3.937 \(\int (b x)^{5/2} (c+d x)^n (e+f x) \, dx\)

Optimal. Leaf size=107 \[ \frac{2 f (b x)^{7/2} (c+d x)^{n+1}}{b d (2 n+9)}-\frac{2 (b x)^{7/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (7 c f-d e (2 n+9)) \, _2F_1\left (\frac{7}{2},-n;\frac{9}{2};-\frac{d x}{c}\right )}{7 b d (2 n+9)} \]

[Out]

(2*f*(b*x)^(7/2)*(c + d*x)^(1 + n))/(b*d*(9 + 2*n)) - (2*(7*c*f - d*e*(9 + 2*n))
*(b*x)^(7/2)*(c + d*x)^n*Hypergeometric2F1[7/2, -n, 9/2, -((d*x)/c)])/(7*b*d*(9
+ 2*n)*(1 + (d*x)/c)^n)

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Rubi [A]  time = 0.125811, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 f (b x)^{7/2} (c+d x)^{n+1}}{b d (2 n+9)}-\frac{2 (b x)^{7/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (7 c f-d e (2 n+9)) \, _2F_1\left (\frac{7}{2},-n;\frac{9}{2};-\frac{d x}{c}\right )}{7 b d (2 n+9)} \]

Antiderivative was successfully verified.

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

[Out]

(2*f*(b*x)^(7/2)*(c + d*x)^(1 + n))/(b*d*(9 + 2*n)) - (2*(7*c*f - d*e*(9 + 2*n))
*(b*x)^(7/2)*(c + d*x)^n*Hypergeometric2F1[7/2, -n, 9/2, -((d*x)/c)])/(7*b*d*(9
+ 2*n)*(1 + (d*x)/c)^n)

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Rubi in Sympy [A]  time = 11.8917, size = 87, normalized size = 0.81 \[ \frac{2 f \left (b x\right )^{\frac{7}{2}} \left (c + d x\right )^{n + 1}}{b d \left (2 n + 9\right )} - \frac{2 \left (b x\right )^{\frac{7}{2}} \left (1 + \frac{d x}{c}\right )^{- n} \left (c + d x\right )^{n} \left (7 c f - d e \left (2 n + 9\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - n, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{- \frac{d x}{c}} \right )}}{7 b d \left (2 n + 9\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*f*(b*x)**(7/2)*(c + d*x)**(n + 1)/(b*d*(2*n + 9)) - 2*(b*x)**(7/2)*(1 + d*x/c)
**(-n)*(c + d*x)**n*(7*c*f - d*e*(2*n + 9))*hyper((-n, 7/2), (9/2,), -d*x/c)/(7*
b*d*(2*n + 9))

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Mathematica [A]  time = 0.069707, size = 73, normalized size = 0.68 \[ \frac{2}{63} x (b x)^{5/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (9 e \, _2F_1\left (\frac{7}{2},-n;\frac{9}{2};-\frac{d x}{c}\right )+7 f x \, _2F_1\left (\frac{9}{2},-n;\frac{11}{2};-\frac{d x}{c}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(b*x)^(5/2)*(c + d*x)^n*(9*e*Hypergeometric2F1[7/2, -n, 9/2, -((d*x)/c)] +
7*f*x*Hypergeometric2F1[9/2, -n, 11/2, -((d*x)/c)]))/(63*(1 + (d*x)/c)^n)

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{{\frac{5}{2}}} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x)^(5/2)*(d*x+c)^n*(f*x+e),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x)^(5/2)*(f*x + e)*(d*x + c)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} f x^{3} + b^{2} e x^{2}\right )} \sqrt{b x}{\left (d x + c\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*f*x^3 + b^2*e*x^2)*sqrt(b*x)*(d*x + c)^n, 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((b*x)**(5/2)*(d*x+c)**n*(f*x+e),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x)^(5/2)*(f*x + e)*(d*x + c)^n, x)