3.766 \(\int (a+b x)^4 \sqrt{a^2-b^2 x^2} \, dx\)

Optimal. Leaf size=173 \[ -\frac{21 a^2 (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}-\frac{3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac{(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac{21 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b}+\frac{21}{16} a^4 x \sqrt{a^2-b^2 x^2}-\frac{7 a^3 \left (a^2-b^2 x^2\right )^{3/2}}{8 b} \]

[Out]

(21*a^4*x*Sqrt[a^2 - b^2*x^2])/16 - (7*a^3*(a^2 - b^2*x^2)^(3/2))/(8*b) - (21*a^
2*(a + b*x)*(a^2 - b^2*x^2)^(3/2))/(40*b) - (3*a*(a + b*x)^2*(a^2 - b^2*x^2)^(3/
2))/(10*b) - ((a + b*x)^3*(a^2 - b^2*x^2)^(3/2))/(6*b) + (21*a^6*ArcTan[(b*x)/Sq
rt[a^2 - b^2*x^2]])/(16*b)

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Rubi [A]  time = 0.215165, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{21 a^2 (a+b x) \left (a^2-b^2 x^2\right )^{3/2}}{40 b}-\frac{3 a (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2}}{10 b}-\frac{(a+b x)^3 \left (a^2-b^2 x^2\right )^{3/2}}{6 b}+\frac{21 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b}+\frac{21}{16} a^4 x \sqrt{a^2-b^2 x^2}-\frac{7 a^3 \left (a^2-b^2 x^2\right )^{3/2}}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4*Sqrt[a^2 - b^2*x^2],x]

[Out]

(21*a^4*x*Sqrt[a^2 - b^2*x^2])/16 - (7*a^3*(a^2 - b^2*x^2)^(3/2))/(8*b) - (21*a^
2*(a + b*x)*(a^2 - b^2*x^2)^(3/2))/(40*b) - (3*a*(a + b*x)^2*(a^2 - b^2*x^2)^(3/
2))/(10*b) - ((a + b*x)^3*(a^2 - b^2*x^2)^(3/2))/(6*b) + (21*a^6*ArcTan[(b*x)/Sq
rt[a^2 - b^2*x^2]])/(16*b)

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Rubi in Sympy [A]  time = 30.9645, size = 148, normalized size = 0.86 \[ \frac{21 a^{6} \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{16 b} + \frac{21 a^{4} x \sqrt{a^{2} - b^{2} x^{2}}}{16} - \frac{7 a^{3} \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b} - \frac{21 a^{2} \left (a + b x\right ) \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{40 b} - \frac{3 a \left (a + b x\right )^{2} \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{10 b} - \frac{\left (a + b x\right )^{3} \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**4*(-b**2*x**2+a**2)**(1/2),x)

[Out]

21*a**6*atan(b*x/sqrt(a**2 - b**2*x**2))/(16*b) + 21*a**4*x*sqrt(a**2 - b**2*x**
2)/16 - 7*a**3*(a**2 - b**2*x**2)**(3/2)/(8*b) - 21*a**2*(a + b*x)*(a**2 - b**2*
x**2)**(3/2)/(40*b) - 3*a*(a + b*x)**2*(a**2 - b**2*x**2)**(3/2)/(10*b) - (a + b
*x)**3*(a**2 - b**2*x**2)**(3/2)/(6*b)

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Mathematica [A]  time = 0.107457, size = 102, normalized size = 0.59 \[ \frac{315 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )+\sqrt{a^2-b^2 x^2} \left (-448 a^5-75 a^4 b x+256 a^3 b^2 x^2+350 a^2 b^3 x^3+192 a b^4 x^4+40 b^5 x^5\right )}{240 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^4*Sqrt[a^2 - b^2*x^2],x]

[Out]

(Sqrt[a^2 - b^2*x^2]*(-448*a^5 - 75*a^4*b*x + 256*a^3*b^2*x^2 + 350*a^2*b^3*x^3
+ 192*a*b^4*x^4 + 40*b^5*x^5) + 315*a^6*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(240*
b)

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Maple [A]  time = 0.031, size = 139, normalized size = 0.8 \[{\frac{21\,{a}^{4}x}{16}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{21\,{a}^{6}}{16}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{{b}^{2}{x}^{3}}{6} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{13\,{a}^{2}x}{8} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{28\,{a}^{3}}{15\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{4\,ab{x}^{2}}{5} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4*(-b^2*x^2+a^2)^(1/2),x)

