3.241 \(\int \frac{A+B x}{x^{7/2} \left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=216 \[ \frac{35 c^3 (8 b B-9 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{11/2}}-\frac{35 c^3 \sqrt{x} (8 b B-9 A c)}{64 b^5 \sqrt{b x+c x^2}}-\frac{35 c^2 (8 b B-9 A c)}{192 b^4 \sqrt{x} \sqrt{b x+c x^2}}+\frac{7 c (8 b B-9 A c)}{96 b^3 x^{3/2} \sqrt{b x+c x^2}}-\frac{8 b B-9 A c}{24 b^2 x^{5/2} \sqrt{b x+c x^2}}-\frac{A}{4 b x^{7/2} \sqrt{b x+c x^2}} \]

[Out]

-A/(4*b*x^(7/2)*Sqrt[b*x + c*x^2]) - (8*b*B - 9*A*c)/(24*b^2*x^(5/2)*Sqrt[b*x +
c*x^2]) + (7*c*(8*b*B - 9*A*c))/(96*b^3*x^(3/2)*Sqrt[b*x + c*x^2]) - (35*c^2*(8*
b*B - 9*A*c))/(192*b^4*Sqrt[x]*Sqrt[b*x + c*x^2]) - (35*c^3*(8*b*B - 9*A*c)*Sqrt
[x])/(64*b^5*Sqrt[b*x + c*x^2]) + (35*c^3*(8*b*B - 9*A*c)*ArcTanh[Sqrt[b*x + c*x
^2]/(Sqrt[b]*Sqrt[x])])/(64*b^(11/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.42837, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{35 c^3 (8 b B-9 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{11/2}}-\frac{35 c^3 \sqrt{x} (8 b B-9 A c)}{64 b^5 \sqrt{b x+c x^2}}-\frac{35 c^2 (8 b B-9 A c)}{192 b^4 \sqrt{x} \sqrt{b x+c x^2}}+\frac{7 c (8 b B-9 A c)}{96 b^3 x^{3/2} \sqrt{b x+c x^2}}-\frac{8 b B-9 A c}{24 b^2 x^{5/2} \sqrt{b x+c x^2}}-\frac{A}{4 b x^{7/2} \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(4*b*x^(7/2)*Sqrt[b*x + c*x^2]) - (8*b*B - 9*A*c)/(24*b^2*x^(5/2)*Sqrt[b*x +
c*x^2]) + (7*c*(8*b*B - 9*A*c))/(96*b^3*x^(3/2)*Sqrt[b*x + c*x^2]) - (35*c^2*(8*
b*B - 9*A*c))/(192*b^4*Sqrt[x]*Sqrt[b*x + c*x^2]) - (35*c^3*(8*b*B - 9*A*c)*Sqrt
[x])/(64*b^5*Sqrt[b*x + c*x^2]) + (35*c^3*(8*b*B - 9*A*c)*ArcTanh[Sqrt[b*x + c*x
^2]/(Sqrt[b]*Sqrt[x])])/(64*b^(11/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.452, size = 209, normalized size = 0.97 \[ - \frac{A}{4 b x^{\frac{7}{2}} \sqrt{b x + c x^{2}}} + \frac{9 A c - 8 B b}{24 b^{2} x^{\frac{5}{2}} \sqrt{b x + c x^{2}}} - \frac{7 c \left (9 A c - 8 B b\right )}{96 b^{3} x^{\frac{3}{2}} \sqrt{b x + c x^{2}}} + \frac{35 c^{2} \left (9 A c - 8 B b\right )}{192 b^{4} \sqrt{x} \sqrt{b x + c x^{2}}} + \frac{35 c^{3} \sqrt{x} \left (9 A c - 8 B b\right )}{64 b^{5} \sqrt{b x + c x^{2}}} - \frac{35 c^{3} \left (9 A c - 8 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{64 b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(4*b*x**(7/2)*sqrt(b*x + c*x**2)) + (9*A*c - 8*B*b)/(24*b**2*x**(5/2)*sqrt(b*
x + c*x**2)) - 7*c*(9*A*c - 8*B*b)/(96*b**3*x**(3/2)*sqrt(b*x + c*x**2)) + 35*c*
*2*(9*A*c - 8*B*b)/(192*b**4*sqrt(x)*sqrt(b*x + c*x**2)) + 35*c**3*sqrt(x)*(9*A*
c - 8*B*b)/(64*b**5*sqrt(b*x + c*x**2)) - 35*c**3*(9*A*c - 8*B*b)*atanh(sqrt(b*x
+ c*x**2)/(sqrt(b)*sqrt(x)))/(64*b**(11/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.211192, size = 153, normalized size = 0.71 \[ \frac{\sqrt{b} \left (A \left (-48 b^4+72 b^3 c x-126 b^2 c^2 x^2+315 b c^3 x^3+945 c^4 x^4\right )-8 b B x \left (8 b^3-14 b^2 c x+35 b c^2 x^2+105 c^3 x^3\right )\right )+105 c^3 x^4 \sqrt{b+c x} (8 b B-9 A c) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{192 b^{11/2} x^{7/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*(-8*b*B*x*(8*b^3 - 14*b^2*c*x + 35*b*c^2*x^2 + 105*c^3*x^3) + A*(-48*b^
4 + 72*b^3*c*x - 126*b^2*c^2*x^2 + 315*b*c^3*x^3 + 945*c^4*x^4)) + 105*c^3*(8*b*
B - 9*A*c)*x^4*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(192*b^(11/2)*x^(7/
2)*Sqrt[x*(b + c*x)])

