3.982 \(\int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^n \, dx\)

Optimal. Leaf size=65 \[ \frac{c \sqrt{c x^2} (d x)^{m+4} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac{b x}{a}\right )}{d^4 (m+4) x} \]

[Out]

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

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Rubi [A]  time = 0.0668694, antiderivative size = 65, 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 \[ \frac{c \sqrt{c x^2} (d x)^{m+4} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac{b x}{a}\right )}{d^4 (m+4) x} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(c*x^2)^(3/2)*(a + b*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 20.0883, size = 53, normalized size = 0.82 \[ \frac{c \sqrt{c x^{2}} \left (d x\right )^{m + 4} \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, m + 4 \\ m + 5 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{d^{4} x \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**2)**(3/2)*(b*x+a)**n,x)

[Out]

c*sqrt(c*x**2)*(d*x)**(m + 4)*(1 + b*x/a)**(-n)*(a + b*x)**n*hyper((-n, m + 4),
(m + 5,), -b*x/a)/(d**4*x*(m + 4))

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Mathematica [A]  time = 0.0574414, size = 57, normalized size = 0.88 \[ \frac{x \left (c x^2\right )^{3/2} (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+4,-n;m+5;-\frac{b x}{a}\right )}{m+4} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*(c*x^2)^(3/2)*(a + b*x)^n,x]

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^2)^(3/2)*(b*x+a)^n,x)

[Out]

int((d*x)^m*(c*x^2)^(3/2)*(b*x+a)^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)*(b*x + a)^n*(d*x)^m,x, algorithm="maxima")

[Out]

integrate((c*x^2)^(3/2)*(b*x + a)^n*(d*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)*(b*x + a)^n*(d*x)^m,x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2)*(b*x + a)^n*(d*x)^m*c*x^2, 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*(c*x**2)**(3/2)*(b*x+a)**n,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{n} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)*(b*x + a)^n*(d*x)^m,x, algorithm="giac")

[Out]

integrate((c*x^2)^(3/2)*(b*x + a)^n*(d*x)^m, x)