[Out]

21/16*a^4*x*(-b^2*x^2+a^2)^(1/2)+21/16*a^6/(b^2)^(1/2)*arctan((b^2)^(1/2)*x/(-b^
2*x^2+a^2)^(1/2))-1/6*b^2*x^3*(-b^2*x^2+a^2)^(3/2)-13/8*a^2*x*(-b^2*x^2+a^2)^(3/
2)-28/15*a^3*(-b^2*x^2+a^2)^(3/2)/b-4/5*a*b*x^2*(-b^2*x^2+a^2)^(3/2)

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Maxima [A]  time = 0.768679, size = 177, normalized size = 1.02 \[ -\frac{1}{6} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} b^{2} x^{3} + \frac{21 \, a^{6} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{16 \, \sqrt{b^{2}}} + \frac{21}{16} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{4} x - \frac{4}{5} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a b x^{2} - \frac{13}{8} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a^{2} x - \frac{28 \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a^{3}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)*(b*x + a)^4,x, algorithm="maxima")

[Out]

-1/6*(-b^2*x^2 + a^2)^(3/2)*b^2*x^3 + 21/16*a^6*arcsin(b^2*x/sqrt(a^2*b^2))/sqrt
(b^2) + 21/16*sqrt(-b^2*x^2 + a^2)*a^4*x - 4/5*(-b^2*x^2 + a^2)^(3/2)*a*b*x^2 -
13/8*(-b^2*x^2 + a^2)^(3/2)*a^2*x - 28/15*(-b^2*x^2 + a^2)^(3/2)*a^3/b

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Fricas [A]  time = 0.221968, size = 593, normalized size = 3.43 \[ -\frac{240 \, a b^{11} x^{11} + 1152 \, a^{2} b^{10} x^{10} + 580 \, a^{3} b^{9} x^{9} - 5760 \, a^{4} b^{8} x^{8} - 11190 \, a^{5} b^{7} x^{7} + 320 \, a^{6} b^{6} x^{6} + 23970 \, a^{7} b^{5} x^{5} + 19200 \, a^{8} b^{4} x^{4} - 16000 \, a^{9} b^{3} x^{3} - 15360 \, a^{10} b^{2} x^{2} + 2400 \, a^{11} b x + 630 \,{\left (a^{6} b^{6} x^{6} - 18 \, a^{8} b^{4} x^{4} + 48 \, a^{10} b^{2} x^{2} - 32 \, a^{12} + 2 \,{\left (3 \, a^{7} b^{4} x^{4} - 16 \, a^{9} b^{2} x^{2} + 16 \, a^{11}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) -{\left (40 \, b^{11} x^{11} + 192 \, a b^{10} x^{10} - 370 \, a^{2} b^{9} x^{9} - 3200 \, a^{3} b^{8} x^{8} - 4455 \, a^{4} b^{7} x^{7} + 4160 \, a^{5} b^{6} x^{6} + 16870 \, a^{6} b^{5} x^{5} + 11520 \, a^{7} b^{4} x^{4} - 14800 \, a^{8} b^{3} x^{3} - 15360 \, a^{9} b^{2} x^{2} + 2400 \, a^{10} b x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{240 \,{\left (b^{7} x^{6} - 18 \, a^{2} b^{5} x^{4} + 48 \, a^{4} b^{3} x^{2} - 32 \, a^{6} b + 2 \,{\left (3 \, a b^{5} x^{4} - 16 \, a^{3} b^{3} x^{2} + 16 \, a^{5} b\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)*(b*x + a)^4,x, algorithm="fricas")

[Out]

-1/240*(240*a*b^11*x^11 + 1152*a^2*b^10*x^10 + 580*a^3*b^9*x^9 - 5760*a^4*b^8*x^
8 - 11190*a^5*b^7*x^7 + 320*a^6*b^6*x^6 + 23970*a^7*b^5*x^5 + 19200*a^8*b^4*x^4
- 16000*a^9*b^3*x^3 - 15360*a^10*b^2*x^2 + 2400*a^11*b*x + 630*(a^6*b^6*x^6 - 18
*a^8*b^4*x^4 + 48*a^10*b^2*x^2 - 32*a^12 + 2*(3*a^7*b^4*x^4 - 16*a^9*b^2*x^2 + 1
6*a^11)*sqrt(-b^2*x^2 + a^2))*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) - (40*b^
11*x^11 + 192*a*b^10*x^10 - 370*a^2*b^9*x^9 - 3200*a^3*b^8*x^8 - 4455*a^4*b^7*x^
7 + 4160*a^5*b^6*x^6 + 16870*a^6*b^5*x^5 + 11520*a^7*b^4*x^4 - 14800*a^8*b^3*x^3
- 15360*a^9*b^2*x^2 + 2400*a^10*b*x)*sqrt(-b^2*x^2 + a^2))/(b^7*x^6 - 18*a^2*b^
5*x^4 + 48*a^4*b^3*x^2 - 32*a^6*b + 2*(3*a*b^5*x^4 - 16*a^3*b^3*x^2 + 16*a^5*b)*
sqrt(-b^2*x^2 + a^2))

