### 3.25 $$\int x (a x+b x^2)^{5/2} \, dx$$

Optimal. Leaf size=139 $-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}+\frac{5 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{1024 b^{9/2}}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}$

[Out]

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

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Rubi [A]  time = 0.0504348, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.267, Rules used = {640, 612, 620, 206} $-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}+\frac{5 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{1024 b^{9/2}}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*(a*x + b*x^2)^(5/2),x]

[Out]

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

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x \left (a x+b x^2\right )^{5/2} \, dx &=\frac{\left (a x+b x^2\right )^{7/2}}{7 b}-\frac{a \int \left (a x+b x^2\right )^{5/2} \, dx}{2 b}\\ &=-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}+\frac{\left (5 a^3\right ) \int \left (a x+b x^2\right )^{3/2} \, dx}{48 b^2}\\ &=\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}-\frac{\left (5 a^5\right ) \int \sqrt{a x+b x^2} \, dx}{256 b^3}\\ &=-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}+\frac{\left (5 a^7\right ) \int \frac{1}{\sqrt{a x+b x^2}} \, dx}{2048 b^4}\\ &=-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}+\frac{\left (5 a^7\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )}{1024 b^4}\\ &=-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b}+\frac{5 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{1024 b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.177274, size = 131, normalized size = 0.94 $\frac{\sqrt{x (a+b x)} \left (\sqrt{b} \left (-56 a^4 b^2 x^2+48 a^3 b^3 x^3+4736 a^2 b^4 x^4+70 a^5 b x-105 a^6+7424 a b^5 x^5+3072 b^6 x^6\right )+\frac{105 a^{13/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{x} \sqrt{\frac{b x}{a}+1}}\right )}{21504 b^{9/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*(a*x + b*x^2)^(5/2),x]

[Out]

(Sqrt[x*(a + b*x)]*(Sqrt[b]*(-105*a^6 + 70*a^5*b*x - 56*a^4*b^2*x^2 + 48*a^3*b^3*x^3 + 4736*a^2*b^4*x^4 + 7424
*a*b^5*x^5 + 3072*b^6*x^6) + (105*a^(13/2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[x]*Sqrt[1 + (b*x)/a])))/(
21504*b^(9/2))

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Maple [A]  time = 0.049, size = 165, normalized size = 1.2 \begin{align*}{\frac{1}{7\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{ax}{12\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}}{24\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{5\,x{a}^{3}}{192\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{4}}{384\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{5}x}{512\,{b}^{3}}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{6}}{1024\,{b}^{4}}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{7}}{2048}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{9}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a*x)^(5/2),x)

[Out]

1/7*(b*x^2+a*x)^(7/2)/b-1/12/b*a*x*(b*x^2+a*x)^(5/2)-1/24/b^2*a^2*(b*x^2+a*x)^(5/2)+5/192/b^2*a^3*(b*x^2+a*x)^
(3/2)*x+5/384/b^3*a^4*(b*x^2+a*x)^(3/2)-5/512/b^3*a^5*(b*x^2+a*x)^(1/2)*x-5/1024/b^4*a^6*(b*x^2+a*x)^(1/2)+5/2
048/b^(9/2)*a^7*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.01484, size = 567, normalized size = 4.08 \begin{align*} \left [\frac{105 \, a^{7} \sqrt{b} \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (3072 \, b^{7} x^{6} + 7424 \, a b^{6} x^{5} + 4736 \, a^{2} b^{5} x^{4} + 48 \, a^{3} b^{4} x^{3} - 56 \, a^{4} b^{3} x^{2} + 70 \, a^{5} b^{2} x - 105 \, a^{6} b\right )} \sqrt{b x^{2} + a x}}{43008 \, b^{5}}, -\frac{105 \, a^{7} \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (3072 \, b^{7} x^{6} + 7424 \, a b^{6} x^{5} + 4736 \, a^{2} b^{5} x^{4} + 48 \, a^{3} b^{4} x^{3} - 56 \, a^{4} b^{3} x^{2} + 70 \, a^{5} b^{2} x - 105 \, a^{6} b\right )} \sqrt{b x^{2} + a x}}{21504 \, b^{5}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a*x)^(5/2),x, algorithm="fricas")

[Out]

[1/43008*(105*a^7*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(3072*b^7*x^6 + 7424*a*b^6*x^5 + 47
36*a^2*b^5*x^4 + 48*a^3*b^4*x^3 - 56*a^4*b^3*x^2 + 70*a^5*b^2*x - 105*a^6*b)*sqrt(b*x^2 + a*x))/b^5, -1/21504*
(105*a^7*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x)) - (3072*b^7*x^6 + 7424*a*b^6*x^5 + 4736*a^2*b^5*x^4
+ 48*a^3*b^4*x^3 - 56*a^4*b^3*x^2 + 70*a^5*b^2*x - 105*a^6*b)*sqrt(b*x^2 + a*x))/b^5]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (x \left (a + b x\right )\right )^{\frac{5}{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a*x)**(5/2),x)

[Out]

Integral(x*(x*(a + b*x))**(5/2), x)

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Giac [A]  time = 1.29223, size = 162, normalized size = 1.17 \begin{align*} -\frac{5 \, a^{7} \log \left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{2048 \, b^{\frac{9}{2}}} - \frac{1}{21504} \, \sqrt{b x^{2} + a x}{\left (\frac{105 \, a^{6}}{b^{4}} - 2 \,{\left (\frac{35 \, a^{5}}{b^{3}} - 4 \,{\left (\frac{7 \, a^{4}}{b^{2}} - 2 \,{\left (\frac{3 \, a^{3}}{b} + 8 \,{\left (37 \, a^{2} + 2 \,{\left (12 \, b^{2} x + 29 \, a b\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a*x)^(5/2),x, algorithm="giac")

[Out]

-5/2048*a^7*log(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a))/b^(9/2) - 1/21504*sqrt(b*x^2 + a*x)*(105*
a^6/b^4 - 2*(35*a^5/b^3 - 4*(7*a^4/b^2 - 2*(3*a^3/b + 8*(37*a^2 + 2*(12*b^2*x + 29*a*b)*x)*x)*x)*x)*x)