3.778 \(\int (a+b x)^2 \left (a^2-b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=131 \[ \frac{7}{24} a^2 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{7 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b}+\frac{7}{16} a^4 x \sqrt{a^2-b^2 x^2} \]

[Out]

(7*a^4*x*Sqrt[a^2 - b^2*x^2])/16 + (7*a^2*x*(a^2 - b^2*x^2)^(3/2))/24 - (7*a*(a^
2 - b^2*x^2)^(5/2))/(30*b) - ((a + b*x)*(a^2 - b^2*x^2)^(5/2))/(6*b) + (7*a^6*Ar
cTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(16*b)

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Rubi [A]  time = 0.125943, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{7}{24} a^2 x \left (a^2-b^2 x^2\right )^{3/2}-\frac{7 a \left (a^2-b^2 x^2\right )^{5/2}}{30 b}-\frac{(a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{6 b}+\frac{7 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b}+\frac{7}{16} a^4 x \sqrt{a^2-b^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(7*a^4*x*Sqrt[a^2 - b^2*x^2])/16 + (7*a^2*x*(a^2 - b^2*x^2)^(3/2))/24 - (7*a*(a^
2 - b^2*x^2)^(5/2))/(30*b) - ((a + b*x)*(a^2 - b^2*x^2)^(5/2))/(6*b) + (7*a^6*Ar
cTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(16*b)

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Rubi in Sympy [A]  time = 21.29, size = 112, normalized size = 0.85 \[ \frac{7 a^{6} \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{16 b} + \frac{7 a^{4} x \sqrt{a^{2} - b^{2} x^{2}}}{16} + \frac{7 a^{2} x \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{24} - \frac{7 a \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{30 b} - \frac{\left (a + b x\right ) \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(-b**2*x**2+a**2)**(3/2),x)

[Out]

7*a**6*atan(b*x/sqrt(a**2 - b**2*x**2))/(16*b) + 7*a**4*x*sqrt(a**2 - b**2*x**2)
/16 + 7*a**2*x*(a**2 - b**2*x**2)**(3/2)/24 - 7*a*(a**2 - b**2*x**2)**(5/2)/(30*
b) - (a + b*x)*(a**2 - b**2*x**2)**(5/2)/(6*b)

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Mathematica [A]  time = 0.124515, size = 102, normalized size = 0.78 \[ \frac{105 a^6 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )+\sqrt{a^2-b^2 x^2} \left (-96 a^5+135 a^4 b x+192 a^3 b^2 x^2+10 a^2 b^3 x^3-96 a b^4 x^4-40 b^5 x^5\right )}{240 b} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a^2 - b^2*x^2]*(-96*a^5 + 135*a^4*b*x + 192*a^3*b^2*x^2 + 10*a^2*b^3*x^3 -
96*a*b^4*x^4 - 40*b^5*x^5) + 105*a^6*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/(240*b)

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Maple [A]  time = 0.01, size = 111, normalized size = 0.9 \[{\frac{7\,{a}^{2}x}{24} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{a}^{4}x}{16}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{7\,{a}^{6}}{16}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{x}{6} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,a}{5\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(-b^2*x^2+a^2)^(3/2),x)

[Out]

7/24*a^2*x*(-b^2*x^2+a^2)^(3/2)+7/16*a^4*x*(-b^2*x^2+a^2)^(1/2)+7/16*a^6/(b^2)^(
1/2)*arctan((b^2)^(1/2)*x/(-b^2*x^2+a^2)^(1/2))-1/6*x*(-b^2*x^2+a^2)^(5/2)-2/5*a
*(-b^2*x^2+a^2)^(5/2)/b

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Maxima [A]  time = 0.767563, size = 139, normalized size = 1.06 \[ \frac{7 \, a^{6} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{16 \, \sqrt{b^{2}}} + \frac{7}{16} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{4} x + \frac{7}{24} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a^{2} x - \frac{1}{6} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} x - \frac{2 \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} a}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/16*a^6*arcsin(b^2*x/sqrt(a^2*b^2))/sqrt(b^2) + 7/16*sqrt(-b^2*x^2 + a^2)*a^4*x
+ 7/24*(-b^2*x^2 + a^2)^(3/2)*a^2*x - 1/6*(-b^2*x^2 + a^2)^(5/2)*x - 2/5*(-b^2*
x^2 + a^2)^(5/2)*a/b