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Sympy [A]  time = 36.654, size = 700, normalized size = 4.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**4*(-b**2*x**2+a**2)**(1/2),x)

[Out]

a**4*Piecewise((-I*a**2*acosh(b*x/a)/(2*b) - I*a*x/(2*sqrt(-1 + b**2*x**2/a**2))
+ I*b**2*x**3/(2*a*sqrt(-1 + b**2*x**2/a**2)), Abs(b**2*x**2/a**2) > 1), (a**2*
asin(b*x/a)/(2*b) + a*x*sqrt(1 - b**2*x**2/a**2)/2, True)) + 4*a**3*b*Piecewise(
(x**2*sqrt(a**2)/2, Eq(b**2, 0)), (-(a**2 - b**2*x**2)**(3/2)/(3*b**2), True)) +
6*a**2*b**2*Piecewise((-I*a**4*acosh(b*x/a)/(8*b**3) + I*a**3*x/(8*b**2*sqrt(-1
+ b**2*x**2/a**2)) - 3*I*a*x**3/(8*sqrt(-1 + b**2*x**2/a**2)) + I*b**2*x**5/(4*
a*sqrt(-1 + b**2*x**2/a**2)), Abs(b**2*x**2/a**2) > 1), (a**4*asin(b*x/a)/(8*b**
3) - a**3*x/(8*b**2*sqrt(1 - b**2*x**2/a**2)) + 3*a*x**3/(8*sqrt(1 - b**2*x**2/a
**2)) - b**2*x**5/(4*a*sqrt(1 - b**2*x**2/a**2)), True)) + 4*a*b**3*Piecewise((-
2*a**4*sqrt(a**2 - b**2*x**2)/(15*b**4) - a**2*x**2*sqrt(a**2 - b**2*x**2)/(15*b
**2) + x**4*sqrt(a**2 - b**2*x**2)/5, Ne(b, 0)), (x**4*sqrt(a**2)/4, True)) + b*
*4*Piecewise((-I*a**6*acosh(b*x/a)/(16*b**5) + I*a**5*x/(16*b**4*sqrt(-1 + b**2*
x**2/a**2)) - I*a**3*x**3/(48*b**2*sqrt(-1 + b**2*x**2/a**2)) - 5*I*a*x**5/(24*s
qrt(-1 + b**2*x**2/a**2)) + I*b**2*x**7/(6*a*sqrt(-1 + b**2*x**2/a**2)), Abs(b**
2*x**2/a**2) > 1), (a**6*asin(b*x/a)/(16*b**5) - a**5*x/(16*b**4*sqrt(1 - b**2*x
**2/a**2)) + a**3*x**3/(48*b**2*sqrt(1 - b**2*x**2/a**2)) + 5*a*x**5/(24*sqrt(1
- b**2*x**2/a**2)) - b**2*x**7/(6*a*sqrt(1 - b**2*x**2/a**2)), True))

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GIAC/XCAS [A]  time = 0.229463, size = 123, normalized size = 0.71 \[ \frac{21 \, a^{6} \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{16 \,{\left | b \right |}} - \frac{1}{240} \,{\left (\frac{448 \, a^{5}}{b} +{\left (75 \, a^{4} - 2 \,{\left (128 \, a^{3} b +{\left (175 \, a^{2} b^{2} + 4 \,{\left (5 \, b^{4} x + 24 \, a b^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-b^{2} x^{2} + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)*(b*x + a)^4,x, algorithm="giac")

[Out]

21/16*a^6*arcsin(b*x/a)*sign(a)*sign(b)/abs(b) - 1/240*(448*a^5/b + (75*a^4 - 2*
(128*a^3*b + (175*a^2*b^2 + 4*(5*b^4*x + 24*a*b^3)*x)*x)*x)*x)*sqrt(-b^2*x^2 + a
^2)