3.339 \(\int \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx\)

Optimal. Leaf size=296 \[ \frac{d x \left (a+b x^2\right )^{p+1} \left (15 a^2 d^2-8 a b c d (p+6)+b^2 c^2 \left (4 p^2+28 p+57\right )\right )}{b^3 (2 p+3) (2 p+5) (2 p+7)}-\frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (15 a^3 d^3-9 a^2 b c d^2 (2 p+7)+3 a b^2 c^2 d \left (4 p^2+24 p+35\right )-b^3 c^3 \left (8 p^3+60 p^2+142 p+105\right )\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{b^3 (2 p+3) (2 p+5) (2 p+7)}-\frac{d x \left (c+d x^2\right ) \left (a+b x^2\right )^{p+1} (5 a d-b c (2 p+11))}{b^2 (2 p+5) (2 p+7)}+\frac{d x \left (c+d x^2\right )^2 \left (a+b x^2\right )^{p+1}}{b (2 p+7)} \]

[Out]

(d*(15*a^2*d^2 - 8*a*b*c*d*(6 + p) + b^2*c^2*(57 + 28*p + 4*p^2))*x*(a + b*x^2)^
(1 + p))/(b^3*(3 + 2*p)*(5 + 2*p)*(7 + 2*p)) - (d*(5*a*d - b*c*(11 + 2*p))*x*(a
+ b*x^2)^(1 + p)*(c + d*x^2))/(b^2*(5 + 2*p)*(7 + 2*p)) + (d*x*(a + b*x^2)^(1 +
p)*(c + d*x^2)^2)/(b*(7 + 2*p)) - ((15*a^3*d^3 - 9*a^2*b*c*d^2*(7 + 2*p) + 3*a*b
^2*c^2*d*(35 + 24*p + 4*p^2) - b^3*c^3*(105 + 142*p + 60*p^2 + 8*p^3))*x*(a + b*
x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(b^3*(3 + 2*p)*(5 + 2*p)*(
7 + 2*p)*(1 + (b*x^2)/a)^p)

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Rubi [A]  time = 0.638722, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{d x \left (a+b x^2\right )^{p+1} \left (15 a^2 d^2-8 a b c d (p+6)+b^2 c^2 \left (4 p^2+28 p+57\right )\right )}{b^3 (2 p+3) (2 p+5) (2 p+7)}-\frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (15 a^3 d^3-9 a^2 b c d^2 (2 p+7)+3 a b^2 c^2 d \left (4 p^2+24 p+35\right )-b^3 c^3 \left (8 p^3+60 p^2+142 p+105\right )\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{b^3 (2 p+3) (2 p+5) (2 p+7)}-\frac{d x \left (c+d x^2\right ) \left (a+b x^2\right )^{p+1} (5 a d-b c (2 p+11))}{b^2 (2 p+5) (2 p+7)}+\frac{d x \left (c+d x^2\right )^2 \left (a+b x^2\right )^{p+1}}{b (2 p+7)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(15*a^2*d^2 - 8*a*b*c*d*(6 + p) + b^2*c^2*(57 + 28*p + 4*p^2))*x*(a + b*x^2)^
(1 + p))/(b^3*(3 + 2*p)*(5 + 2*p)*(7 + 2*p)) - (d*(5*a*d - b*c*(11 + 2*p))*x*(a
+ b*x^2)^(1 + p)*(c + d*x^2))/(b^2*(5 + 2*p)*(7 + 2*p)) + (d*x*(a + b*x^2)^(1 +
p)*(c + d*x^2)^2)/(b*(7 + 2*p)) - ((15*a^3*d^3 - 9*a^2*b*c*d^2*(7 + 2*p) + 3*a*b
^2*c^2*d*(35 + 24*p + 4*p^2) - b^3*c^3*(105 + 142*p + 60*p^2 + 8*p^3))*x*(a + b*
x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)])/(b^3*(3 + 2*p)*(5 + 2*p)*(
7 + 2*p)*(1 + (b*x^2)/a)^p)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0975513, size = 136, normalized size = 0.46 \[ \frac{1}{35} x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (35 c^3 \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+d x^2 \left (35 c^2 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )+d x^2 \left (21 c \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )+5 d x^2 \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};-\frac{b x^2}{a}\right )\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^2)^p*(35*c^3*Hypergeometric2F1[1/2, -p, 3/2, -((b*x^2)/a)] + d*x^2*(
35*c^2*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)] + d*x^2*(21*c*Hypergeometri
c2F1[5/2, -p, 7/2, -((b*x^2)/a)] + 5*d*x^2*Hypergeometric2F1[7/2, -p, 9/2, -((b*
x^2)/a)]))))/(35*(1 + (b*x^2)/a)^p)

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^p*(d*x^2+c)^3,x)

[Out]

int((b*x^2+a)^p*(d*x^2+c)^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x^2 + c)^3*(b*x^2 + a)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}\right )}{\left (b x^{2} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3)*(b*x^2 + a)^p, 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**2+a)**p*(d*x**2+c)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x^2 + c)^3*(b*x^2 + a)^p, x)