3.97 \(\int \frac{\left (a+b x^2\right )^3}{\left (c+d x^2\right )^{11/2}} \, dx\)

Optimal. Leaf size=224 \[ \frac{16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt{c+d x^2} (b c-a d)}+\frac{8 a^2 x \left (a+b x^2\right ) (9 b c-8 a d)}{315 c^4 \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{2 a x \left (a+b x^2\right )^2 (9 b c-8 a d)}{105 c^3 \left (c+d x^2\right )^{5/2} (b c-a d)}+\frac{x \left (a+b x^2\right )^3 (9 b c-8 a d)}{63 c^2 \left (c+d x^2\right )^{7/2} (b c-a d)}-\frac{d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)} \]

[Out]

-(d*x*(a + b*x^2)^4)/(9*c*(b*c - a*d)*(c + d*x^2)^(9/2)) + ((9*b*c - 8*a*d)*x*(a
+ b*x^2)^3)/(63*c^2*(b*c - a*d)*(c + d*x^2)^(7/2)) + (2*a*(9*b*c - 8*a*d)*x*(a
+ b*x^2)^2)/(105*c^3*(b*c - a*d)*(c + d*x^2)^(5/2)) + (8*a^2*(9*b*c - 8*a*d)*x*(
a + b*x^2))/(315*c^4*(b*c - a*d)*(c + d*x^2)^(3/2)) + (16*a^3*(9*b*c - 8*a*d)*x)
/(315*c^5*(b*c - a*d)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.279336, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{16 a^3 x (9 b c-8 a d)}{315 c^5 \sqrt{c+d x^2} (b c-a d)}+\frac{8 a^2 x \left (a+b x^2\right ) (9 b c-8 a d)}{315 c^4 \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{2 a x \left (a+b x^2\right )^2 (9 b c-8 a d)}{105 c^3 \left (c+d x^2\right )^{5/2} (b c-a d)}+\frac{x \left (a+b x^2\right )^3 (9 b c-8 a d)}{63 c^2 \left (c+d x^2\right )^{7/2} (b c-a d)}-\frac{d x \left (a+b x^2\right )^4}{9 c \left (c+d x^2\right )^{9/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/(c + d*x^2)^(11/2),x]

[Out]

-(d*x*(a + b*x^2)^4)/(9*c*(b*c - a*d)*(c + d*x^2)^(9/2)) + ((9*b*c - 8*a*d)*x*(a
+ b*x^2)^3)/(63*c^2*(b*c - a*d)*(c + d*x^2)^(7/2)) + (2*a*(9*b*c - 8*a*d)*x*(a
+ b*x^2)^2)/(105*c^3*(b*c - a*d)*(c + d*x^2)^(5/2)) + (8*a^2*(9*b*c - 8*a*d)*x*(
a + b*x^2))/(315*c^4*(b*c - a*d)*(c + d*x^2)^(3/2)) + (16*a^3*(9*b*c - 8*a*d)*x)
/(315*c^5*(b*c - a*d)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 41.4738, size = 204, normalized size = 0.91 \[ \frac{16 a^{3} x \left (8 a d - 9 b c\right )}{315 c^{5} \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{8 a^{2} x \left (a + b x^{2}\right ) \left (8 a d - 9 b c\right )}{315 c^{4} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{2 a x \left (a + b x^{2}\right )^{2} \left (8 a d - 9 b c\right )}{105 c^{3} \left (c + d x^{2}\right )^{\frac{5}{2}} \left (a d - b c\right )} + \frac{d x \left (a + b x^{2}\right )^{4}}{9 c \left (c + d x^{2}\right )^{\frac{9}{2}} \left (a d - b c\right )} + \frac{x \left (a + b x^{2}\right )^{3} \left (8 a d - 9 b c\right )}{63 c^{2} \left (c + d x^{2}\right )^{\frac{7}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3/(d*x**2+c)**(11/2),x)

[Out]

