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

Optimal. Leaf size=164 \[ -\frac{3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{9 a^7 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b}+\frac{9}{16} a^5 x \sqrt{a^2-b^2 x^2}+\frac{3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2} \]

[Out]

(9*a^5*x*Sqrt[a^2 - b^2*x^2])/16 + (3*a^3*x*(a^2 - b^2*x^2)^(3/2))/8 - (3*a^2*(a
^2 - b^2*x^2)^(5/2))/(10*b) - (3*a*(a + b*x)*(a^2 - b^2*x^2)^(5/2))/(14*b) - ((a
+ b*x)^2*(a^2 - b^2*x^2)^(5/2))/(7*b) + (9*a^7*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]
])/(16*b)

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Rubi [A]  time = 0.18079, antiderivative size = 164, 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 \[ -\frac{3 a^2 \left (a^2-b^2 x^2\right )^{5/2}}{10 b}-\frac{3 a (a+b x) \left (a^2-b^2 x^2\right )^{5/2}}{14 b}-\frac{(a+b x)^2 \left (a^2-b^2 x^2\right )^{5/2}}{7 b}+\frac{9 a^7 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{16 b}+\frac{9}{16} a^5 x \sqrt{a^2-b^2 x^2}+\frac{3}{8} a^3 x \left (a^2-b^2 x^2\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(9*a^5*x*Sqrt[a^2 - b^2*x^2])/16 + (3*a^3*x*(a^2 - b^2*x^2)^(3/2))/8 - (3*a^2*(a
^2 - b^2*x^2)^(5/2))/(10*b) - (3*a*(a + b*x)*(a^2 - b^2*x^2)^(5/2))/(14*b) - ((a
+ b*x)^2*(a^2 - b^2*x^2)^(5/2))/(7*b) + (9*a^7*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]
])/(16*b)

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Rubi in Sympy [A]  time = 27.9647, size = 141, normalized size = 0.86 \[ \frac{9 a^{7} \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{16 b} + \frac{9 a^{5} x \sqrt{a^{2} - b^{2} x^{2}}}{16} + \frac{3 a^{3} x \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{8} - \frac{3 a^{2} \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{10 b} - \frac{3 a \left (a + b x\right ) \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{14 b} - \frac{\left (a + b x\right )^{2} \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*a**7*atan(b*x/sqrt(a**2 - b**2*x**2))/(16*b) + 9*a**5*x*sqrt(a**2 - b**2*x**2)
/16 + 3*a**3*x*(a**2 - b**2*x**2)**(3/2)/8 - 3*a**2*(a**2 - b**2*x**2)**(5/2)/(1
0*b) - 3*a*(a + b*x)*(a**2 - b**2*x**2)**(5/2)/(14*b) - (a + b*x)**2*(a**2 - b**
2*x**2)**(5/2)/(7*b)

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Mathematica [A]  time = 0.118589, size = 113, normalized size = 0.69 \[ \frac{315 a^7 \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )+\sqrt{a^2-b^2 x^2} \left (-368 a^6+245 a^5 b x+656 a^4 b^2 x^2+350 a^3 b^3 x^3-208 a^2 b^4 x^4-280 a b^5 x^5-80 b^6 x^6\right )}{560 b} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a^2 - b^2*x^2]*(-368*a^6 + 245*a^5*b*x + 656*a^4*b^2*x^2 + 350*a^3*b^3*x^3
- 208*a^2*b^4*x^4 - 280*a*b^5*x^5 - 80*b^6*x^6) + 315*a^7*ArcTan[(b*x)/Sqrt[a^2
- b^2*x^2]])/(560*b)

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Maple [A]  time = 0.023, size = 134, normalized size = 0.8 \[{\frac{3\,{a}^{3}x}{8} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{a}^{5}x}{16}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}+{\frac{9\,{a}^{7}}{16}\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{b{x}^{2}}{7} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{23\,{a}^{2}}{35\,b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{ax}{2} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*a^3*x*(-b^2*x^2+a^2)^(3/2)+9/16*a^5*x*(-b^2*x^2+a^2)^(1/2)+9/16*a^7/(b^2)^(1
/2)*arctan((b^2)^(1/2)*x/(-b^2*x^2+a^2)^(1/2))-1/7*b*x^2*(-b^2*x^2+a^2)^(5/2)-23
/35*a^2*(-b^2*x^2+a^2)^(5/2)/b-1/2*a*x*(-b^2*x^2+a^2)^(5/2)

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Maxima [A]  time = 0.773981, size = 170, normalized size = 1.04 \[ \frac{9 \, a^{7} \arcsin \left (\frac{b^{2} x}{\sqrt{a^{2} b^{2}}}\right )}{16 \, \sqrt{b^{2}}} + \frac{9}{16} \, \sqrt{-b^{2} x^{2} + a^{2}} a^{5} x + \frac{3}{8} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{3}{2}} a^{3} x - \frac{1}{7} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} b x^{2} - \frac{1}{2} \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} a x - \frac{23 \,{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac{5}{2}} a^{2}}{35 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/16*a^7*arcsin(b^2*x/sqrt(a^2*b^2))/sqrt(b^2) + 9/16*sqrt(-b^2*x^2 + a^2)*a^5*x
+ 3/8*(-b^2*x^2 + a^2)^(3/2)*a^3*x - 1/7*(-b^2*x^2 + a^2)^(5/2)*b*x^2 - 1/2*(-b
^2*x^2 + a^2)^(5/2)*a*x - 23/35*(-b^2*x^2 + a^2)^(5/2)*a^2/b

