3.409 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^7} \, dx\)

Optimal. Leaf size=208 \[ \frac{b^5 (5 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{7/2}}-\frac{b^4 \sqrt{a+b x} (5 A b-12 a B)}{512 a^3 x}+\frac{b^3 \sqrt{a+b x} (5 A b-12 a B)}{768 a^2 x^2}+\frac{b^2 \sqrt{a+b x} (5 A b-12 a B)}{192 a x^3}+\frac{(a+b x)^{5/2} (5 A b-12 a B)}{60 a x^5}+\frac{b (a+b x)^{3/2} (5 A b-12 a B)}{96 a x^4}-\frac{A (a+b x)^{7/2}}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.280252, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{b^5 (5 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{7/2}}-\frac{b^4 \sqrt{a+b x} (5 A b-12 a B)}{512 a^3 x}+\frac{b^3 \sqrt{a+b x} (5 A b-12 a B)}{768 a^2 x^2}+\frac{b^2 \sqrt{a+b x} (5 A b-12 a B)}{192 a x^3}+\frac{(a+b x)^{5/2} (5 A b-12 a B)}{60 a x^5}+\frac{b (a+b x)^{3/2} (5 A b-12 a B)}{96 a x^4}-\frac{A (a+b x)^{7/2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 26.0781, size = 192, normalized size = 0.92 \[ - \frac{A \left (a + b x\right )^{\frac{7}{2}}}{6 a x^{6}} + \frac{b^{2} \sqrt{a + b x} \left (5 A b - 12 B a\right )}{192 a x^{3}} + \frac{b \left (a + b x\right )^{\frac{3}{2}} \left (5 A b - 12 B a\right )}{96 a x^{4}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (5 A b - 12 B a\right )}{60 a x^{5}} + \frac{b^{3} \sqrt{a + b x} \left (5 A b - 12 B a\right )}{768 a^{2} x^{2}} - \frac{b^{4} \sqrt{a + b x} \left (5 A b - 12 B a\right )}{512 a^{3} x} + \frac{b^{5} \left (5 A b - 12 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{512 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**7,x)

[Out]

-A*(a + b*x)**(7/2)/(6*a*x**6) + b**2*sqrt(a + b*x)*(5*A*b - 12*B*a)/(192*a*x**3
) + b*(a + b*x)**(3/2)*(5*A*b - 12*B*a)/(96*a*x**4) + (a + b*x)**(5/2)*(5*A*b -
12*B*a)/(60*a*x**5) + b**3*sqrt(a + b*x)*(5*A*b - 12*B*a)/(768*a**2*x**2) - b**4
*sqrt(a + b*x)*(5*A*b - 12*B*a)/(512*a**3*x) + b**5*(5*A*b - 12*B*a)*atanh(sqrt(
a + b*x)/sqrt(a))/(512*a**(7/2))

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Mathematica [A]  time = 0.248139, size = 148, normalized size = 0.71 \[ \frac{b^5 (5 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{7/2}}-\frac{\sqrt{a+b x} \left (256 a^5 (5 A+6 B x)+64 a^4 b x (50 A+63 B x)+48 a^3 b^2 x^2 (45 A+62 B x)+40 a^2 b^3 x^3 (A+3 B x)-10 a b^4 x^4 (5 A+18 B x)+75 A b^5 x^5\right )}{7680 a^3 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^7,x]

[Out]

-(Sqrt[a + b*x]*(75*A*b^5*x^5 + 40*a^2*b^3*x^3*(A + 3*B*x) + 256*a^5*(5*A + 6*B*
x) - 10*a*b^4*x^4*(5*A + 18*B*x) + 48*a^3*b^2*x^2*(45*A + 62*B*x) + 64*a^4*b*x*(
50*A + 63*B*x)))/(7680*a^3*x^6) + (b^5*(5*A*b - 12*a*B)*ArcTanh[Sqrt[a + b*x]/Sq
rt[a]])/(512*a^(7/2))

