3.986 \(\int \frac{(d x)^m (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{d^4 x (d x)^{m-4} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m-4,-n;m-3;-\frac{b x}{a}\right )}{c^2 (4-m) \sqrt{c x^2}} \]

[Out]

-((d^4*x*(d*x)^(-4 + m)*(a + b*x)^n*Hypergeometric2F1[-4 + m, -n, -3 + m, -((b*x
)/a)])/(c^2*(4 - m)*Sqrt[c*x^2]*(1 + (b*x)/a)^n))

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Rubi [A]  time = 0.0736946, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{d^4 x (d x)^{m-4} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m-4,-n;m-3;-\frac{b x}{a}\right )}{c^2 (4-m) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((d*x)^m*(a + b*x)^n)/(c*x^2)^(5/2),x]

[Out]

-((d^4*x*(d*x)^(-4 + m)*(a + b*x)^n*Hypergeometric2F1[-4 + m, -n, -3 + m, -((b*x
)/a)])/(c^2*(4 - m)*Sqrt[c*x^2]*(1 + (b*x)/a)^n))

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Rubi in Sympy [A]  time = 21.0274, size = 56, normalized size = 0.82 \[ - \frac{d^{4} \sqrt{c x^{2}} \left (d x\right )^{m - 4} \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, m - 4 \\ m - 3 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{c^{3} x \left (- m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(b*x+a)**n/(c*x**2)**(5/2),x)

[Out]

-d**4*sqrt(c*x**2)*(d*x)**(m - 4)*(1 + b*x/a)**(-n)*(a + b*x)**n*hyper((-n, m -
4), (m - 3,), -b*x/a)/(c**3*x*(-m + 4))

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Mathematica [A]  time = 0.087477, size = 57, normalized size = 0.84 \[ \frac{x (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m-4,-n;m-3;-\frac{b x}{a}\right )}{(m-4) \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d*x)^m*(a + b*x)^n*Hypergeometric2F1[-4 + m, -n, -3 + m, -((b*x)/a)])/((-4 +
m)*(c*x^2)^(5/2)*(1 + (b*x)/a)^n)

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{ \left ( dx \right ) ^{m} \left ( bx+a \right ) ^{n} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(b*x+a)^n/(c*x^2)^(5/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*(d*x)^m/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\sqrt{c x^{2}} c^{2} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*(d*x)^m/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

integral((b*x + a)^n*(d*x)^m/(sqrt(c*x^2)*c^2*x^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(b*x+a)**n/(c*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*(d*x)^m/(c*x^2)^(5/2),x, algorithm="giac")

[Out]

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