_______________________________________________________________________________________

Maple [A]  time = 0.037, size = 174, normalized size = 0.8 \[ -{\frac{1}{192\,cx+192\,b}\sqrt{x \left ( cx+b \right ) } \left ( 945\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{4}{c}^{4}+64\,B{b}^{9/2}x-112\,B{b}^{7/2}{x}^{2}c+280\,B{b}^{5/2}{x}^{3}{c}^{2}+840\,B{b}^{3/2}{x}^{4}{c}^{3}-840\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{4}b{c}^{3}+48\,A{b}^{9/2}-72\,A{b}^{7/2}xc+126\,A{b}^{5/2}{x}^{2}{c}^{2}-315\,A{b}^{3/2}{x}^{3}{c}^{3}-945\,A\sqrt{b}{x}^{4}{c}^{4} \right ){x}^{-{\frac{9}{2}}}{b}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192/x^(9/2)*(x*(c*x+b))^(1/2)*(945*A*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(
1/2)*x^4*c^4+64*B*b^(9/2)*x-112*B*b^(7/2)*x^2*c+280*B*b^(5/2)*x^3*c^2+840*B*b^(3
/2)*x^4*c^3-840*B*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x^4*b*c^3+48*A*b^
(9/2)-72*A*b^(7/2)*x*c+126*A*b^(5/2)*x^2*c^2-315*A*b^(3/2)*x^3*c^3-945*A*b^(1/2)
*x^4*c^4)/(c*x+b)/b^(11/2)

_______________________________________________________________________________________

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)/((c*x^2 + b*x)^(3/2)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.324182, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (48 \, A b^{4} + 105 \,{\left (8 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} + 35 \,{\left (8 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} - 14 \,{\left (8 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2} + 8 \,{\left (8 \, B b^{4} - 9 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x} + 105 \,{\left ({\left (8 \, B b c^{4} - 9 \, A c^{5}\right )} x^{6} +{\left (8 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} x^{5}\right )} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{384 \,{\left (b^{5} c x^{6} + b^{6} x^{5}\right )} \sqrt{b}}, -\frac{{\left (48 \, A b^{4} + 105 \,{\left (8 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} + 35 \,{\left (8 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} - 14 \,{\left (8 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2} + 8 \,{\left (8 \, B b^{4} - 9 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x} - 105 \,{\left ({\left (8 \, B b c^{4} - 9 \, A c^{5}\right )} x^{6} +{\left (8 \, B b^{2} c^{3} - 9 \, A b c^{4}\right )} x^{5}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{192 \,{\left (b^{5} c x^{6} + b^{6} x^{5}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/384*(2*(48*A*b^4 + 105*(8*B*b*c^3 - 9*A*c^4)*x^4 + 35*(8*B*b^2*c^2 - 9*A*b*c
^3)*x^3 - 14*(8*B*b^3*c - 9*A*b^2*c^2)*x^2 + 8*(8*B*b^4 - 9*A*b^3*c)*x)*sqrt(c*x
^2 + b*x)*sqrt(b)*sqrt(x) + 105*((8*B*b*c^4 - 9*A*c^5)*x^6 + (8*B*b^2*c^3 - 9*A*
b*c^4)*x^5)*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x^2))/
((b^5*c*x^6 + b^6*x^5)*sqrt(b)), -1/192*((48*A*b^4 + 105*(8*B*b*c^3 - 9*A*c^4)*x
^4 + 35*(8*B*b^2*c^2 - 9*A*b*c^3)*x^3 - 14*(8*B*b^3*c - 9*A*b^2*c^2)*x^2 + 8*(8*
B*b^4 - 9*A*b^3*c)*x)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x) - 105*((8*B*b*c^4 - 9*A
*c^5)*x^6 + (8*B*b^2*c^3 - 9*A*b*c^4)*x^5)*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 +
b*x)))/((b^5*c*x^6 + b^6*x^5)*sqrt(-b))]

_______________________________________________________________________________________

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)/x**(7/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.576353, size = 266, normalized size = 1.23 \[ -\frac{35 \,{\left (8 \, B b c^{3} - 9 \, A c^{4}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{64 \, \sqrt{-b} b^{5}} - \frac{2 \,{\left (B b c^{3} - A c^{4}\right )}}{\sqrt{c x + b} b^{5}} - \frac{456 \,{\left (c x + b\right )}^{\frac{7}{2}} B b c^{3} - 1544 \,{\left (c x + b\right )}^{\frac{5}{2}} B b^{2} c^{3} + 1784 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{3} c^{3} - 696 \, \sqrt{c x + b} B b^{4} c^{3} - 561 \,{\left (c x + b\right )}^{\frac{7}{2}} A c^{4} + 1929 \,{\left (c x + b\right )}^{\frac{5}{2}} A b c^{4} - 2295 \,{\left (c x + b\right )}^{\frac{3}{2}} A b^{2} c^{4} + 975 \, \sqrt{c x + b} A b^{3} c^{4}}{192 \, b^{5} c^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-35/64*(8*B*b*c^3 - 9*A*c^4)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^5) - 2*(
B*b*c^3 - A*c^4)/(sqrt(c*x + b)*b^5) - 1/192*(456*(c*x + b)^(7/2)*B*b*c^3 - 1544
*(c*x + b)^(5/2)*B*b^2*c^3 + 1784*(c*x + b)^(3/2)*B*b^3*c^3 - 696*sqrt(c*x + b)*
B*b^4*c^3 - 561*(c*x + b)^(7/2)*A*c^4 + 1929*(c*x + b)^(5/2)*A*b*c^4 - 2295*(c*x
+ b)^(3/2)*A*b^2*c^4 + 975*sqrt(c*x + b)*A*b^3*c^4)/(b^5*c^4*x^4)