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Fricas [A]  time = 0.234354, size = 593, normalized size = 4.53 \[ \frac{240 \, a b^{11} x^{11} + 576 \, a^{2} b^{10} x^{10} - 1580 \, a^{3} b^{9} x^{9} - 4800 \, a^{4} b^{8} x^{8} + 2130 \, a^{5} b^{7} x^{7} + 13920 \, a^{6} b^{6} x^{6} + 3210 \, a^{7} b^{5} x^{5} - 17280 \, a^{8} b^{4} x^{4} - 8320 \, a^{9} b^{3} x^{3} + 7680 \, a^{10} b^{2} x^{2} + 4320 \, a^{11} b x - 210 \,{\left (a^{6} b^{6} x^{6} - 18 \, a^{8} b^{4} x^{4} + 48 \, a^{10} b^{2} x^{2} - 32 \, a^{12} + 2 \,{\left (3 \, a^{7} b^{4} x^{4} - 16 \, a^{9} b^{2} x^{2} + 16 \, a^{11}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) -{\left (40 \, b^{11} x^{11} + 96 \, a b^{10} x^{10} - 730 \, a^{2} b^{9} x^{9} - 1920 \, a^{3} b^{8} x^{8} + 1965 \, a^{4} b^{7} x^{7} + 8160 \, a^{5} b^{6} x^{6} + 670 \, a^{6} b^{5} x^{5} - 13440 \, a^{7} b^{4} x^{4} - 6160 \, a^{8} b^{3} x^{3} + 7680 \, a^{9} b^{2} x^{2} + 4320 \, a^{10} b x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{240 \,{\left (b^{7} x^{6} - 18 \, a^{2} b^{5} x^{4} + 48 \, a^{4} b^{3} x^{2} - 32 \, a^{6} b + 2 \,{\left (3 \, a b^{5} x^{4} - 16 \, a^{3} b^{3} x^{2} + 16 \, a^{5} b\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/240*(240*a*b^11*x^11 + 576*a^2*b^10*x^10 - 1580*a^3*b^9*x^9 - 4800*a^4*b^8*x^8
+ 2130*a^5*b^7*x^7 + 13920*a^6*b^6*x^6 + 3210*a^7*b^5*x^5 - 17280*a^8*b^4*x^4 -
8320*a^9*b^3*x^3 + 7680*a^10*b^2*x^2 + 4320*a^11*b*x - 210*(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 + 16*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 + 96*a*b^10*x^10 - 730*a^2*b^9*x^9 - 1920*a^3*b^8*x^8 + 1965*a^4*b^7*x^7 +
8160*a^5*b^6*x^6 + 670*a^6*b^5*x^5 - 13440*a^7*b^4*x^4 - 6160*a^8*b^3*x^3 + 7680
*a^9*b^2*x^2 + 4320*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 = 26.9588, size = 495, normalized size = 3.78 \[ a^{4} \left (\begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{2 b} - \frac{i a x}{2 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{3}}{2 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{a^{2} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{2 b} + \frac{a x \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + 2 a^{3} b \left (\begin{cases} \frac{x^{2} \sqrt{a^{2}}}{2} & \text{for}\: b^{2} = 0 \\- \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} & \text{otherwise} \end{cases}\right ) - 2 a b^{3} \left (\begin{cases} - \frac{2 a^{4} \sqrt{a^{2} - b^{2} x^{2}}}{15 b^{4}} - \frac{a^{2} x^{2} \sqrt{a^{2} - b^{2} x^{2}}}{15 b^{2}} + \frac{x^{4} \sqrt{a^{2} - b^{2} x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{x^{4} \sqrt{a^{2}}}{4} & \text{otherwise} \end{cases}\right ) - b^{4} \left (\begin{cases} - \frac{i a^{6} \operatorname{acosh}{\left (\frac{b x}{a} \right )}}{16 b^{5}} + \frac{i a^{5} x}{16 b^{4} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{i a^{3} x^{3}}{48 b^{2} \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} - \frac{5 i a x^{5}}{24 \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} + \frac{i b^{2} x^{7}}{6 a \sqrt{-1 + \frac{b^{2} x^{2}}{a^{2}}}} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{a^{6} \operatorname{asin}{\left (\frac{b x}{a} \right )}}{16 b^{5}} - \frac{a^{5} x}{16 b^{4} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{a^{3} x^{3}}{48 b^{2} \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} + \frac{5 a x^{5}}{24 \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} - \frac{b^{2} x^{7}}{6 a \sqrt{1 - \frac{b^{2} x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2*(-b**2*x**2+a**2)**(3/2),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)) + 2*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)) -
2*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*sqr
t(a**2)/4, True)) - b**4*Piecewise((-I*a**6*acosh(b*x/a)/(16*b**5) + I*a**5*x/(1
6*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*sqrt(-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/(1
6*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 = 0.237591, size = 124, normalized size = 0.95 \[ \frac{7 \, a^{6} \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{16 \,{\left | b \right |}} - \frac{1}{240} \,{\left (\frac{96 \, a^{5}}{b} -{\left (135 \, a^{4} + 2 \,{\left (96 \, a^{3} b +{\left (5 \, a^{2} b^{2} - 4 \,{\left (5 \, b^{4} x + 12 \, a b^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-b^{2} x^{2} + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/16*a^6*arcsin(b*x/a)*sign(a)*sign(b)/abs(b) - 1/240*(96*a^5/b - (135*a^4 + 2*(
96*a^3*b + (5*a^2*b^2 - 4*(5*b^4*x + 12*a*b^3)*x)*x)*x)*x)*sqrt(-b^2*x^2 + a^2)