16*a**3*x*(8*a*d - 9*b*c)/(315*c**5*sqrt(c + d*x**2)*(a*d - b*c)) + 8*a**2*x*(a
+ b*x**2)*(8*a*d - 9*b*c)/(315*c**4*(c + d*x**2)**(3/2)*(a*d - b*c)) + 2*a*x*(a
+ b*x**2)**2*(8*a*d - 9*b*c)/(105*c**3*(c + d*x**2)**(5/2)*(a*d - b*c)) + d*x*(a
+ b*x**2)**4/(9*c*(c + d*x**2)**(9/2)*(a*d - b*c)) + x*(a + b*x**2)**3*(8*a*d -
9*b*c)/(63*c**2*(c + d*x**2)**(7/2)*(a*d - b*c))

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Mathematica [A]  time = 0.157624, size = 163, normalized size = 0.73 \[ \frac{a^3 \left (315 c^4 x+840 c^3 d x^3+1008 c^2 d^2 x^5+576 c d^3 x^7+128 d^4 x^9\right )+3 a^2 b c x^3 \left (105 c^3+126 c^2 d x^2+72 c d^2 x^4+16 d^3 x^6\right )+3 a b^2 c^2 x^5 \left (63 c^2+36 c d x^2+8 d^2 x^4\right )+5 b^3 c^3 x^7 \left (9 c+2 d x^2\right )}{315 c^5 \left (c+d x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/(c + d*x^2)^(11/2),x]

[Out]

(5*b^3*c^3*x^7*(9*c + 2*d*x^2) + 3*a*b^2*c^2*x^5*(63*c^2 + 36*c*d*x^2 + 8*d^2*x^
4) + 3*a^2*b*c*x^3*(105*c^3 + 126*c^2*d*x^2 + 72*c*d^2*x^4 + 16*d^3*x^6) + a^3*(
315*c^4*x + 840*c^3*d*x^3 + 1008*c^2*d^2*x^5 + 576*c*d^3*x^7 + 128*d^4*x^9))/(31
5*c^5*(c + d*x^2)^(9/2))

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Maple [A]  time = 0.011, size = 190, normalized size = 0.9 \[{\frac{x \left ( 128\,{a}^{3}{d}^{4}{x}^{8}+48\,{a}^{2}bc{d}^{3}{x}^{8}+24\,a{b}^{2}{c}^{2}{d}^{2}{x}^{8}+10\,{b}^{3}{c}^{3}d{x}^{8}+576\,{a}^{3}c{d}^{3}{x}^{6}+216\,{a}^{2}b{c}^{2}{d}^{2}{x}^{6}+108\,a{b}^{2}{c}^{3}d{x}^{6}+45\,{b}^{3}{c}^{4}{x}^{6}+1008\,{a}^{3}{c}^{2}{d}^{2}{x}^{4}+378\,{a}^{2}b{c}^{3}d{x}^{4}+189\,a{b}^{2}{c}^{4}{x}^{4}+840\,{a}^{3}{c}^{3}d{x}^{2}+315\,{a}^{2}b{c}^{4}{x}^{2}+315\,{a}^{3}{c}^{4} \right ) }{315\,{c}^{5}} \left ( d{x}^{2}+c \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/(d*x^2+c)^(11/2),x)

[Out]

1/315*x*(128*a^3*d^4*x^8+48*a^2*b*c*d^3*x^8+24*a*b^2*c^2*d^2*x^8+10*b^3*c^3*d*x^
8+576*a^3*c*d^3*x^6+216*a^2*b*c^2*d^2*x^6+108*a*b^2*c^3*d*x^6+45*b^3*c^4*x^6+100
8*a^3*c^2*d^2*x^4+378*a^2*b*c^3*d*x^4+189*a*b^2*c^4*x^4+840*a^3*c^3*d*x^2+315*a^
2*b*c^4*x^2+315*a^3*c^4)/(d*x^2+c)^(9/2)/c^5