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Fricas [A]  time = 0.229217, size = 682, normalized size = 4.16 \[ -\frac{80 \, b^{14} x^{14} + 280 \, a b^{13} x^{13} - 1792 \, a^{2} b^{12} x^{12} - 7350 \, a^{3} b^{11} x^{11} + 2464 \, a^{4} b^{10} x^{10} + 37625 \, a^{5} b^{9} x^{9} + 26880 \, a^{6} b^{8} x^{8} - 70595 \, a^{7} b^{7} x^{7} - 99680 \, a^{8} b^{6} x^{6} + 42840 \, a^{9} b^{5} x^{5} + 125440 \, a^{10} b^{4} x^{4} + 12880 \, a^{11} b^{3} x^{3} - 53760 \, a^{12} b^{2} x^{2} - 15680 \, a^{13} b x + 630 \,{\left (7 \, a^{8} b^{6} x^{6} - 56 \, a^{10} b^{4} x^{4} + 112 \, a^{12} b^{2} x^{2} - 64 \, a^{14} -{\left (a^{7} b^{6} x^{6} - 24 \, a^{9} b^{4} x^{4} + 80 \, a^{11} b^{2} x^{2} - 64 \, a^{13}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) + 7 \,{\left (80 \, a b^{12} x^{12} + 280 \, a^{2} b^{11} x^{11} - 432 \, a^{3} b^{10} x^{10} - 2590 \, a^{4} b^{9} x^{9} - 1040 \, a^{5} b^{8} x^{8} + 7035 \, a^{6} b^{7} x^{7} + 8160 \, a^{7} b^{6} x^{6} - 6200 \, a^{8} b^{5} x^{5} - 14080 \, a^{9} b^{4} x^{4} - 720 \, a^{10} b^{3} x^{3} + 7680 \, a^{11} b^{2} x^{2} + 2240 \, a^{12} b x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{560 \,{\left (7 \, a b^{7} x^{6} - 56 \, a^{3} b^{5} x^{4} + 112 \, a^{5} b^{3} x^{2} - 64 \, a^{7} b -{\left (b^{7} x^{6} - 24 \, a^{2} b^{5} x^{4} + 80 \, a^{4} b^{3} x^{2} - 64 \, a^{6} b\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/560*(80*b^14*x^14 + 280*a*b^13*x^13 - 1792*a^2*b^12*x^12 - 7350*a^3*b^11*x^11
+ 2464*a^4*b^10*x^10 + 37625*a^5*b^9*x^9 + 26880*a^6*b^8*x^8 - 70595*a^7*b^7*x^
7 - 99680*a^8*b^6*x^6 + 42840*a^9*b^5*x^5 + 125440*a^10*b^4*x^4 + 12880*a^11*b^3
*x^3 - 53760*a^12*b^2*x^2 - 15680*a^13*b*x + 630*(7*a^8*b^6*x^6 - 56*a^10*b^4*x^
4 + 112*a^12*b^2*x^2 - 64*a^14 - (a^7*b^6*x^6 - 24*a^9*b^4*x^4 + 80*a^11*b^2*x^2
- 64*a^13)*sqrt(-b^2*x^2 + a^2))*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) + 7*
(80*a*b^12*x^12 + 280*a^2*b^11*x^11 - 432*a^3*b^10*x^10 - 2590*a^4*b^9*x^9 - 104
0*a^5*b^8*x^8 + 7035*a^6*b^7*x^7 + 8160*a^7*b^6*x^6 - 6200*a^8*b^5*x^5 - 14080*a
^9*b^4*x^4 - 720*a^10*b^3*x^3 + 7680*a^11*b^2*x^2 + 2240*a^12*b*x)*sqrt(-b^2*x^2
+ a^2))/(7*a*b^7*x^6 - 56*a^3*b^5*x^4 + 112*a^5*b^3*x^2 - 64*a^7*b - (b^7*x^6 -
24*a^2*b^5*x^4 + 80*a^4*b^3*x^2 - 64*a^6*b)*sqrt(-b^2*x^2 + a^2))

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Sympy [A]  time = 38.8924, size = 816, normalized size = 4.98 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**5*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)) + 3*a**4*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**3*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)) - 2*a**2*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)/(1
5*b**2) + x**4*sqrt(a**2 - b**2*x**2)/5, Ne(b, 0)), (x**4*sqrt(a**2)/4, True)) -
3*a*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*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/(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)) - b
**5*Piecewise((-8*a**6*sqrt(a**2 - b**2*x**2)/(105*b**6) - 4*a**4*x**2*sqrt(a**2
- b**2*x**2)/(105*b**4) - a**2*x**4*sqrt(a**2 - b**2*x**2)/(35*b**2) + x**6*sqr
t(a**2 - b**2*x**2)/7, Ne(b, 0)), (x**6*sqrt(a**2)/6, True))

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GIAC/XCAS [A]  time = 0.236326, size = 140, normalized size = 0.85 \[ \frac{9 \, a^{7} \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{16 \,{\left | b \right |}} - \frac{1}{560} \,{\left (\frac{368 \, a^{6}}{b} -{\left (245 \, a^{5} + 2 \,{\left (328 \, a^{4} b +{\left (175 \, a^{3} b^{2} - 4 \,{\left (26 \, a^{2} b^{3} + 5 \,{\left (2 \, b^{5} x + 7 \, a b^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-b^{2} x^{2} + a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/16*a^7*arcsin(b*x/a)*sign(a)*sign(b)/abs(b) - 1/560*(368*a^6/b - (245*a^5 + 2*
(328*a^4*b + (175*a^3*b^2 - 4*(26*a^2*b^3 + 5*(2*b^5*x + 7*a*b^4)*x)*x)*x)*x)*x)
*sqrt(-b^2*x^2 + a^2)