3.496 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^{9/2}} \, dx\)

Optimal. Leaf size=111 \[ -\frac{2 A (a+b x)^{7/2}}{7 a x^{7/2}}+2 b^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 b^2 B \sqrt{a+b x}}{\sqrt{x}}-\frac{2 B (a+b x)^{5/2}}{5 x^{5/2}}-\frac{2 b B (a+b x)^{3/2}}{3 x^{3/2}} \]

[Out]

(-2*b^2*B*Sqrt[a + b*x])/Sqrt[x] - (2*b*B*(a + b*x)^(3/2))/(3*x^(3/2)) - (2*B*(a
+ b*x)^(5/2))/(5*x^(5/2)) - (2*A*(a + b*x)^(7/2))/(7*a*x^(7/2)) + 2*b^(5/2)*B*A
rcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]]

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Rubi [A]  time = 0.107823, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 A (a+b x)^{7/2}}{7 a x^{7/2}}+2 b^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 b^2 B \sqrt{a+b x}}{\sqrt{x}}-\frac{2 B (a+b x)^{5/2}}{5 x^{5/2}}-\frac{2 b B (a+b x)^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b^2*B*Sqrt[a + b*x])/Sqrt[x] - (2*b*B*(a + b*x)^(3/2))/(3*x^(3/2)) - (2*B*(a
+ b*x)^(5/2))/(5*x^(5/2)) - (2*A*(a + b*x)^(7/2))/(7*a*x^(7/2)) + 2*b^(5/2)*B*A
rcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]]

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Rubi in Sympy [A]  time = 12.146, size = 107, normalized size = 0.96 \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{7 a x^{\frac{7}{2}}} + 2 B b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )} - \frac{2 B b^{2} \sqrt{a + b x}}{\sqrt{x}} - \frac{2 B b \left (a + b x\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} - \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(a + b*x)**(7/2)/(7*a*x**(7/2)) + 2*B*b**(5/2)*atanh(sqrt(a + b*x)/(sqrt(b)
*sqrt(x))) - 2*B*b**2*sqrt(a + b*x)/sqrt(x) - 2*B*b*(a + b*x)**(3/2)/(3*x**(3/2)
) - 2*B*(a + b*x)**(5/2)/(5*x**(5/2))

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Mathematica [A]  time = 0.216594, size = 107, normalized size = 0.96 \[ 2 b^{5/2} B \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{2 \sqrt{a+b x} \left (3 a^3 (5 A+7 B x)+a^2 b x (45 A+77 B x)+a b^2 x^2 (45 A+161 B x)+15 A b^3 x^3\right )}{105 a x^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b*x]*(15*A*b^3*x^3 + 3*a^3*(5*A + 7*B*x) + a^2*b*x*(45*A + 77*B*x)
+ a*b^2*x^2*(45*A + 161*B*x)))/(105*a*x^(7/2)) + 2*b^(5/2)*B*Log[b*Sqrt[x] + Sqr
t[b]*Sqrt[a + b*x]]

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Maple [B]  time = 0.021, size = 185, normalized size = 1.7 \[ -{\frac{1}{105\,a}\sqrt{bx+a} \left ( -105\,B{b}^{5/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) a{x}^{4}+30\,A{x}^{3}{b}^{3}\sqrt{x \left ( bx+a \right ) }+322\,B{x}^{3}a{b}^{2}\sqrt{x \left ( bx+a \right ) }+90\,A{x}^{2}a{b}^{2}\sqrt{x \left ( bx+a \right ) }+154\,B{x}^{2}{a}^{2}b\sqrt{x \left ( bx+a \right ) }+90\,Ax{a}^{2}b\sqrt{x \left ( bx+a \right ) }+42\,Bx{a}^{3}\sqrt{x \left ( bx+a \right ) }+30\,A{a}^{3}\sqrt{x \left ( bx+a \right ) } \right ){x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(b*x+a)^(1/2)/x^(7/2)*(-105*B*b^(5/2)*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)
+2*b*x+a)/b^(1/2))*a*x^4+30*A*x^3*b^3*(x*(b*x+a))^(1/2)+322*B*x^3*a*b^2*(x*(b*x+
a))^(1/2)+90*A*x^2*a*b^2*(x*(b*x+a))^(1/2)+154*B*x^2*a^2*b*(x*(b*x+a))^(1/2)+90*
A*x*a^2*b*(x*(b*x+a))^(1/2)+42*B*x*a^3*(x*(b*x+a))^(1/2)+30*A*a^3*(x*(b*x+a))^(1
/2))/a/(x*(b*x+a))^(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((B*x + A)*(b*x + a)^(5/2)/x^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228677, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B a b^{\frac{5}{2}} x^{4} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (15 \, A a^{3} +{\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} x^{3} +{\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} x^{2} + 3 \,{\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{105 \, a x^{4}}, \frac{2 \,{\left (105 \, B a \sqrt{-b} b^{2} x^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) -{\left (15 \, A a^{3} +{\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} x^{3} +{\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} x^{2} + 3 \,{\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{x}\right )}}{105 \, a x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/105*(105*B*a*b^(5/2)*x^4*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2
*(15*A*a^3 + (161*B*a*b^2 + 15*A*b^3)*x^3 + (77*B*a^2*b + 45*A*a*b^2)*x^2 + 3*(7
*B*a^3 + 15*A*a^2*b)*x)*sqrt(b*x + a)*sqrt(x))/(a*x^4), 2/105*(105*B*a*sqrt(-b)*
b^2*x^4*arctan(sqrt(b*x + a)/(sqrt(-b)*sqrt(x))) - (15*A*a^3 + (161*B*a*b^2 + 15
*A*b^3)*x^3 + (77*B*a^2*b + 45*A*a*b^2)*x^2 + 3*(7*B*a^3 + 15*A*a^2*b)*x)*sqrt(b
*x + a)*sqrt(x))/(a*x^4)]

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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+a)**(5/2)*(B*x+A)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError