3.1488 \(\int \frac{(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(7/2))/(7*(b*c - a*d)*(a + b*x)^(7/2))

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Rubi [A]  time = 0.0219742, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(7*(b*c - a*d)*(a + b*x)^(7/2))

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Rubi in Sympy [A]  time = 3.91702, size = 26, normalized size = 0.81 \[ \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{7 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/(b*x+a)**(9/2),x)

[Out]

2*(c + d*x)**(7/2)/(7*(a + b*x)**(7/2)*(a*d - b*c))

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Mathematica [A]  time = 0.115367, size = 32, normalized size = 1. \[ -\frac{2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(7/2))/(7*(b*c - a*d)*(a + b*x)^(7/2))

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Maple [A]  time = 0.007, size = 27, normalized size = 0.8 \[{\frac{2}{7\,ad-7\,bc} \left ( dx+c \right ) ^{{\frac{7}{2}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/(b*x+a)^(9/2),x)

[Out]

2/7/(b*x+a)^(7/2)*(d*x+c)^(7/2)/(a*d-b*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.670846, size = 186, normalized size = 5.81 \[ -\frac{2 \,{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sqrt{b x + a} \sqrt{d x + c}}{7 \,{\left (a^{4} b c - a^{5} d +{\left (b^{5} c - a b^{4} d\right )} x^{4} + 4 \,{\left (a b^{4} c - a^{2} b^{3} d\right )} x^{3} + 6 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2} + 4 \,{\left (a^{3} b^{2} c - a^{4} b d\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^(9/2),x, algorithm="fricas")

[Out]

-2/7*(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*sqrt(b*x + a)*sqrt(d*x + c)/(a^4*
b*c - a^5*d + (b^5*c - a*b^4*d)*x^4 + 4*(a*b^4*c - a^2*b^3*d)*x^3 + 6*(a^2*b^3*c
- a^3*b^2*d)*x^2 + 4*(a^3*b^2*c - a^4*b*d)*x)

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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((d*x+c)**(5/2)/(b*x+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.435799, size = 953, normalized size = 29.78 \[ -\frac{4 \,{\left (\sqrt{b d} b^{12} c^{6} d^{3}{\left | b \right |} - 6 \, \sqrt{b d} a b^{11} c^{5} d^{4}{\left | b \right |} + 15 \, \sqrt{b d} a^{2} b^{10} c^{4} d^{5}{\left | b \right |} - 20 \, \sqrt{b d} a^{3} b^{9} c^{3} d^{6}{\left | b \right |} + 15 \, \sqrt{b d} a^{4} b^{8} c^{2} d^{7}{\left | b \right |} - 6 \, \sqrt{b d} a^{5} b^{7} c d^{8}{\left | b \right |} + \sqrt{b d} a^{6} b^{6} d^{9}{\left | b \right |} + 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{8} c^{4} d^{3}{\left | b \right |} - 84 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{7} c^{3} d^{4}{\left | b \right |} + 126 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{6} c^{2} d^{5}{\left | b \right |} - 84 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{5} c d^{6}{\left | b \right |} + 21 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{4} b^{4} d^{7}{\left | b \right |} + 35 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{8} b^{4} c^{2} d^{3}{\left | b \right |} - 70 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{8} a b^{3} c d^{4}{\left | b \right |} + 35 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{8} a^{2} b^{2} d^{5}{\left | b \right |} + 7 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{12} d^{3}{\left | b \right |}\right )}}{7 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{7} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(b*x + a)^(9/2),x, algorithm="giac")

[Out]

-4/7*(sqrt(b*d)*b^12*c^6*d^3*abs(b) - 6*sqrt(b*d)*a*b^11*c^5*d^4*abs(b) + 15*sqr
t(b*d)*a^2*b^10*c^4*d^5*abs(b) - 20*sqrt(b*d)*a^3*b^9*c^3*d^6*abs(b) + 15*sqrt(b
*d)*a^4*b^8*c^2*d^7*abs(b) - 6*sqrt(b*d)*a^5*b^7*c*d^8*abs(b) + sqrt(b*d)*a^6*b^
6*d^9*abs(b) + 21*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*
d - a*b*d))^4*b^8*c^4*d^3*abs(b) - 84*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(
b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^7*c^3*d^4*abs(b) + 126*sqrt(b*d)*(sqrt(b*d
)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^6*c^2*d^5*abs(b)
- 84*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4
*a^3*b^5*c*d^6*abs(b) + 21*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*
x + a)*b*d - a*b*d))^4*a^4*b^4*d^7*abs(b) + 35*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*b^4*c^2*d^3*abs(b) - 70*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^8*a*b^3*c*d^4*abs(b
) + 35*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))
^8*a^2*b^2*d^5*abs(b) + 7*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^12*d^3*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)^7*b^4)