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

Optimal. Leaf size=195 \[ \frac{2 (b x)^{7/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (63 c^2 f^2-14 c d e f (2 n+11)+d^2 e^2 \left (4 n^2+40 n+99\right )\right ) \, _2F_1\left (\frac{7}{2},-n;\frac{9}{2};-\frac{d x}{c}\right )}{7 b d^2 (2 n+9) (2 n+11)}-\frac{2 f (b x)^{7/2} (c+d x)^{n+1} (9 c f-d e (2 n+13))}{b d^2 (2 n+9) (2 n+11)}+\frac{2 f (b x)^{7/2} (e+f x) (c+d x)^{n+1}}{b d (2 n+11)} \]

[Out]

(-2*f*(9*c*f - d*e*(13 + 2*n))*(b*x)^(7/2)*(c + d*x)^(1 + n))/(b*d^2*(9 + 2*n)*(
11 + 2*n)) + (2*f*(b*x)^(7/2)*(c + d*x)^(1 + n)*(e + f*x))/(b*d*(11 + 2*n)) + (2
*(63*c^2*f^2 - 14*c*d*e*f*(11 + 2*n) + d^2*e^2*(99 + 40*n + 4*n^2))*(b*x)^(7/2)*
(c + d*x)^n*Hypergeometric2F1[7/2, -n, 9/2, -((d*x)/c)])/(7*b*d^2*(9 + 2*n)*(11
+ 2*n)*(1 + (d*x)/c)^n)

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Rubi [A]  time = 0.327715, antiderivative size = 195, 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 \[ \frac{2 (b x)^{7/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (63 c^2 f^2-14 c d e f (2 n+11)+d^2 e^2 \left (4 n^2+40 n+99\right )\right ) \, _2F_1\left (\frac{7}{2},-n;\frac{9}{2};-\frac{d x}{c}\right )}{7 b d^2 (2 n+9) (2 n+11)}-\frac{2 f (b x)^{7/2} (c+d x)^{n+1} (9 c f-d e (2 n+13))}{b d^2 (2 n+9) (2 n+11)}+\frac{2 f (b x)^{7/2} (e+f x) (c+d x)^{n+1}}{b d (2 n+11)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*f*(9*c*f - d*e*(13 + 2*n))*(b*x)^(7/2)*(c + d*x)^(1 + n))/(b*d^2*(9 + 2*n)*(
11 + 2*n)) + (2*f*(b*x)^(7/2)*(c + d*x)^(1 + n)*(e + f*x))/(b*d*(11 + 2*n)) + (2
*(63*c^2*f^2 - 14*c*d*e*f*(11 + 2*n) + d^2*e^2*(99 + 40*n + 4*n^2))*(b*x)^(7/2)*
(c + d*x)^n*Hypergeometric2F1[7/2, -n, 9/2, -((d*x)/c)])/(7*b*d^2*(9 + 2*n)*(11
+ 2*n)*(1 + (d*x)/c)^n)

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Rubi in Sympy [A]  time = 33.2964, size = 173, normalized size = 0.89 \[ \frac{2 f \left (b x\right )^{\frac{7}{2}} \left (c + d x\right )^{n + 1} \left (e + f x\right )}{b d \left (2 n + 11\right )} - \frac{2 f \left (b x\right )^{\frac{7}{2}} \left (c + d x\right )^{n + 1} \left (9 c f - d e \left (2 n + 13\right )\right )}{b d^{2} \left (2 n + 9\right ) \left (2 n + 11\right )} + \frac{2 \left (b x\right )^{\frac{7}{2}} \left (1 + \frac{d x}{c}\right )^{- n} \left (c + d x\right )^{n} \left (7 c f \left (9 c f - d e \left (2 n + 13\right )\right ) - d e \left (2 n + 9\right ) \left (7 c f - d e \left (2 n + 11\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - n, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{- \frac{d x}{c}} \right )}}{7 b d^{2} \left (2 n + 9\right ) \left (2 n + 11\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*f*(b*x)**(7/2)*(c + d*x)**(n + 1)*(e + f*x)/(b*d*(2*n + 11)) - 2*f*(b*x)**(7/2
)*(c + d*x)**(n + 1)*(9*c*f - d*e*(2*n + 13))/(b*d**2*(2*n + 9)*(2*n + 11)) + 2*
(b*x)**(7/2)*(1 + d*x/c)**(-n)*(c + d*x)**n*(7*c*f*(9*c*f - d*e*(2*n + 13)) - d*
e*(2*n + 9)*(7*c*f - d*e*(2*n + 11)))*hyper((-n, 7/2), (9/2,), -d*x/c)/(7*b*d**2
*(2*n + 9)*(2*n + 11))

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Mathematica [A]  time = 0.133555, size = 100, normalized size = 0.51 \[ \frac{2}{693} x (b x)^{5/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (99 e^2 \, _2F_1\left (\frac{7}{2},-n;\frac{9}{2};-\frac{d x}{c}\right )+7 f x \left (22 e \, _2F_1\left (\frac{9}{2},-n;\frac{11}{2};-\frac{d x}{c}\right )+9 f x \, _2F_1\left (\frac{11}{2},-n;\frac{13}{2};-\frac{d x}{c}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(b*x)^(5/2)*(c + d*x)^n*(99*e^2*Hypergeometric2F1[7/2, -n, 9/2, -((d*x)/c)]
+ 7*f*x*(22*e*Hypergeometric2F1[9/2, -n, 11/2, -((d*x)/c)] + 9*f*x*Hypergeometr
ic2F1[11/2, -n, 13/2, -((d*x)/c)])))/(693*(1 + (d*x)/c)^n)

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{{\frac{5}{2}}} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b^{2} f^{2} x^{4} + 2 \, b^{2} e f x^{3} + b^{2} e^{2} x^{2}\right )} \sqrt{b x}{\left (d x + c\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*f^2*x^4 + 2*b^2*e*f*x^3 + b^2*e^2*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)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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