3.490 \(\int x^{3/2} (a+b x)^{5/2} (A+B x) \, dx\)

Optimal. Leaf size=225 \[ \frac{a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{7/2}}-\frac{a^4 \sqrt{x} \sqrt{a+b x} (12 A b-5 a B)}{512 b^3}+\frac{a^3 x^{3/2} \sqrt{a+b x} (12 A b-5 a B)}{768 b^2}+\frac{a^2 x^{5/2} \sqrt{a+b x} (12 A b-5 a B)}{192 b}+\frac{a x^{5/2} (a+b x)^{3/2} (12 A b-5 a B)}{96 b}+\frac{x^{5/2} (a+b x)^{5/2} (12 A b-5 a B)}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b} \]

[Out]

-(a^4*(12*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(512*b^3) + (a^3*(12*A*b - 5*a*B)*
x^(3/2)*Sqrt[a + b*x])/(768*b^2) + (a^2*(12*A*b - 5*a*B)*x^(5/2)*Sqrt[a + b*x])/
(192*b) + (a*(12*A*b - 5*a*B)*x^(5/2)*(a + b*x)^(3/2))/(96*b) + ((12*A*b - 5*a*B
)*x^(5/2)*(a + b*x)^(5/2))/(60*b) + (B*x^(5/2)*(a + b*x)^(7/2))/(6*b) + (a^5*(12
*A*b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(512*b^(7/2))

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Rubi [A]  time = 0.268074, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{7/2}}-\frac{a^4 \sqrt{x} \sqrt{a+b x} (12 A b-5 a B)}{512 b^3}+\frac{a^3 x^{3/2} \sqrt{a+b x} (12 A b-5 a B)}{768 b^2}+\frac{a^2 x^{5/2} \sqrt{a+b x} (12 A b-5 a B)}{192 b}+\frac{a x^{5/2} (a+b x)^{3/2} (12 A b-5 a B)}{96 b}+\frac{x^{5/2} (a+b x)^{5/2} (12 A b-5 a B)}{60 b}+\frac{B x^{5/2} (a+b x)^{7/2}}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

-(a^4*(12*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(512*b^3) + (a^3*(12*A*b - 5*a*B)*
x^(3/2)*Sqrt[a + b*x])/(768*b^2) + (a^2*(12*A*b - 5*a*B)*x^(5/2)*Sqrt[a + b*x])/
(192*b) + (a*(12*A*b - 5*a*B)*x^(5/2)*(a + b*x)^(3/2))/(96*b) + ((12*A*b - 5*a*B
)*x^(5/2)*(a + b*x)^(5/2))/(60*b) + (B*x^(5/2)*(a + b*x)^(7/2))/(6*b) + (a^5*(12
*A*b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(512*b^(7/2))

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Rubi in Sympy [A]  time = 26.2375, size = 214, normalized size = 0.95 \[ \frac{B x^{\frac{5}{2}} \left (a + b x\right )^{\frac{7}{2}}}{6 b} + \frac{a^{5} \left (12 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{512 b^{\frac{7}{2}}} + \frac{a^{4} \sqrt{x} \sqrt{a + b x} \left (12 A b - 5 B a\right )}{512 b^{3}} + \frac{a^{3} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (12 A b - 5 B a\right )}{768 b^{3}} + \frac{a^{2} \sqrt{x} \left (a + b x\right )^{\frac{5}{2}} \left (12 A b - 5 B a\right )}{960 b^{3}} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{7}{2}} \left (12 A b - 5 B a\right )}{160 b^{3}} + \frac{x^{\frac{3}{2}} \left (a + b x\right )^{\frac{7}{2}} \left (12 A b - 5 B a\right )}{60 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

B*x**(5/2)*(a + b*x)**(7/2)/(6*b) + a**5*(12*A*b - 5*B*a)*atanh(sqrt(b)*sqrt(x)/
sqrt(a + b*x))/(512*b**(7/2)) + a**4*sqrt(x)*sqrt(a + b*x)*(12*A*b - 5*B*a)/(512
*b**3) + a**3*sqrt(x)*(a + b*x)**(3/2)*(12*A*b - 5*B*a)/(768*b**3) + a**2*sqrt(x
)*(a + b*x)**(5/2)*(12*A*b - 5*B*a)/(960*b**3) - a*sqrt(x)*(a + b*x)**(7/2)*(12*
A*b - 5*B*a)/(160*b**3) + x**(3/2)*(a + b*x)**(7/2)*(12*A*b - 5*B*a)/(60*b**2)

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Mathematica [A]  time = 0.195294, size = 157, normalized size = 0.7 \[ \frac{15 a^5 (12 A b-5 a B) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (75 a^5 B-10 a^4 b (18 A+5 B x)+40 a^3 b^2 x (3 A+B x)+48 a^2 b^3 x^2 (62 A+45 B x)+64 a b^4 x^3 (63 A+50 B x)+256 b^5 x^4 (6 A+5 B x)\right )}{7680 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(75*a^5*B + 40*a^3*b^2*x*(3*A + B*x) + 256*b^5*x^
4*(6*A + 5*B*x) - 10*a^4*b*(18*A + 5*B*x) + 48*a^2*b^3*x^2*(62*A + 45*B*x) + 64*
a*b^4*x^3*(63*A + 50*B*x)) + 15*a^5*(12*A*b - 5*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqr
t[a + b*x]])/(7680*b^(7/2))

