3.304 \(\int (d+e x) \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=175 \[ -\frac{5 b^6 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{9/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac{e \left (b x+c x^2\right )^{7/2}}{7 c} \]

[Out]

(5*b^4*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^4) - (5*b^2*(2*c*d -
b*e)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(384*c^3) + ((2*c*d - b*e)*(b + 2*c*x)*(b
*x + c*x^2)^(5/2))/(24*c^2) + (e*(b*x + c*x^2)^(7/2))/(7*c) - (5*b^6*(2*c*d - b*
e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(1024*c^(9/2))

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Rubi [A]  time = 0.196294, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{5 b^6 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{9/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{1024 c^4}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{384 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{24 c^2}+\frac{e \left (b x+c x^2\right )^{7/2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^4*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^4) - (5*b^2*(2*c*d -
b*e)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(384*c^3) + ((2*c*d - b*e)*(b + 2*c*x)*(b
*x + c*x^2)^(5/2))/(24*c^2) + (e*(b*x + c*x^2)^(7/2))/(7*c) - (5*b^6*(2*c*d - b*
e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(1024*c^(9/2))

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Rubi in Sympy [A]  time = 21.7263, size = 163, normalized size = 0.93 \[ \frac{5 b^{6} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{1024 c^{\frac{9}{2}}} - \frac{5 b^{4} \left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{1024 c^{4}} + \frac{5 b^{2} \left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{384 c^{3}} + \frac{e \left (b x + c x^{2}\right )^{\frac{7}{2}}}{7 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{24 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(c*x**2+b*x)**(5/2),x)

[Out]

5*b**6*(b*e - 2*c*d)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(1024*c**(9/2)) - 5*b**
4*(b + 2*c*x)*(b*e - 2*c*d)*sqrt(b*x + c*x**2)/(1024*c**4) + 5*b**2*(b + 2*c*x)*
(b*e - 2*c*d)*(b*x + c*x**2)**(3/2)/(384*c**3) + e*(b*x + c*x**2)**(7/2)/(7*c) -
(b + 2*c*x)*(b*e - 2*c*d)*(b*x + c*x**2)**(5/2)/(24*c**2)

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Mathematica [A]  time = 0.363869, size = 186, normalized size = 1.06 \[ \frac{\sqrt{x (b+c x)} \left (\frac{105 b^6 (b e-2 c d) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-105 b^6 e+70 b^5 c (3 d+e x)-28 b^4 c^2 x (5 d+2 e x)+16 b^3 c^3 x^2 (7 d+3 e x)+32 b^2 c^4 x^3 (189 d+148 e x)+256 b c^5 x^4 (35 d+29 e x)+512 c^6 x^5 (7 d+6 e x)\right )\right )}{21504 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^6*e + 70*b^5*c*(3*d + e*x) - 28*b^4*c^2*x*(5
*d + 2*e*x) + 16*b^3*c^3*x^2*(7*d + 3*e*x) + 512*c^6*x^5*(7*d + 6*e*x) + 256*b*c
^5*x^4*(35*d + 29*e*x) + 32*b^2*c^4*x^3*(189*d + 148*e*x)) + (105*b^6*(-2*c*d +
b*e)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(21504*c^
(9/2))

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Maple [B]  time = 0.01, size = 321, normalized size = 1.8 \[{\frac{dx}{6} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{bd}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}dx}{96\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,d{b}^{3}}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,d{b}^{4}x}{256\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,d{b}^{5}}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,d{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{e}{7\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{bex}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}e}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{5\,e{b}^{3}x}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,e{b}^{4}}{384\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,e{b}^{5}x}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,e{b}^{6}}{1024\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,e{b}^{7}}{2048}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x)^(5/2),x)

[Out]