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Maxima [A]  time = 1.36496, size = 628, normalized size = 2.8 \[ -\frac{b^{3} x^{5}}{4 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d} - \frac{5 \, b^{3} c x^{3}}{24 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d^{2}} - \frac{a b^{2} x^{3}}{2 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d} + \frac{128 \, a^{3} x}{315 \, \sqrt{d x^{2} + c} c^{5}} + \frac{64 \, a^{3} x}{315 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{4}} + \frac{16 \, a^{3} x}{105 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c^{3}} + \frac{8 \, a^{3} x}{63 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} c^{2}} + \frac{a^{3} x}{9 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} c} + \frac{b^{3} x}{84 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} d^{3}} + \frac{2 \, b^{3} x}{63 \, \sqrt{d x^{2} + c} c^{2} d^{3}} + \frac{b^{3} x}{63 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c d^{3}} + \frac{5 \, b^{3} c x}{504 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} d^{3}} - \frac{5 \, b^{3} c^{2} x}{72 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d^{3}} + \frac{a b^{2} x}{42 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} d^{2}} + \frac{8 \, a b^{2} x}{105 \, \sqrt{d x^{2} + c} c^{3} d^{2}} + \frac{4 \, a b^{2} x}{105 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{2} d^{2}} + \frac{a b^{2} x}{35 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c d^{2}} - \frac{a b^{2} c x}{6 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d^{2}} - \frac{a^{2} b x}{3 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}} d} + \frac{16 \, a^{2} b x}{105 \, \sqrt{d x^{2} + c} c^{4} d} + \frac{8 \, a^{2} b x}{105 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{3} d} + \frac{2 \, a^{2} b x}{35 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c^{2} d} + \frac{a^{2} b x}{21 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/(d*x^2 + c)^(11/2),x, algorithm="maxima")

[Out]

-1/4*b^3*x^5/((d*x^2 + c)^(9/2)*d) - 5/24*b^3*c*x^3/((d*x^2 + c)^(9/2)*d^2) - 1/
2*a*b^2*x^3/((d*x^2 + c)^(9/2)*d) + 128/315*a^3*x/(sqrt(d*x^2 + c)*c^5) + 64/315
*a^3*x/((d*x^2 + c)^(3/2)*c^4) + 16/105*a^3*x/((d*x^2 + c)^(5/2)*c^3) + 8/63*a^3
*x/((d*x^2 + c)^(7/2)*c^2) + 1/9*a^3*x/((d*x^2 + c)^(9/2)*c) + 1/84*b^3*x/((d*x^
2 + c)^(5/2)*d^3) + 2/63*b^3*x/(sqrt(d*x^2 + c)*c^2*d^3) + 1/63*b^3*x/((d*x^2 +
c)^(3/2)*c*d^3) + 5/504*b^3*c*x/((d*x^2 + c)^(7/2)*d^3) - 5/72*b^3*c^2*x/((d*x^2
+ c)^(9/2)*d^3) + 1/42*a*b^2*x/((d*x^2 + c)^(7/2)*d^2) + 8/105*a*b^2*x/(sqrt(d*
x^2 + c)*c^3*d^2) + 4/105*a*b^2*x/((d*x^2 + c)^(3/2)*c^2*d^2) + 1/35*a*b^2*x/((d
*x^2 + c)^(5/2)*c*d^2) - 1/6*a*b^2*c*x/((d*x^2 + c)^(9/2)*d^2) - 1/3*a^2*b*x/((d
*x^2 + c)^(9/2)*d) + 16/105*a^2*b*x/(sqrt(d*x^2 + c)*c^4*d) + 8/105*a^2*b*x/((d*
x^2 + c)^(3/2)*c^3*d) + 2/35*a^2*b*x/((d*x^2 + c)^(5/2)*c^2*d) + 1/21*a^2*b*x/((
d*x^2 + c)^(7/2)*c*d)

