3.462 \(\int \frac{x^{7/2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=95 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}-\frac{35 a \sqrt{x}}{4 b^4}-\frac{7 x^{5/2}}{4 b^2 (a+b x)}-\frac{x^{7/2}}{2 b (a+b x)^2}+\frac{35 x^{3/2}}{12 b^3} \]

[Out]

(-35*a*Sqrt[x])/(4*b^4) + (35*x^(3/2))/(12*b^3) - x^(7/2)/(2*b*(a + b*x)^2) - (7
*x^(5/2))/(4*b^2*(a + b*x)) + (35*a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*
b^(9/2))

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Rubi [A]  time = 0.0768414, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}-\frac{35 a \sqrt{x}}{4 b^4}-\frac{7 x^{5/2}}{4 b^2 (a+b x)}-\frac{x^{7/2}}{2 b (a+b x)^2}+\frac{35 x^{3/2}}{12 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/(a + b*x)^3,x]

[Out]

(-35*a*Sqrt[x])/(4*b^4) + (35*x^(3/2))/(12*b^3) - x^(7/2)/(2*b*(a + b*x)^2) - (7
*x^(5/2))/(4*b^2*(a + b*x)) + (35*a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*
b^(9/2))

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Rubi in Sympy [A]  time = 15.6919, size = 87, normalized size = 0.92 \[ \frac{35 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{9}{2}}} - \frac{35 a \sqrt{x}}{4 b^{4}} - \frac{x^{\frac{7}{2}}}{2 b \left (a + b x\right )^{2}} - \frac{7 x^{\frac{5}{2}}}{4 b^{2} \left (a + b x\right )} + \frac{35 x^{\frac{3}{2}}}{12 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)/(b*x+a)**3,x)

[Out]

35*a**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(9/2)) - 35*a*sqrt(x)/(4*b**4) -
x**(7/2)/(2*b*(a + b*x)**2) - 7*x**(5/2)/(4*b**2*(a + b*x)) + 35*x**(3/2)/(12*b
**3)

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Mathematica [A]  time = 0.0691787, size = 81, normalized size = 0.85 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2}}+\frac{\sqrt{x} \left (-105 a^3-175 a^2 b x-56 a b^2 x^2+8 b^3 x^3\right )}{12 b^4 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/(a + b*x)^3,x]

[Out]

(Sqrt[x]*(-105*a^3 - 175*a^2*b*x - 56*a*b^2*x^2 + 8*b^3*x^3))/(12*b^4*(a + b*x)^
2) + (35*a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(9/2))

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Maple [A]  time = 0.018, size = 79, normalized size = 0.8 \[{\frac{2}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-6\,{\frac{a\sqrt{x}}{{b}^{4}}}-{\frac{13\,{a}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{11\,{a}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{35\,{a}^{2}}{4\,{b}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(b*x+a)^3,x)

[Out]

2/3*x^(3/2)/b^3-6*a*x^(1/2)/b^4-13/4/b^3*a^2/(b*x+a)^2*x^(3/2)-11/4/b^4*a^3/(b*x
+a)^2*x^(1/2)+35/4/b^4*a^2/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))

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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(x^(7/2)/(b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236324, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt{x}}{24 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac{105 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt{x}}{12 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(b*x + a)^3,x, algorithm="fricas")

[Out]

[1/24*(105*(a*b^2*x^2 + 2*a^2*b*x + a^3)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(
-a/b) - a)/(b*x + a)) + 2*(8*b^3*x^3 - 56*a*b^2*x^2 - 175*a^2*b*x - 105*a^3)*sqr
t(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), 1/12*(105*(a*b^2*x^2 + 2*a^2*b*x + a^3)*s
qrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) + (8*b^3*x^3 - 56*a*b^2*x^2 - 175*a^2*b*x - 1
05*a^3)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]

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Sympy [A]  time = 34.5948, size = 746, normalized size = 7.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)/(b*x+a)**3,x)

[Out]

105*a**(129/2)*b**27*x**(63/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(12*a**63*b**(63/2)
*x**(63/2) + 36*a**62*b**(65/2)*x**(65/2) + 36*a**61*b**(67/2)*x**(67/2) + 12*a*
*60*b**(69/2)*x**(69/2)) + 315*a**(127/2)*b**28*x**(65/2)*atan(sqrt(b)*sqrt(x)/s
qrt(a))/(12*a**63*b**(63/2)*x**(63/2) + 36*a**62*b**(65/2)*x**(65/2) + 36*a**61*
b**(67/2)*x**(67/2) + 12*a**60*b**(69/2)*x**(69/2)) + 315*a**(125/2)*b**29*x**(6
7/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(12*a**63*b**(63/2)*x**(63/2) + 36*a**62*b**(
65/2)*x**(65/2) + 36*a**61*b**(67/2)*x**(67/2) + 12*a**60*b**(69/2)*x**(69/2)) +
105*a**(123/2)*b**30*x**(69/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(12*a**63*b**(63/2
)*x**(63/2) + 36*a**62*b**(65/2)*x**(65/2) + 36*a**61*b**(67/2)*x**(67/2) + 12*a
**60*b**(69/2)*x**(69/2)) - 105*a**64*b**(55/2)*x**32/(12*a**63*b**(63/2)*x**(63
/2) + 36*a**62*b**(65/2)*x**(65/2) + 36*a**61*b**(67/2)*x**(67/2) + 12*a**60*b**
(69/2)*x**(69/2)) - 280*a**63*b**(57/2)*x**33/(12*a**63*b**(63/2)*x**(63/2) + 36
*a**62*b**(65/2)*x**(65/2) + 36*a**61*b**(67/2)*x**(67/2) + 12*a**60*b**(69/2)*x
**(69/2)) - 231*a**62*b**(59/2)*x**34/(12*a**63*b**(63/2)*x**(63/2) + 36*a**62*b
**(65/2)*x**(65/2) + 36*a**61*b**(67/2)*x**(67/2) + 12*a**60*b**(69/2)*x**(69/2)
) - 48*a**61*b**(61/2)*x**35/(12*a**63*b**(63/2)*x**(63/2) + 36*a**62*b**(65/2)*
x**(65/2) + 36*a**61*b**(67/2)*x**(67/2) + 12*a**60*b**(69/2)*x**(69/2)) + 8*a**
60*b**(63/2)*x**36/(12*a**63*b**(63/2)*x**(63/2) + 36*a**62*b**(65/2)*x**(65/2)
+ 36*a**61*b**(67/2)*x**(67/2) + 12*a**60*b**(69/2)*x**(69/2))

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GIAC/XCAS [A]  time = 0.205487, size = 104, normalized size = 1.09 \[ \frac{35 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{4}} - \frac{13 \, a^{2} b x^{\frac{3}{2}} + 11 \, a^{3} \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{4}} + \frac{2 \,{\left (b^{6} x^{\frac{3}{2}} - 9 \, a b^{5} \sqrt{x}\right )}}{3 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(b*x + a)^3,x, algorithm="giac")

[Out]

35/4*a^2*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) - 1/4*(13*a^2*b*x^(3/2) + 1
1*a^3*sqrt(x))/((b*x + a)^2*b^4) + 2/3*(b^6*x^(3/2) - 9*a*b^5*sqrt(x))/b^9