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

Optimal. Leaf size=192 \[ \frac{7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{11/2}}-\frac{7 a^3 \sqrt{x} \sqrt{a+b x} (10 A b-9 a B)}{128 b^5}+\frac{7 a^2 x^{3/2} \sqrt{a+b x} (10 A b-9 a B)}{192 b^4}-\frac{7 a x^{5/2} \sqrt{a+b x} (10 A b-9 a B)}{240 b^3}+\frac{x^{7/2} \sqrt{a+b x} (10 A b-9 a B)}{40 b^2}+\frac{B x^{9/2} \sqrt{a+b x}}{5 b} \]

[Out]

(-7*a^3*(10*A*b - 9*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^5) + (7*a^2*(10*A*b - 9*a
*B)*x^(3/2)*Sqrt[a + b*x])/(192*b^4) - (7*a*(10*A*b - 9*a*B)*x^(5/2)*Sqrt[a + b*
x])/(240*b^3) + ((10*A*b - 9*a*B)*x^(7/2)*Sqrt[a + b*x])/(40*b^2) + (B*x^(9/2)*S
qrt[a + b*x])/(5*b) + (7*a^4*(10*A*b - 9*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a +
b*x]])/(128*b^(11/2))

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Rubi [A]  time = 0.247322, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{11/2}}-\frac{7 a^3 \sqrt{x} \sqrt{a+b x} (10 A b-9 a B)}{128 b^5}+\frac{7 a^2 x^{3/2} \sqrt{a+b x} (10 A b-9 a B)}{192 b^4}-\frac{7 a x^{5/2} \sqrt{a+b x} (10 A b-9 a B)}{240 b^3}+\frac{x^{7/2} \sqrt{a+b x} (10 A b-9 a B)}{40 b^2}+\frac{B x^{9/2} \sqrt{a+b x}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(-7*a^3*(10*A*b - 9*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^5) + (7*a^2*(10*A*b - 9*a
*B)*x^(3/2)*Sqrt[a + b*x])/(192*b^4) - (7*a*(10*A*b - 9*a*B)*x^(5/2)*Sqrt[a + b*
x])/(240*b^3) + ((10*A*b - 9*a*B)*x^(7/2)*Sqrt[a + b*x])/(40*b^2) + (B*x^(9/2)*S
qrt[a + b*x])/(5*b) + (7*a^4*(10*A*b - 9*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a +
b*x]])/(128*b^(11/2))

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Rubi in Sympy [A]  time = 22.29, size = 189, normalized size = 0.98 \[ \frac{B x^{\frac{9}{2}} \sqrt{a + b x}}{5 b} + \frac{7 a^{4} \left (10 A b - 9 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{128 b^{\frac{11}{2}}} - \frac{7 a^{3} \sqrt{x} \sqrt{a + b x} \left (10 A b - 9 B a\right )}{128 b^{5}} + \frac{7 a^{2} x^{\frac{3}{2}} \sqrt{a + b x} \left (10 A b - 9 B a\right )}{192 b^{4}} - \frac{7 a x^{\frac{5}{2}} \sqrt{a + b x} \left (10 A b - 9 B a\right )}{240 b^{3}} + \frac{x^{\frac{7}{2}} \sqrt{a + b x} \left (10 A b - 9 B a\right )}{40 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x+A)/(b*x+a)**(1/2),x)

[Out]

B*x**(9/2)*sqrt(a + b*x)/(5*b) + 7*a**4*(10*A*b - 9*B*a)*atanh(sqrt(a + b*x)/(sq
rt(b)*sqrt(x)))/(128*b**(11/2)) - 7*a**3*sqrt(x)*sqrt(a + b*x)*(10*A*b - 9*B*a)/
(128*b**5) + 7*a**2*x**(3/2)*sqrt(a + b*x)*(10*A*b - 9*B*a)/(192*b**4) - 7*a*x**
(5/2)*sqrt(a + b*x)*(10*A*b - 9*B*a)/(240*b**3) + x**(7/2)*sqrt(a + b*x)*(10*A*b
- 9*B*a)/(40*b**2)

