3.1478 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{4 d (c+d x)^{5/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

[Out]

(-2*(c + d*x)^(5/2))/(7*(b*c - a*d)*(a + b*x)^(7/2)) + (4*d*(c + d*x)^(5/2))/(35
*(b*c - a*d)^2*(a + b*x)^(5/2))

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Rubi [A]  time = 0.0485677, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{4 d (c+d x)^{5/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 (c+d x)^{5/2}}{7 (a+b x)^{7/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2)/(a + b*x)^(9/2),x]

[Out]

(-2*(c + d*x)^(5/2))/(7*(b*c - a*d)*(a + b*x)^(7/2)) + (4*d*(c + d*x)^(5/2))/(35
*(b*c - a*d)^2*(a + b*x)^(5/2))

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Rubi in Sympy [A]  time = 8.0947, size = 56, normalized size = 0.85 \[ \frac{4 d \left (c + d x\right )^{\frac{5}{2}}}{35 \left (a + b x\right )^{\frac{5}{2}} \left (a d - b c\right )^{2}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{7 \left (a + b x\right )^{\frac{7}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2)/(b*x+a)**(9/2),x)

[Out]

4*d*(c + d*x)**(5/2)/(35*(a + b*x)**(5/2)*(a*d - b*c)**2) + 2*(c + d*x)**(5/2)/(
7*(a + b*x)**(7/2)*(a*d - b*c))

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Mathematica [A]  time = 0.0981874, size = 46, normalized size = 0.7 \[ \frac{2 (c+d x)^{5/2} (7 a d-5 b c+2 b d x)}{35 (a+b x)^{7/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2)/(a + b*x)^(9/2),x]

[Out]

(2*(c + d*x)^(5/2)*(-5*b*c + 7*a*d + 2*b*d*x))/(35*(b*c - a*d)^2*(a + b*x)^(7/2)
)

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Maple [A]  time = 0.008, size = 54, normalized size = 0.8 \[{\frac{4\,bdx+14\,ad-10\,bc}{35\,{a}^{2}{d}^{2}-70\,abcd+35\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)^(9/2),x)

[Out]

2/35*(d*x+c)^(5/2)*(2*b*d*x+7*a*d-5*b*c)/(b*x+a)^(7/2)/(a^2*d^2-2*a*b*c*d+b^2*c^
2)

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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)^(3/2)/(b*x + a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.687278, size = 317, normalized size = 4.8 \[ \frac{2 \,{\left (2 \, b d^{3} x^{3} - 5 \, b c^{3} + 7 \, a c^{2} d -{\left (b c d^{2} - 7 \, a d^{3}\right )} x^{2} - 2 \,{\left (4 \, b c^{2} d - 7 \, a c d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{35 \,{\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 4 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{3} + 6 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{2} + 4 \,{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(b*x + a)^(9/2),x, algorithm="fricas")

[Out]

2/35*(2*b*d^3*x^3 - 5*b*c^3 + 7*a*c^2*d - (b*c*d^2 - 7*a*d^3)*x^2 - 2*(4*b*c^2*d
- 7*a*c*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^
2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^4 + 4*(a*b^5*c^2 - 2*a^2*b^4*c*d + a
^3*b^3*d^2)*x^3 + 6*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x^2 + 4*(a^3*b^3
*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*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)**(3/2)/(b*x+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.357325, size = 1, normalized size = 0.02 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(b*x + a)^(9/2),x, algorithm="giac")

[Out]

Done