3.38 \(\int \frac{(a x+b x^2)^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=152 \[ \frac{512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac{64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}} \]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(17*a*x^12) + (4*b*(a*x + b*x^2)^(7/2))/(51*a^2*x^11) - (32*b^2*(a*x + b*x^2)^(7/2))/
(663*a^3*x^10) + (64*b^3*(a*x + b*x^2)^(7/2))/(2431*a^4*x^9) - (256*b^4*(a*x + b*x^2)^(7/2))/(21879*a^5*x^8) +
(512*b^5*(a*x + b*x^2)^(7/2))/(153153*a^6*x^7)

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Rubi [A]  time = 0.0730637, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ \frac{512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac{64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^2)^(5/2)/x^12,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(17*a*x^12) + (4*b*(a*x + b*x^2)^(7/2))/(51*a^2*x^11) - (32*b^2*(a*x + b*x^2)^(7/2))/
(663*a^3*x^10) + (64*b^3*(a*x + b*x^2)^(7/2))/(2431*a^4*x^9) - (256*b^4*(a*x + b*x^2)^(7/2))/(21879*a^5*x^8) +
(512*b^5*(a*x + b*x^2)^(7/2))/(153153*a^6*x^7)

Rule 658

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{\left (a x+b x^2\right )^{5/2}}{x^{12}} \, dx &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}-\frac{(10 b) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^{11}} \, dx}{17 a}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}+\frac{\left (16 b^2\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx}{51 a^2}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}-\frac{\left (32 b^3\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{221 a^3}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac{64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}+\frac{\left (128 b^4\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{2431 a^4}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac{64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}-\frac{\left (256 b^5\right ) \int \frac{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{21879 a^5}\\ &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{17 a x^{12}}+\frac{4 b \left (a x+b x^2\right )^{7/2}}{51 a^2 x^{11}}-\frac{32 b^2 \left (a x+b x^2\right )^{7/2}}{663 a^3 x^{10}}+\frac{64 b^3 \left (a x+b x^2\right )^{7/2}}{2431 a^4 x^9}-\frac{256 b^4 \left (a x+b x^2\right )^{7/2}}{21879 a^5 x^8}+\frac{512 b^5 \left (a x+b x^2\right )^{7/2}}{153153 a^6 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0188533, size = 80, normalized size = 0.53 \[ \frac{2 (a+b x)^3 \sqrt{x (a+b x)} \left (-3696 a^3 b^2 x^2+2016 a^2 b^3 x^3+6006 a^4 b x-9009 a^5-896 a b^4 x^4+256 b^5 x^5\right )}{153153 a^6 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^12,x]

[Out]

(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-9009*a^5 + 6006*a^4*b*x - 3696*a^3*b^2*x^2 + 2016*a^2*b^3*x^3 - 896*a*b^4*x
^4 + 256*b^5*x^5))/(153153*a^6*x^9)

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Maple [A]  time = 0.053, size = 77, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -256\,{b}^{5}{x}^{5}+896\,{b}^{4}{x}^{4}a-2016\,{b}^{3}{x}^{3}{a}^{2}+3696\,{b}^{2}{x}^{2}{a}^{3}-6006\,bx{a}^{4}+9009\,{a}^{5} \right ) }{153153\,{x}^{11}{a}^{6}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a*x)^(5/2)/x^12,x)

[Out]

-2/153153*(b*x+a)*(-256*b^5*x^5+896*a*b^4*x^4-2016*a^2*b^3*x^3+3696*a^3*b^2*x^2-6006*a^4*b*x+9009*a^5)*(b*x^2+
a*x)^(5/2)/x^11/a^6

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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((b*x^2+a*x)^(5/2)/x^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89062, size = 247, normalized size = 1.62 \begin{align*} \frac{2 \,{\left (256 \, b^{8} x^{8} - 128 \, a b^{7} x^{7} + 96 \, a^{2} b^{6} x^{6} - 80 \, a^{3} b^{5} x^{5} + 70 \, a^{4} b^{4} x^{4} - 63 \, a^{5} b^{3} x^{3} - 12705 \, a^{6} b^{2} x^{2} - 21021 \, a^{7} b x - 9009 \, a^{8}\right )} \sqrt{b x^{2} + a x}}{153153 \, a^{6} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^12,x, algorithm="fricas")

[Out]

2/153153*(256*b^8*x^8 - 128*a*b^7*x^7 + 96*a^2*b^6*x^6 - 80*a^3*b^5*x^5 + 70*a^4*b^4*x^4 - 63*a^5*b^3*x^3 - 12
705*a^6*b^2*x^2 - 21021*a^7*b*x - 9009*a^8)*sqrt(b*x^2 + a*x)/(a^6*x^9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{12}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a*x)**(5/2)/x**12,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**12, x)

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Giac [B]  time = 1.25247, size = 458, normalized size = 3.01 \begin{align*} \frac{2 \,{\left (816816 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{11} b^{\frac{11}{2}} + 5951088 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{10} a b^{5} + 19909890 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9} a^{2} b^{\frac{9}{2}} + 40160120 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{8} a^{3} b^{4} + 54063009 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} a^{4} b^{\frac{7}{2}} + 50860719 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a^{5} b^{3} + 34051017 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{6} b^{\frac{5}{2}} + 16198875 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{7} b^{2} + 5360355 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{8} b^{\frac{3}{2}} + 1174173 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{9} b + 153153 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{10} \sqrt{b} + 9009 \, a^{11}\right )}}{153153 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^12,x, algorithm="giac")

[Out]

2/153153*(816816*(sqrt(b)*x - sqrt(b*x^2 + a*x))^11*b^(11/2) + 5951088*(sqrt(b)*x - sqrt(b*x^2 + a*x))^10*a*b^
5 + 19909890*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*a^2*b^(9/2) + 40160120*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a^3*b^
4 + 54063009*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^4*b^(7/2) + 50860719*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^5*b^
3 + 34051017*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^6*b^(5/2) + 16198875*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^7*b^
2 + 5360355*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^8*b^(3/2) + 1174173*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^9*b +
153153*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^10*sqrt(b) + 9009*a^11)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^17