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Maple [A]  time = 0.023, size = 161, normalized size = 0.8 \[ 2\,{b}^{5} \left ({\frac{1}{{x}^{6}{b}^{6}} \left ( -{\frac{ \left ( 5\,Ab-12\,Ba \right ) \left ( bx+a \right ) ^{11/2}}{1024\,{a}^{3}}}+{\frac{ \left ( 85\,Ab-204\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{3072\,{a}^{2}}}-{\frac{ \left ( 165\,Ab+116\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{2560\,a}}+ \left ( -{\frac{33\,Ab}{512}}+{\frac{99\,Ba}{640}} \right ) \left ( bx+a \right ) ^{5/2}+{\frac{17\,a \left ( 5\,Ab-12\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{3072}}-{\frac{{a}^{2} \left ( 5\,Ab-12\,Ba \right ) \sqrt{bx+a}}{1024}} \right ) }+{\frac{5\,Ab-12\,Ba}{1024\,{a}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^7,x)

[Out]

2*b^5*((-1/1024*(5*A*b-12*B*a)/a^3*(b*x+a)^(11/2)+17/3072/a^2*(5*A*b-12*B*a)*(b*
x+a)^(9/2)-1/2560*(165*A*b+116*B*a)/a*(b*x+a)^(7/2)+(-33/512*A*b+99/640*B*a)*(b*
x+a)^(5/2)+17/3072*a*(5*A*b-12*B*a)*(b*x+a)^(3/2)-1/1024*a^2*(5*A*b-12*B*a)*(b*x
+a)^(1/2))/x^6/b^6+1/1024*(5*A*b-12*B*a)/a^(7/2)*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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219935, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (12 \, B a b^{5} - 5 \, A b^{6}\right )} x^{6} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (1280 \, A a^{5} - 15 \,{\left (12 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 10 \,{\left (12 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 8 \,{\left (372 \, B a^{3} b^{2} + 5 \, A a^{2} b^{3}\right )} x^{3} + 144 \,{\left (28 \, B a^{4} b + 15 \, A a^{3} b^{2}\right )} x^{2} + 128 \,{\left (12 \, B a^{5} + 25 \, A a^{4} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{15360 \, a^{\frac{7}{2}} x^{6}}, \frac{15 \,{\left (12 \, B a b^{5} - 5 \, A b^{6}\right )} x^{6} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (1280 \, A a^{5} - 15 \,{\left (12 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 10 \,{\left (12 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 8 \,{\left (372 \, B a^{3} b^{2} + 5 \, A a^{2} b^{3}\right )} x^{3} + 144 \,{\left (28 \, B a^{4} b + 15 \, A a^{3} b^{2}\right )} x^{2} + 128 \,{\left (12 \, B a^{5} + 25 \, A a^{4} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{7680 \, \sqrt{-a} a^{3} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^7,x, algorithm="fricas")

[Out]

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

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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**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218313, size = 324, normalized size = 1.56 \[ \frac{\frac{15 \,{\left (12 \, B a b^{6} - 5 \, A b^{7}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{180 \,{\left (b x + a\right )}^{\frac{11}{2}} B a b^{6} - 1020 \,{\left (b x + a\right )}^{\frac{9}{2}} B a^{2} b^{6} - 696 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{3} b^{6} + 2376 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{4} b^{6} - 1020 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{5} b^{6} + 180 \, \sqrt{b x + a} B a^{6} b^{6} - 75 \,{\left (b x + a\right )}^{\frac{11}{2}} A b^{7} + 425 \,{\left (b x + a\right )}^{\frac{9}{2}} A a b^{7} - 990 \,{\left (b x + a\right )}^{\frac{7}{2}} A a^{2} b^{7} - 990 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{3} b^{7} + 425 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{4} b^{7} - 75 \, \sqrt{b x + a} A a^{5} b^{7}}{a^{3} b^{6} x^{6}}}{7680 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^7,x, algorithm="giac")

[Out]

1/7680*(15*(12*B*a*b^6 - 5*A*b^7)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3)
+ (180*(b*x + a)^(11/2)*B*a*b^6 - 1020*(b*x + a)^(9/2)*B*a^2*b^6 - 696*(b*x + a)
^(7/2)*B*a^3*b^6 + 2376*(b*x + a)^(5/2)*B*a^4*b^6 - 1020*(b*x + a)^(3/2)*B*a^5*b
^6 + 180*sqrt(b*x + a)*B*a^6*b^6 - 75*(b*x + a)^(11/2)*A*b^7 + 425*(b*x + a)^(9/
2)*A*a*b^7 - 990*(b*x + a)^(7/2)*A*a^2*b^7 - 990*(b*x + a)^(5/2)*A*a^3*b^7 + 425
*(b*x + a)^(3/2)*A*a^4*b^7 - 75*sqrt(b*x + a)*A*a^5*b^7)/(a^3*b^6*x^6))/b