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Maple [A]  time = 0.02, size = 302, normalized size = 1.3 \[{\frac{1}{15360}\sqrt{x}\sqrt{bx+a} \left ( 2560\,B{x}^{5}{b}^{11/2}\sqrt{x \left ( bx+a \right ) }+3072\,A{x}^{4}{b}^{11/2}\sqrt{x \left ( bx+a \right ) }+6400\,B{x}^{4}a{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+8064\,A{x}^{3}a{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+4320\,B{x}^{3}{a}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+5952\,A{x}^{2}{a}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+80\,B{x}^{2}{a}^{3}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+240\,{a}^{3}\sqrt{x \left ( bx+a \right ) }xA{b}^{5/2}-100\,{a}^{4}\sqrt{x \left ( bx+a \right ) }xB{b}^{3/2}+180\,{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) Ab-360\,{a}^{4}\sqrt{x \left ( bx+a \right ) }A{b}^{3/2}-75\,{a}^{6}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) B+150\,{a}^{5}\sqrt{x \left ( bx+a \right ) }B\sqrt{b} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(b*x+a)^(5/2)*(B*x+A),x)

[Out]

1/15360*x^(1/2)*(b*x+a)^(1/2)/b^(7/2)*(2560*B*x^5*b^(11/2)*(x*(b*x+a))^(1/2)+307
2*A*x^4*b^(11/2)*(x*(b*x+a))^(1/2)+6400*B*x^4*a*b^(9/2)*(x*(b*x+a))^(1/2)+8064*A
*x^3*a*b^(9/2)*(x*(b*x+a))^(1/2)+4320*B*x^3*a^2*b^(7/2)*(x*(b*x+a))^(1/2)+5952*A
*x^2*a^2*b^(7/2)*(x*(b*x+a))^(1/2)+80*B*x^2*a^3*b^(5/2)*(x*(b*x+a))^(1/2)+240*a^
3*(x*(b*x+a))^(1/2)*x*A*b^(5/2)-100*a^4*(x*(b*x+a))^(1/2)*x*B*b^(3/2)+180*a^5*ln
(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*A*b-360*a^4*(x*(b*x+a))^(1/2
)*A*b^(3/2)-75*a^6*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*B+150*a
^5*(x*(b*x+a))^(1/2)*B*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)*(b*x + a)^(5/2)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242619, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, B b^{5} x^{5} + 75 \, B a^{5} - 180 \, A a^{4} b + 128 \,{\left (25 \, B a b^{4} + 12 \, A b^{5}\right )} x^{4} + 144 \,{\left (15 \, B a^{2} b^{3} + 28 \, A a b^{4}\right )} x^{3} + 8 \,{\left (5 \, B a^{3} b^{2} + 372 \, A a^{2} b^{3}\right )} x^{2} - 10 \,{\left (5 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 15 \,{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{15360 \, b^{\frac{7}{2}}}, \frac{{\left (1280 \, B b^{5} x^{5} + 75 \, B a^{5} - 180 \, A a^{4} b + 128 \,{\left (25 \, B a b^{4} + 12 \, A b^{5}\right )} x^{4} + 144 \,{\left (15 \, B a^{2} b^{3} + 28 \, A a b^{4}\right )} x^{3} + 8 \,{\left (5 \, B a^{3} b^{2} + 372 \, A a^{2} b^{3}\right )} x^{2} - 10 \,{\left (5 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} - 15 \,{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{7680 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*x^(3/2),x, algorithm="fricas")

[Out]

[1/15360*(2*(1280*B*b^5*x^5 + 75*B*a^5 - 180*A*a^4*b + 128*(25*B*a*b^4 + 12*A*b^
5)*x^4 + 144*(15*B*a^2*b^3 + 28*A*a*b^4)*x^3 + 8*(5*B*a^3*b^2 + 372*A*a^2*b^3)*x
^2 - 10*(5*B*a^4*b - 12*A*a^3*b^2)*x)*sqrt(b*x + a)*sqrt(b)*sqrt(x) - 15*(5*B*a^
6 - 12*A*a^5*b)*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)))/b^(7/2), 1
/7680*((1280*B*b^5*x^5 + 75*B*a^5 - 180*A*a^4*b + 128*(25*B*a*b^4 + 12*A*b^5)*x^
4 + 144*(15*B*a^2*b^3 + 28*A*a*b^4)*x^3 + 8*(5*B*a^3*b^2 + 372*A*a^2*b^3)*x^2 -
10*(5*B*a^4*b - 12*A*a^3*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) - 15*(5*B*a^6 -
12*A*a^5*b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^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(x**(3/2)*(b*x+a)**(5/2)*(B*x+A),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)*(b*x + a)^(5/2)*x^(3/2),x, algorithm="giac")

[Out]

Timed out