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Mathematica [A]  time = 0.182079, size = 139, normalized size = 0.72 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (945 a^4 B-210 a^3 b (5 A+3 B x)+28 a^2 b^2 x (25 A+18 B x)-16 a b^3 x^2 (35 A+27 B x)+96 b^4 x^3 (5 A+4 B x)\right )-105 a^4 (9 a B-10 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{1920 b^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(945*a^4*B - 210*a^3*b*(5*A + 3*B*x) + 96*b^4*x^3
*(5*A + 4*B*x) + 28*a^2*b^2*x*(25*A + 18*B*x) - 16*a*b^3*x^2*(35*A + 27*B*x)) -
105*a^4*(-10*A*b + 9*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(1920*b^(11/2)
)

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Maple [A]  time = 0.027, size = 260, normalized size = 1.4 \[{\frac{1}{3840}\sqrt{x}\sqrt{bx+a} \left ( 768\,B{x}^{4}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+960\,A{x}^{3}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-864\,B{x}^{3}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-1120\,A{x}^{2}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+1008\,B{x}^{2}{a}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+1400\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-1260\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+1050\,A{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-2100\,A{a}^{3}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-945\,B{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +1890\,B{a}^{4}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)/(b*x+a)^(1/2),x)

[Out]

1/3840*x^(1/2)*(b*x+a)^(1/2)/b^(11/2)*(768*B*x^4*b^(9/2)*(x*(b*x+a))^(1/2)+960*A
*x^3*b^(9/2)*(x*(b*x+a))^(1/2)-864*B*x^3*a*b^(7/2)*(x*(b*x+a))^(1/2)-1120*A*x^2*
a*b^(7/2)*(x*(b*x+a))^(1/2)+1008*B*x^2*a^2*b^(5/2)*(x*(b*x+a))^(1/2)+1400*A*a^2*
(x*(b*x+a))^(1/2)*x*b^(5/2)-1260*B*a^3*(x*(b*x+a))^(1/2)*x*b^(3/2)+1050*A*a^4*ln
(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-2100*A*a^3*(x*(b*x+a))^(1/
2)*b^(3/2)-945*B*a^5*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+1890*
B*a^4*(x*(b*x+a))^(1/2)*b^(1/2))/(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)*x^(7/2)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249897, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (384 \, B b^{4} x^{4} + 945 \, B a^{4} - 1050 \, A a^{3} b - 48 \,{\left (9 \, B a b^{3} - 10 \, A b^{4}\right )} x^{3} + 56 \,{\left (9 \, B a^{2} b^{2} - 10 \, A a b^{3}\right )} x^{2} - 70 \,{\left (9 \, B a^{3} b - 10 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 105 \,{\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{3840 \, b^{\frac{11}{2}}}, \frac{{\left (384 \, B b^{4} x^{4} + 945 \, B a^{4} - 1050 \, A a^{3} b - 48 \,{\left (9 \, B a b^{3} - 10 \, A b^{4}\right )} x^{3} + 56 \,{\left (9 \, B a^{2} b^{2} - 10 \, A a b^{3}\right )} x^{2} - 70 \,{\left (9 \, B a^{3} b - 10 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} - 105 \,{\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{1920 \, \sqrt{-b} b^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3840*(2*(384*B*b^4*x^4 + 945*B*a^4 - 1050*A*a^3*b - 48*(9*B*a*b^3 - 10*A*b^4)
*x^3 + 56*(9*B*a^2*b^2 - 10*A*a*b^3)*x^2 - 70*(9*B*a^3*b - 10*A*a^2*b^2)*x)*sqrt
(b*x + a)*sqrt(b)*sqrt(x) - 105*(9*B*a^5 - 10*A*a^4*b)*log(2*sqrt(b*x + a)*b*sqr
t(x) + (2*b*x + a)*sqrt(b)))/b^(11/2), 1/1920*((384*B*b^4*x^4 + 945*B*a^4 - 1050
*A*a^3*b - 48*(9*B*a*b^3 - 10*A*b^4)*x^3 + 56*(9*B*a^2*b^2 - 10*A*a*b^3)*x^2 - 7
0*(9*B*a^3*b - 10*A*a^2*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) - 105*(9*B*a^5 -
10*A*a^4*b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^5)]

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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(x**(7/2)*(B*x+A)/(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [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)*x^(7/2)/sqrt(b*x + a),x, algorithm="giac")

[Out]

Timed out