3.99 \(\int \frac{a+b x^2}{\left (c+d x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=91 \[ \frac{2 x (4 a d+b c)}{15 c^3 d \sqrt{c+d x^2}}+\frac{x (4 a d+b c)}{15 c^2 d \left (c+d x^2\right )^{3/2}}-\frac{x (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]

[Out]

-((b*c - a*d)*x)/(5*c*d*(c + d*x^2)^(5/2)) + ((b*c + 4*a*d)*x)/(15*c^2*d*(c + d*
x^2)^(3/2)) + (2*(b*c + 4*a*d)*x)/(15*c^3*d*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.0816296, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 x (4 a d+b c)}{15 c^3 d \sqrt{c+d x^2}}+\frac{x (4 a d+b c)}{15 c^2 d \left (c+d x^2\right )^{3/2}}-\frac{x (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/(c + d*x^2)^(7/2),x]

[Out]

-((b*c - a*d)*x)/(5*c*d*(c + d*x^2)^(5/2)) + ((b*c + 4*a*d)*x)/(15*c^2*d*(c + d*
x^2)^(3/2)) + (2*(b*c + 4*a*d)*x)/(15*c^3*d*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 10.6007, size = 78, normalized size = 0.86 \[ \frac{x \left (a d - b c\right )}{5 c d \left (c + d x^{2}\right )^{\frac{5}{2}}} + \frac{x \left (4 a d + b c\right )}{15 c^{2} d \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{2 x \left (4 a d + b c\right )}{15 c^{3} d \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/(d*x**2+c)**(7/2),x)

[Out]

x*(a*d - b*c)/(5*c*d*(c + d*x**2)**(5/2)) + x*(4*a*d + b*c)/(15*c**2*d*(c + d*x*
*2)**(3/2)) + 2*x*(4*a*d + b*c)/(15*c**3*d*sqrt(c + d*x**2))

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Mathematica [A]  time = 0.0599623, size = 59, normalized size = 0.65 \[ \frac{a \left (15 c^2 x+20 c d x^3+8 d^2 x^5\right )+b c x^3 \left (5 c+2 d x^2\right )}{15 c^3 \left (c+d x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)/(c + d*x^2)^(7/2),x]

[Out]

(b*c*x^3*(5*c + 2*d*x^2) + a*(15*c^2*x + 20*c*d*x^3 + 8*d^2*x^5))/(15*c^3*(c + d
*x^2)^(5/2))

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Maple [A]  time = 0.006, size = 57, normalized size = 0.6 \[{\frac{x \left ( 8\,a{d}^{2}{x}^{4}+2\,bcd{x}^{4}+20\,acd{x}^{2}+5\,b{c}^{2}{x}^{2}+15\,{c}^{2}a \right ) }{15\,{c}^{3}} \left ( d{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/(d*x^2+c)^(7/2),x)

[Out]

1/15*x*(8*a*d^2*x^4+2*b*c*d*x^4+20*a*c*d*x^2+5*b*c^2*x^2+15*a*c^2)/(d*x^2+c)^(5/
2)/c^3

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Maxima [A]  time = 1.3498, size = 139, normalized size = 1.53 \[ \frac{8 \, a x}{15 \, \sqrt{d x^{2} + c} c^{3}} + \frac{4 \, a x}{15 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{2}} + \frac{a x}{5 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} c} - \frac{b x}{5 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} d} + \frac{2 \, b x}{15 \, \sqrt{d x^{2} + c} c^{2} d} + \frac{b x}{15 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(d*x^2 + c)^(7/2),x, algorithm="maxima")

[Out]

8/15*a*x/(sqrt(d*x^2 + c)*c^3) + 4/15*a*x/((d*x^2 + c)^(3/2)*c^2) + 1/5*a*x/((d*
x^2 + c)^(5/2)*c) - 1/5*b*x/((d*x^2 + c)^(5/2)*d) + 2/15*b*x/(sqrt(d*x^2 + c)*c^
2*d) + 1/15*b*x/((d*x^2 + c)^(3/2)*c*d)

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Fricas [A]  time = 0.228327, size = 117, normalized size = 1.29 \[ \frac{{\left (2 \,{\left (b c d + 4 \, a d^{2}\right )} x^{5} + 15 \, a c^{2} x + 5 \,{\left (b c^{2} + 4 \, a c d\right )} x^{3}\right )} \sqrt{d x^{2} + c}}{15 \,{\left (c^{3} d^{3} x^{6} + 3 \, c^{4} d^{2} x^{4} + 3 \, c^{5} d x^{2} + c^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(d*x^2 + c)^(7/2),x, algorithm="fricas")

