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

Optimal. Leaf size=86 \[ -2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+2 a^2 A \sqrt{a+b x}+\frac{2}{5} A (a+b x)^{5/2}+\frac{2}{3} a A (a+b x)^{3/2}+\frac{2 B (a+b x)^{7/2}}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.107123, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+2 a^2 A \sqrt{a+b x}+\frac{2}{5} A (a+b x)^{5/2}+\frac{2}{3} a A (a+b x)^{3/2}+\frac{2 B (a+b x)^{7/2}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.0702, size = 82, normalized size = 0.95 \[ - 2 A a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 A a^{2} \sqrt{a + b x} + \frac{2 A a \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{2 B \left (a + b x\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.136087, size = 91, normalized size = 1.06 \[ \frac{2 \sqrt{a+b x} \left (15 a^3 B+a^2 b (161 A+45 B x)+a b^2 x (77 A+45 B x)+3 b^3 x^2 (7 A+5 B x)\right )}{105 b}-2 a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 72, normalized size = 0.8 \[ 2\,{\frac{1}{b} \left ( 1/7\,B \left ( bx+a \right ) ^{7/2}+1/5\,Ab \left ( bx+a \right ) ^{5/2}+1/3\,Aab \left ( bx+a \right ) ^{3/2}+{a}^{2}bA\sqrt{bx+a}-A{a}^{5/2}b{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x,x)

[Out]

2/b*(1/7*B*(b*x+a)^(7/2)+1/5*A*b*(b*x+a)^(5/2)+1/3*A*a*b*(b*x+a)^(3/2)+a^2*b*A*(
b*x+a)^(1/2)-A*a^(5/2)*b*arctanh((b*x+a)^(1/2)/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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224334, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, A a^{\frac{5}{2}} b \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (15 \, B b^{3} x^{3} + 15 \, B a^{3} + 161 \, A a^{2} b + 3 \,{\left (15 \, B a b^{2} + 7 \, A b^{3}\right )} x^{2} +{\left (45 \, B a^{2} b + 77 \, A a b^{2}\right )} x\right )} \sqrt{b x + a}}{105 \, b}, -\frac{2 \,{\left (105 \, A \sqrt{-a} a^{2} b \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (15 \, B b^{3} x^{3} + 15 \, B a^{3} + 161 \, A a^{2} b + 3 \,{\left (15 \, B a b^{2} + 7 \, A b^{3}\right )} x^{2} +{\left (45 \, B a^{2} b + 77 \, A a b^{2}\right )} x\right )} \sqrt{b x + a}\right )}}{105 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 18.3216, size = 144, normalized size = 1.67 \[ - 2 A a^{3} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 A a^{2} \sqrt{a + b x} + \frac{2 A a \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{2 B \left (a + b x\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**3*Piecewise((-atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqr
t(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (a < a + b*x)), (atanh(sqrt(a + b*x)/sqr
t(a))/sqrt(a), (-a < 0) & (a > a + b*x))) + 2*A*a**2*sqrt(a + b*x) + 2*A*a*(a +
b*x)**(3/2)/3 + 2*A*(a + b*x)**(5/2)/5 + 2*B*(a + b*x)**(7/2)/(7*b)

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GIAC/XCAS [A]  time = 0.214688, size = 119, normalized size = 1.38 \[ \frac{2 \, A a^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} B b^{6} + 21 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{7} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{7} + 105 \, \sqrt{b x + a} A a^{2} b^{7}\right )}}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a^3*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 2/105*(15*(b*x + a)^(7/2)*B*b^
6 + 21*(b*x + a)^(5/2)*A*b^7 + 35*(b*x + a)^(3/2)*A*a*b^7 + 105*sqrt(b*x + a)*A*
a^2*b^7)/b^7