3.33 \(\int \frac {(a x+b x^2)^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=23 \[ -\frac {2 \left (a x+b x^2\right )^{7/2}}{7 a x^7} \]

[Out]

-2/7*(b*x^2+a*x)^(7/2)/a/x^7

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {650} \[ -\frac {2 \left (a x+b x^2\right )^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^2)^(5/2)/x^7,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(7*a*x^7)

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^7} \, dx &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{7 a x^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.91 \[ -\frac {2 (x (a+b x))^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^2)^(5/2)/x^7,x]

[Out]

(-2*(x*(a + b*x))^(7/2))/(7*a*x^7)

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fricas [B]  time = 0.93, size = 46, normalized size = 2.00 \[ -\frac {2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {b x^{2} + a x}}{7 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^7,x, algorithm="fricas")

[Out]

-2/7*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(b*x^2 + a*x)/(a*x^4)

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giac [B]  time = 0.21, size = 192, normalized size = 8.35 \[ \frac {2 \, {\left (7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} b^{3} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a b^{\frac {5}{2}} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{2} b^{2} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{3} b^{\frac {3}{2}} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{4} b + 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{5} \sqrt {b} + a^{6}\right )}}{7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^7,x, algorithm="giac")

[Out]

2/7*(7*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*b^3 + 21*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a*b^(5/2) + 35*(sqrt(b)*x
- sqrt(b*x^2 + a*x))^4*a^2*b^2 + 35*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^3*b^(3/2) + 21*(sqrt(b)*x - sqrt(b*x^2
+ a*x))^2*a^4*b + 7*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^5*sqrt(b) + a^6)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^7

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maple [A]  time = 0.04, size = 25, normalized size = 1.09 \[ -\frac {2 \left (b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 a \,x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a*x)^(5/2)/x^7,x)

[Out]

-2/7*(b*x+a)/x^6/a*(b*x^2+a*x)^(5/2)

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maxima [B]  time = 1.31, size = 112, normalized size = 4.87 \[ -\frac {2 \, \sqrt {b x^{2} + a x} b^{3}}{7 \, a x} + \frac {\sqrt {b x^{2} + a x} b^{2}}{7 \, x^{2}} - \frac {3 \, \sqrt {b x^{2} + a x} a b}{28 \, x^{3}} - \frac {15 \, \sqrt {b x^{2} + a x} a^{2}}{28 \, x^{4}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{4 \, x^{5}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a*x)^(5/2)/x^7,x, algorithm="maxima")

[Out]

-2/7*sqrt(b*x^2 + a*x)*b^3/(a*x) + 1/7*sqrt(b*x^2 + a*x)*b^2/x^2 - 3/28*sqrt(b*x^2 + a*x)*a*b/x^3 - 15/28*sqrt
(b*x^2 + a*x)*a^2/x^4 + 5/4*(b*x^2 + a*x)^(3/2)*a/x^5 - (b*x^2 + a*x)^(5/2)/x^6

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mupad [B]  time = 0.78, size = 79, normalized size = 3.43 \[ -\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{7\,x^4}-\frac {6\,b^2\,\sqrt {b\,x^2+a\,x}}{7\,x^2}-\frac {2\,b^3\,\sqrt {b\,x^2+a\,x}}{7\,a\,x}-\frac {6\,a\,b\,\sqrt {b\,x^2+a\,x}}{7\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^2)^(5/2)/x^7,x)

[Out]

- (2*a^2*(a*x + b*x^2)^(1/2))/(7*x^4) - (6*b^2*(a*x + b*x^2)^(1/2))/(7*x^2) - (2*b^3*(a*x + b*x^2)^(1/2))/(7*a
*x) - (6*a*b*(a*x + b*x^2)^(1/2))/(7*x^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a*x)**(5/2)/x**7,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**7, x)

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