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

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

[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 = 0.0543799, 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, Rules used = {612, 620, 206} \[ -\frac{35 b^6 (b+2 c x) \sqrt{b x+c x^2}}{16384 c^4}+\frac{35 b^4 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^3}-\frac{7 b^2 (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^2}+\frac{35 b^8 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{9/2}}+\frac{(b+2 c x) \left (b x+c x^2\right )^{7/2}}{16 c} \]

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))

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 \left (b x+c x^2\right )^{7/2} \, dx &=\frac{(b+2 c x) \left (b x+c x^2\right )^{7/2}}{16 c}-\frac{\left (7 b^2\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{32 c}\\ &=-\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}+\frac{\left (35 b^4\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{768 c^2}\\ &=\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}-\frac{\left (35 b^6\right ) \int \sqrt{b x+c x^2} \, dx}{4096 c^3}\\ &=-\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}+\frac{\left (35 b^8\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{32768 c^4}\\ &=-\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}+\frac{\left (35 b^8\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{16384 c^4}\\ &=-\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}+\frac{35 b^8 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.198729, size = 142, normalized size = 0.97 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-56 b^5 c^2 x^2+48 b^4 c^3 x^3+10880 b^3 c^4 x^4+25856 b^2 c^5 x^5+70 b^6 c x-105 b^7+21504 b c^6 x^6+6144 c^7 x^7\right )+\frac{105 b^{15/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{49152 c^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(7/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-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 + 258
56*b^2*c^5*x^5 + 21504*b*c^6*x^6 + 6144*c^7*x^7) + (105*b^(15/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*
Sqrt[1 + (c*x)/b])))/(49152*c^(9/2))

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Maple [A]  time = 0.05, size = 173, normalized size = 1.2 \begin{align*}{\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 ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \end{align*}

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.28795, size = 626, normalized size = 4.26 \begin{align*} \left [\frac{105 \, b^{8} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (6144 \, c^{8} x^{7} + 21504 \, b c^{7} x^{6} + 25856 \, b^{2} c^{6} x^{5} + 10880 \, b^{3} c^{5} x^{4} + 48 \, b^{4} c^{4} x^{3} - 56 \, b^{5} c^{3} x^{2} + 70 \, b^{6} c^{2} x - 105 \, b^{7} c\right )} \sqrt{c x^{2} + b x}}{98304 \, c^{5}}, -\frac{105 \, b^{8} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (6144 \, c^{8} x^{7} + 21504 \, b c^{7} x^{6} + 25856 \, b^{2} c^{6} x^{5} + 10880 \, b^{3} c^{5} x^{4} + 48 \, b^{4} c^{4} x^{3} - 56 \, b^{5} c^{3} x^{2} + 70 \, b^{6} c^{2} x - 105 \, b^{7} c\right )} \sqrt{c x^{2} + b x}}{49152 \, c^{5}}\right ] \end{align*}

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*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(6144*c^8*x^7 + 21504*b*c^7*x^6 + 2
5856*b^2*c^6*x^5 + 10880*b^3*c^5*x^4 + 48*b^4*c^4*x^3 - 56*b^5*c^3*x^2 + 70*b^6*c^2*x - 105*b^7*c)*sqrt(c*x^2
+ b*x))/c^5, -1/49152*(105*b^8*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (6144*c^8*x^7 + 21504*b*c^7
*x^6 + 25856*b^2*c^6*x^5 + 10880*b^3*c^5*x^4 + 48*b^4*c^4*x^3 - 56*b^5*c^3*x^2 + 70*b^6*c^2*x - 105*b^7*c)*sqr
t(c*x^2 + b*x))/c^5]

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

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 [A]  time = 1.25027, size = 178, normalized size = 1.21 \begin{align*} -\frac{35 \, b^{8} \log \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} \end{align*}

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*log(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)