} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, A a^{4} b x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 2 \, A a^{5} \sqrt{x}{\rm sign}\left (b x + a\right ) \]

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)/sqrt(x),x, algorithm="giac")

[Out]

2/13*B*b^5*x^(13/2)*sign(b*x + a) + 10/11*B*a*b^4*x^(11/2)*sign(b*x + a) + 2/11*
A*b^5*x^(11/2)*sign(b*x + a) + 20/9*B*a^2*b^3*x^(9/2)*sign(b*x + a) + 10/9*A*a*b
^4*x^(9/2)*sign(b*x + a) + 20/7*B*a^3*b^2*x^(7/2)*sign(b*x + a) + 20/7*A*a^2*b^3
*x^(7/2)*sign(b*x + a) + 2*B*a^4*b*x^(5/2)*sign(b*x + a) + 4*A*a^3*b^2*x^(5/2)*s
ign(b*x + a) + 2/3*B*a^5*x^(3/2)*sign(b*x + a) + 10/3*A*a^4*b*x^(3/2)*sign(b*x +
a) + 2*A*a^5*sqrt(x)*sign(b*x + a)