3.123 \(\int \frac{(d x)^m}{(b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{2 c x^2 (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c x}{b}+1\right )}{3 b^2 \left (b x+c x^2\right )^{5/2}} \]

[Out]

(-2*c*x^2*(-((c*x)/b))^(1/2 - m)*(d*x)^m*(b + c*x)*Hypergeometric2F1[-3/2, 5/2 - m, -1/2, 1 + (c*x)/b])/(3*b^2
*(b*x + c*x^2)^(5/2))

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Rubi [A]  time = 0.0281655, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {674, 67, 65} \[ -\frac{2 c x^2 (b+c x) (d x)^m \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c x}{b}+1\right )}{3 b^2 \left (b x+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*c*x^2*(-((c*x)/b))^(1/2 - m)*(d*x)^m*(b + c*x)*Hypergeometric2F1[-3/2, 5/2 - m, -1/2, 1 + (c*x)/b])/(3*b^2
*(b*x + c*x^2)^(5/2))

Rule 674

Int[((e_.)*(x_))^(m_)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((e*x)^m*(b*x + c*x^2)^p)/(x^(m + p)
*(b + c*x)^p), Int[x^(m + p)*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, m}, x] &&  !IntegerQ[p]

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-((b*c)/d))^IntPart[m]*(b*x)^FracPart[m])/
(-((d*x)/c))^FracPart[m], Int[(-((d*x)/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin{align*} \int \frac{(d x)^m}{\left (b x+c x^2\right )^{5/2}} \, dx &=\frac{\left (x^{\frac{5}{2}-m} (d x)^m (b+c x)^{5/2}\right ) \int \frac{x^{-\frac{5}{2}+m}}{(b+c x)^{5/2}} \, dx}{\left (b x+c x^2\right )^{5/2}}\\ &=\frac{\left (c^2 x^2 \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} (d x)^m (b+c x)^{5/2}\right ) \int \frac{\left (-\frac{c x}{b}\right )^{-\frac{5}{2}+m}}{(b+c x)^{5/2}} \, dx}{b^2 \left (b x+c x^2\right )^{5/2}}\\ &=-\frac{2 c x^2 \left (-\frac{c x}{b}\right )^{\frac{1}{2}-m} (d x)^m (b+c x) \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};1+\frac{c x}{b}\right )}{3 b^2 \left (b x+c x^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.108664, size = 60, normalized size = 0.85 \[ \frac{2 (d x)^m \left (-\frac{c x}{b}\right )^{\frac{3}{2}-m} \, _2F_1\left (-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{c x}{b}+1\right )}{3 b (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-((c*x)/b))^(3/2 - m)*(d*x)^m*Hypergeometric2F1[-3/2, 5/2 - m, -1/2, 1 + (c*x)/b])/(3*b*(x*(b + c*x))^(3/2
))

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Maple [F]  time = 0.425, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m/(c*x^2+b*x)^(5/2),x)

[Out]

int((d*x)^m/(c*x^2+b*x)^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x} \left (d x\right )^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + b*x)*(d*x)^m/(c^3*x^6 + 3*b*c^2*x^5 + 3*b^2*c*x^4 + b^3*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m/(c*x**2+b*x)**(5/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^(5/2), x)