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

Optimal. Leaf size=89 \[ -\frac{2 A (a+b x)^{5/2}}{5 a x^{5/2}}+2 b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b B \sqrt{a+b x}}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0901926, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 A (a+b x)^{5/2}}{5 a x^{5/2}}+2 b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b B \sqrt{a+b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.42584, size = 85, normalized size = 0.96 \[ - \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{5 a x^{\frac{5}{2}}} + 2 B b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )} - \frac{2 B b \sqrt{a + b x}}{\sqrt{x}} - \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.146091, size = 89, normalized size = 1. \[ 2 b^{3/2} B \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{2 \sqrt{a+b x} \left (a^2 (3 A+5 B x)+2 a b x (3 A+10 B x)+3 A b^2 x^2\right )}{15 a x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + b*x]*(3*A*b^2*x^2 + a^2*(3*A + 5*B*x) + 2*a*b*x*(3*A + 10*B*x)))/(1
5*a*x^(5/2)) + 2*b^(3/2)*B*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]]

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Maple [B]  time = 0.02, size = 143, normalized size = 1.6 \[ -{\frac{1}{15\,a}\sqrt{bx+a} \left ( -15\,B{b}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) a{x}^{3}+6\,A{x}^{2}{b}^{2}\sqrt{x \left ( bx+a \right ) }+40\,B{x}^{2}ab\sqrt{x \left ( bx+a \right ) }+12\,Axab\sqrt{x \left ( bx+a \right ) }+10\,Bx{a}^{2}\sqrt{x \left ( bx+a \right ) }+6\,A{a}^{2}\sqrt{x \left ( bx+a \right ) } \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(b*x+a)^(1/2)/x^(5/2)*(-15*B*b^(3/2)*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2
*b*x+a)/b^(1/2))*a*x^3+6*A*x^2*b^2*(x*(b*x+a))^(1/2)+40*B*x^2*a*b*(x*(b*x+a))^(1
/2)+12*A*x*a*b*(x*(b*x+a))^(1/2)+10*B*x*a^2*(x*(b*x+a))^(1/2)+6*A*a^2*(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)^(3/2)/x^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247758, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B a b^{\frac{3}{2}} x^{3} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (3 \, A a^{2} +{\left (20 \, B a b + 3 \, A b^{2}\right )} x^{2} +{\left (5 \, B a^{2} + 6 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{15 \, a x^{3}}, \frac{2 \,{\left (15 \, B a \sqrt{-b} b x^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) -{\left (3 \, A a^{2} +{\left (20 \, B a b + 3 \, A b^{2}\right )} x^{2} +{\left (5 \, B a^{2} + 6 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{x}\right )}}{15 \, a x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15*(15*B*a*b^(3/2)*x^3*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(
3*A*a^2 + (20*B*a*b + 3*A*b^2)*x^2 + (5*B*a^2 + 6*A*a*b)*x)*sqrt(b*x + a)*sqrt(x
))/(a*x^3), 2/15*(15*B*a*sqrt(-b)*b*x^3*arctan(sqrt(b*x + a)/(sqrt(-b)*sqrt(x)))
- (3*A*a^2 + (20*B*a*b + 3*A*b^2)*x^2 + (5*B*a^2 + 6*A*a*b)*x)*sqrt(b*x + a)*sq
rt(x))/(a*x^3)]

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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)**(3/2)*(B*x+A)/x**(7/2),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)^(3/2)/x^(7/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError