3.1 \(\int \left (b x+c x^2\right )^{7/2} \, dx\)

Optimal. Leaf size=147 \[ \frac{35 b^8 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{9/2}}-\frac{35 b^6 (b+2 c x) \sqrt{b x+c x^2}}{16384 c^4}+\frac{35 b^4 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^3}-\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{7/2}}{16 c} \]

[Out]

(-35*b^6*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^4) + (35*b^4*(b + 2*c*x)*(b*x +
c*x^2)^(3/2))/(6144*c^3) - (7*b^2*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^2) +
((b + 2*c*x)*(b*x + c*x^2)^(7/2))/(16*c) + (35*b^8*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/(16384*c^(9/2))

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Rubi [A]  time = 0.129437, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{35 b^8 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{9/2}}-\frac{35 b^6 (b+2 c x) \sqrt{b x+c x^2}}{16384 c^4}+\frac{35 b^4 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^3}-\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{7/2}}{16 c} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(7/2),x]

[Out]

(-35*b^6*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^4) + (35*b^4*(b + 2*c*x)*(b*x +
c*x^2)^(3/2))/(6144*c^3) - (7*b^2*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^2) +
((b + 2*c*x)*(b*x + c*x^2)^(7/2))/(16*c) + (35*b^8*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/(16384*c^(9/2))

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Rubi in Sympy [A]  time = 16.0857, size = 139, normalized size = 0.95 \[ \frac{35 b^{8} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{16384 c^{\frac{9}{2}}} - \frac{35 b^{6} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{16384 c^{4}} + \frac{35 b^{4} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{6144 c^{3}} - \frac{7 b^{2} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{384 c^{2}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{16 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35*b**8*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(16384*c**(9/2)) - 35*b**6*(b + 2*c*
x)*sqrt(b*x + c*x**2)/(16384*c**4) + 35*b**4*(b + 2*c*x)*(b*x + c*x**2)**(3/2)/(
6144*c**3) - 7*b**2*(b + 2*c*x)*(b*x + c*x**2)**(5/2)/(384*c**2) + (b + 2*c*x)*(
b*x + c*x**2)**(7/2)/(16*c)

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Mathematica [A]  time = 0.2166, size = 158, normalized size = 1.07 \[ \frac{\sqrt{x} \sqrt{b+c x} \left (105 b^8 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )+\sqrt{c} \sqrt{x} \sqrt{b+c x} \left (-105 b^7+70 b^6 c x-56 b^5 c^2 x^2+48 b^4 c^3 x^3+10880 b^3 c^4 x^4+25856 b^2 c^5 x^5+21504 b c^6 x^6+6144 c^7 x^7\right )\right )}{49152 c^{9/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(-105*b^7 + 70*b^6*c*x - 5
6*b^5*c^2*x^2 + 48*b^4*c^3*x^3 + 10880*b^3*c^4*x^4 + 25856*b^2*c^5*x^5 + 21504*b
*c^6*x^6 + 6144*c^7*x^7) + 105*b^8*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]]))/(491
52*c^(9/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.007, size = 173, normalized size = 1.2 \[{\frac{2\,cx+b}{16\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{7\,{b}^{2}x}{192\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{b}^{3}}{384\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{b}^{4}x}{3072\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{b}^{5}}{6144\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{35\,{b}^{6}x}{8192\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{35\,{b}^{7}}{16384\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{35\,{b}^{8}}{32768}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(7/2),x)

[Out]

1/16*(2*c*x+b)*(c*x^2+b*x)^(7/2)/c-7/192*b^2/c*(c*x^2+b*x)^(5/2)*x-7/384*b^3/c^2
*(c*x^2+b*x)^(5/2)+35/3072*b^4/c^2*(c*x^2+b*x)^(3/2)*x+35/6144*b^5/c^3*(c*x^2+b*
x)^(3/2)-35/8192*b^6/c^3*(c*x^2+b*x)^(1/2)*x-35/16384*b^7/c^4*(c*x^2+b*x)^(1/2)+
35/32768*b^8/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(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((c*x^2 + b*x)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242545, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{8} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (6144 \, c^{7} x^{7} + 21504 \, b c^{6} x^{6} + 25856 \, b^{2} c^{5} x^{5} + 10880 \, b^{3} c^{4} x^{4} + 48 \, b^{4} c^{3} x^{3} - 56 \, b^{5} c^{2} x^{2} + 70 \, b^{6} c x - 105 \, b^{7}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{98304 \, c^{\frac{9}{2}}}, \frac{105 \, b^{8} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (6144 \, c^{7} x^{7} + 21504 \, b c^{6} x^{6} + 25856 \, b^{2} c^{5} x^{5} + 10880 \, b^{3} c^{4} x^{4} + 48 \, b^{4} c^{3} x^{3} - 56 \, b^{5} c^{2} x^{2} + 70 \, b^{6} c x - 105 \, b^{7}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{49152 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/98304*(105*b^8*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c) + 2*(6144*c^7
*x^7 + 21504*b*c^6*x^6 + 25856*b^2*c^5*x^5 + 10880*b^3*c^4*x^4 + 48*b^4*c^3*x^3
- 56*b^5*c^2*x^2 + 70*b^6*c*x - 105*b^7)*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2), 1/4
9152*(105*b^8*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (6144*c^7*x^7 + 21504*b
*c^6*x^6 + 25856*b^2*c^5*x^5 + 10880*b^3*c^4*x^4 + 48*b^4*c^3*x^3 - 56*b^5*c^2*x
^2 + 70*b^6*c*x - 105*b^7)*sqrt(c*x^2 + b*x)*sqrt(-c))/(sqrt(-c)*c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b x + c x^{2}\right )^{\frac{7}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(7/2),x)

[Out]

Integral((b*x + c*x**2)**(7/2), x)

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GIAC/XCAS [A]  time = 0.226363, size = 178, normalized size = 1.21 \[ -\frac{35 \, b^{8}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{9}{2}}} - \frac{1}{49152} \,{\left (\frac{105 \, b^{7}}{c^{4}} - 2 \,{\left (\frac{35 \, b^{6}}{c^{3}} - 4 \,{\left (\frac{7 \, b^{5}}{c^{2}} - 2 \,{\left (\frac{3 \, b^{4}}{c} + 8 \,{\left (85 \, b^{3} + 2 \,{\left (101 \, b^{2} c + 12 \,{\left (2 \, c^{3} x + 7 \, b c^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-35/32768*b^8*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2) -
1/49152*(105*b^7/c^4 - 2*(35*b^6/c^3 - 4*(7*b^5/c^2 - 2*(3*b^4/c + 8*(85*b^3 + 2
*(101*b^2*c + 12*(2*c^3*x + 7*b*c^2)*x)*x)*x)*x)*x)*x)*sqrt(c*x^2 + b*x)