1/6*d*(c*x^2+b*x)^(5/2)*x+1/12*d/c*(c*x^2+b*x)^(5/2)*b-5/96*d*b^2/c*(c*x^2+b*x)^
(3/2)*x-5/192*d*b^3/c^2*(c*x^2+b*x)^(3/2)+5/256*d*b^4/c^2*(c*x^2+b*x)^(1/2)*x+5/
512*d*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*d*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x)^(1/2))+1/7*e*(c*x^2+b*x)^(7/2)/c-1/12*e*b/c*(c*x^2+b*x)^(5/2)*x-1/24*e*b
^2/c^2*(c*x^2+b*x)^(5/2)+5/192*e*b^3/c^2*(c*x^2+b*x)^(3/2)*x+5/384*e*b^4/c^3*(c*
x^2+b*x)^(3/2)-5/512*e*b^5/c^3*(c*x^2+b*x)^(1/2)*x-5/1024*e*b^6/c^4*(c*x^2+b*x)^
(1/2)+5/2048*e*b^7/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)^(5/2)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237754, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3072 \, c^{6} e x^{6} + 210 \, b^{5} c d - 105 \, b^{6} e + 256 \,{\left (14 \, c^{6} d + 29 \, b c^{5} e\right )} x^{5} + 128 \,{\left (70 \, b c^{5} d + 37 \, b^{2} c^{4} e\right )} x^{4} + 48 \,{\left (126 \, b^{2} c^{4} d + b^{3} c^{3} e\right )} x^{3} + 56 \,{\left (2 \, b^{3} c^{3} d - b^{4} c^{2} e\right )} x^{2} - 70 \,{\left (2 \, b^{4} c^{2} d - b^{5} c e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 105 \,{\left (2 \, b^{6} c d - b^{7} e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{43008 \, c^{\frac{9}{2}}}, \frac{{\left (3072 \, c^{6} e x^{6} + 210 \, b^{5} c d - 105 \, b^{6} e + 256 \,{\left (14 \, c^{6} d + 29 \, b c^{5} e\right )} x^{5} + 128 \,{\left (70 \, b c^{5} d + 37 \, b^{2} c^{4} e\right )} x^{4} + 48 \,{\left (126 \, b^{2} c^{4} d + b^{3} c^{3} e\right )} x^{3} + 56 \,{\left (2 \, b^{3} c^{3} d - b^{4} c^{2} e\right )} x^{2} - 70 \,{\left (2 \, b^{4} c^{2} d - b^{5} c e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 105 \,{\left (2 \, b^{6} c d - b^{7} e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{21504 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(e*x + d),x, algorithm="fricas")

[Out]

[1/43008*(2*(3072*c^6*e*x^6 + 210*b^5*c*d - 105*b^6*e + 256*(14*c^6*d + 29*b*c^5
*e)*x^5 + 128*(70*b*c^5*d + 37*b^2*c^4*e)*x^4 + 48*(126*b^2*c^4*d + b^3*c^3*e)*x
^3 + 56*(2*b^3*c^3*d - b^4*c^2*e)*x^2 - 70*(2*b^4*c^2*d - b^5*c*e)*x)*sqrt(c*x^2
+ b*x)*sqrt(c) - 105*(2*b^6*c*d - b^7*e)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2
+ b*x)*c))/c^(9/2), 1/21504*((3072*c^6*e*x^6 + 210*b^5*c*d - 105*b^6*e + 256*(1
4*c^6*d + 29*b*c^5*e)*x^5 + 128*(70*b*c^5*d + 37*b^2*c^4*e)*x^4 + 48*(126*b^2*c^
4*d + b^3*c^3*e)*x^3 + 56*(2*b^3*c^3*d - b^4*c^2*e)*x^2 - 70*(2*b^4*c^2*d - b^5*
c*e)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 105*(2*b^6*c*d - b^7*e)*arctan(sqrt(c*x^2 +
b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (d + e x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(c*x**2+b*x)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*(d + e*x), x)

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GIAC/XCAS [A]  time = 0.232996, size = 315, normalized size = 1.8 \[ \frac{1}{21504} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \, c^{2} x e + \frac{14 \, c^{8} d + 29 \, b c^{7} e}{c^{6}}\right )} x + \frac{70 \, b c^{7} d + 37 \, b^{2} c^{6} e}{c^{6}}\right )} x + \frac{3 \,{\left (126 \, b^{2} c^{6} d + b^{3} c^{5} e\right )}}{c^{6}}\right )} x + \frac{7 \,{\left (2 \, b^{3} c^{5} d - b^{4} c^{4} e\right )}}{c^{6}}\right )} x - \frac{35 \,{\left (2 \, b^{4} c^{4} d - b^{5} c^{3} e\right )}}{c^{6}}\right )} x + \frac{105 \,{\left (2 \, b^{5} c^{3} d - b^{6} c^{2} e\right )}}{c^{6}}\right )} + \frac{5 \,{\left (2 \, b^{6} c d - b^{7} e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(e*x + d),x, algorithm="giac")

[Out]

1/21504*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*c^2*x*e + (14*c^8*d + 29*b*c^7*e)/c
^6)*x + (70*b*c^7*d + 37*b^2*c^6*e)/c^6)*x + 3*(126*b^2*c^6*d + b^3*c^5*e)/c^6)*
x + 7*(2*b^3*c^5*d - b^4*c^4*e)/c^6)*x - 35*(2*b^4*c^4*d - b^5*c^3*e)/c^6)*x + 1
05*(2*b^5*c^3*d - b^6*c^2*e)/c^6) + 5/2048*(2*b^6*c*d - b^7*e)*ln(abs(-2*(sqrt(c
)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2)