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Fricas [A]  time = 0.80248, size = 309, normalized size = 1.38 \[ \frac{{\left (2 \,{\left (5 \, b^{3} c^{3} d + 12 \, a b^{2} c^{2} d^{2} + 24 \, a^{2} b c d^{3} + 64 \, a^{3} d^{4}\right )} x^{9} + 315 \, a^{3} c^{4} x + 9 \,{\left (5 \, b^{3} c^{4} + 12 \, a b^{2} c^{3} d + 24 \, a^{2} b c^{2} d^{2} + 64 \, a^{3} c d^{3}\right )} x^{7} + 63 \,{\left (3 \, a b^{2} c^{4} + 6 \, a^{2} b c^{3} d + 16 \, a^{3} c^{2} d^{2}\right )} x^{5} + 105 \,{\left (3 \, a^{2} b c^{4} + 8 \, a^{3} c^{3} d\right )} x^{3}\right )} \sqrt{d x^{2} + c}}{315 \,{\left (c^{5} d^{5} x^{10} + 5 \, c^{6} d^{4} x^{8} + 10 \, c^{7} d^{3} x^{6} + 10 \, c^{8} d^{2} x^{4} + 5 \, c^{9} d x^{2} + c^{10}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/(d*x^2 + c)^(11/2),x, algorithm="fricas")

[Out]

1/315*(2*(5*b^3*c^3*d + 12*a*b^2*c^2*d^2 + 24*a^2*b*c*d^3 + 64*a^3*d^4)*x^9 + 31
5*a^3*c^4*x + 9*(5*b^3*c^4 + 12*a*b^2*c^3*d + 24*a^2*b*c^2*d^2 + 64*a^3*c*d^3)*x
^7 + 63*(3*a*b^2*c^4 + 6*a^2*b*c^3*d + 16*a^3*c^2*d^2)*x^5 + 105*(3*a^2*b*c^4 +
8*a^3*c^3*d)*x^3)*sqrt(d*x^2 + c)/(c^5*d^5*x^10 + 5*c^6*d^4*x^8 + 10*c^7*d^3*x^6
+ 10*c^8*d^2*x^4 + 5*c^9*d*x^2 + c^10)

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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**2+a)**3/(d*x**2+c)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23366, size = 294, normalized size = 1.31 \[ \frac{{\left ({\left ({\left (x^{2}{\left (\frac{2 \,{\left (5 \, b^{3} c^{3} d^{5} + 12 \, a b^{2} c^{2} d^{6} + 24 \, a^{2} b c d^{7} + 64 \, a^{3} d^{8}\right )} x^{2}}{c^{5} d^{4}} + \frac{9 \,{\left (5 \, b^{3} c^{4} d^{4} + 12 \, a b^{2} c^{3} d^{5} + 24 \, a^{2} b c^{2} d^{6} + 64 \, a^{3} c d^{7}\right )}}{c^{5} d^{4}}\right )} + \frac{63 \,{\left (3 \, a b^{2} c^{4} d^{4} + 6 \, a^{2} b c^{3} d^{5} + 16 \, a^{3} c^{2} d^{6}\right )}}{c^{5} d^{4}}\right )} x^{2} + \frac{105 \,{\left (3 \, a^{2} b c^{4} d^{4} + 8 \, a^{3} c^{3} d^{5}\right )}}{c^{5} d^{4}}\right )} x^{2} + \frac{315 \, a^{3}}{c}\right )} x}{315 \,{\left (d x^{2} + c\right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/(d*x^2 + c)^(11/2),x, algorithm="giac")

[Out]

1/315*(((x^2*(2*(5*b^3*c^3*d^5 + 12*a*b^2*c^2*d^6 + 24*a^2*b*c*d^7 + 64*a^3*d^8)
*x^2/(c^5*d^4) + 9*(5*b^3*c^4*d^4 + 12*a*b^2*c^3*d^5 + 24*a^2*b*c^2*d^6 + 64*a^3
*c*d^7)/(c^5*d^4)) + 63*(3*a*b^2*c^4*d^4 + 6*a^2*b*c^3*d^5 + 16*a^3*c^2*d^6)/(c^
5*d^4))*x^2 + 105*(3*a^2*b*c^4*d^4 + 8*a^3*c^3*d^5)/(c^5*d^4))*x^2 + 315*a^3/c)*
x/(d*x^2 + c)^(9/2)