[Out]

1/15*(2*(b*c*d + 4*a*d^2)*x^5 + 15*a*c^2*x + 5*(b*c^2 + 4*a*c*d)*x^3)*sqrt(d*x^2
+ c)/(c^3*d^3*x^6 + 3*c^4*d^2*x^4 + 3*c^5*d*x^2 + c^6)

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Sympy [A]  time = 127.945, size = 566, normalized size = 6.22 \[ a \left (\frac{15 c^{5} x}{15 c^{\frac{17}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{15}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{13}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{11}{2}} d^{3} x^{6} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 c^{4} d x^{3}}{15 c^{\frac{17}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{15}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{13}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{11}{2}} d^{3} x^{6} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{28 c^{3} d^{2} x^{5}}{15 c^{\frac{17}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{15}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{13}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{11}{2}} d^{3} x^{6} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{8 c^{2} d^{3} x^{7}}{15 c^{\frac{17}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{15}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 45 c^{\frac{13}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{11}{2}} d^{3} x^{6} \sqrt{1 + \frac{d x^{2}}{c}}}\right ) + b \left (\frac{5 c x^{3}}{15 c^{\frac{9}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 30 c^{\frac{7}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{5}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{2 d x^{5}}{15 c^{\frac{9}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 30 c^{\frac{7}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}} + 15 c^{\frac{5}{2}} d^{2} x^{4} \sqrt{1 + \frac{d x^{2}}{c}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/(d*x**2+c)**(7/2),x)

[Out]

a*(15*c**5*x/(15*c**(17/2)*sqrt(1 + d*x**2/c) + 45*c**(15/2)*d*x**2*sqrt(1 + d*x
**2/c) + 45*c**(13/2)*d**2*x**4*sqrt(1 + d*x**2/c) + 15*c**(11/2)*d**3*x**6*sqrt
(1 + d*x**2/c)) + 35*c**4*d*x**3/(15*c**(17/2)*sqrt(1 + d*x**2/c) + 45*c**(15/2)
*d*x**2*sqrt(1 + d*x**2/c) + 45*c**(13/2)*d**2*x**4*sqrt(1 + d*x**2/c) + 15*c**(
11/2)*d**3*x**6*sqrt(1 + d*x**2/c)) + 28*c**3*d**2*x**5/(15*c**(17/2)*sqrt(1 + d
*x**2/c) + 45*c**(15/2)*d*x**2*sqrt(1 + d*x**2/c) + 45*c**(13/2)*d**2*x**4*sqrt(
1 + d*x**2/c) + 15*c**(11/2)*d**3*x**6*sqrt(1 + d*x**2/c)) + 8*c**2*d**3*x**7/(1
5*c**(17/2)*sqrt(1 + d*x**2/c) + 45*c**(15/2)*d*x**2*sqrt(1 + d*x**2/c) + 45*c**
(13/2)*d**2*x**4*sqrt(1 + d*x**2/c) + 15*c**(11/2)*d**3*x**6*sqrt(1 + d*x**2/c))
) + b*(5*c*x**3/(15*c**(9/2)*sqrt(1 + d*x**2/c) + 30*c**(7/2)*d*x**2*sqrt(1 + d*
x**2/c) + 15*c**(5/2)*d**2*x**4*sqrt(1 + d*x**2/c)) + 2*d*x**5/(15*c**(9/2)*sqrt
(1 + d*x**2/c) + 30*c**(7/2)*d*x**2*sqrt(1 + d*x**2/c) + 15*c**(5/2)*d**2*x**4*s
qrt(1 + d*x**2/c)))

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GIAC/XCAS [A]  time = 0.229023, size = 97, normalized size = 1.07 \[ \frac{{\left (x^{2}{\left (\frac{2 \,{\left (b c d^{3} + 4 \, a d^{4}\right )} x^{2}}{c^{3} d^{2}} + \frac{5 \,{\left (b c^{2} d^{2} + 4 \, a c d^{3}\right )}}{c^{3} d^{2}}\right )} + \frac{15 \, a}{c}\right )} x}{15 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(d*x^2 + c)^(7/2),x, algorithm="giac")

[Out]

1/15*(x^2*(2*(b*c*d^3 + 4*a*d^4)*x^2/(c^3*d^2) + 5*(b*c^2*d^2 + 4*a*c*d^3)/(c^3*
d^2)) + 15*a/c)*x/(d*x^2 + c)^(5/2)