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

Optimal. Leaf size=240 \[ \frac{x \left (c+d x^2\right )^{5/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{480 d^2}+\frac{c x \left (c+d x^2\right )^{3/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{384 d^2}+\frac{c^2 x \sqrt{c+d x^2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{256 d^2}+\frac{c^3 \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{256 d^{5/2}}-\frac{3 b x \left (c+d x^2\right )^{7/2} (b c-4 a d)}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d} \]

[Out]

(c^2*(3*b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*x*Sqrt[c + d*x^2])/(256*d^2) + (c*(3*
b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*x*(c + d*x^2)^(3/2))/(384*d^2) + ((3*b^2*c^2
- 20*a*b*c*d + 80*a^2*d^2)*x*(c + d*x^2)^(5/2))/(480*d^2) - (3*b*(b*c - 4*a*d)*x
*(c + d*x^2)^(7/2))/(80*d^2) + (b*x*(a + b*x^2)*(c + d*x^2)^(7/2))/(10*d) + (c^3
*(3*b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(25
6*d^(5/2))

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Rubi [A]  time = 0.328643, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \left (c+d x^2\right )^{5/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{480 d^2}+\frac{c x \left (c+d x^2\right )^{3/2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{384 d^2}+\frac{c^2 x \sqrt{c+d x^2} \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right )}{256 d^2}+\frac{c^3 \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{256 d^{5/2}}-\frac{3 b x \left (c+d x^2\right )^{7/2} (b c-4 a d)}{80 d^2}+\frac{b x \left (a+b x^2\right ) \left (c+d x^2\right )^{7/2}}{10 d} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(3*b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*x*Sqrt[c + d*x^2])/(256*d^2) + (c*(3*
b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*x*(c + d*x^2)^(3/2))/(384*d^2) + ((3*b^2*c^2
- 20*a*b*c*d + 80*a^2*d^2)*x*(c + d*x^2)^(5/2))/(480*d^2) - (3*b*(b*c - 4*a*d)*x
*(c + d*x^2)^(7/2))/(80*d^2) + (b*x*(a + b*x^2)*(c + d*x^2)^(7/2))/(10*d) + (c^3
*(3*b^2*c^2 - 20*a*b*c*d + 80*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(25
6*d^(5/2))

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Rubi in Sympy [A]  time = 32.1846, size = 238, normalized size = 0.99 \[ \frac{b x \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{7}{2}}}{10 d} + \frac{3 b x \left (c + d x^{2}\right )^{\frac{7}{2}} \left (4 a d - b c\right )}{80 d^{2}} + \frac{c^{3} \left (80 a^{2} d^{2} - 20 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{256 d^{\frac{5}{2}}} + \frac{c^{2} x \sqrt{c + d x^{2}} \left (80 a^{2} d^{2} - 20 a b c d + 3 b^{2} c^{2}\right )}{256 d^{2}} + \frac{c x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (80 a^{2} d^{2} - 20 a b c d + 3 b^{2} c^{2}\right )}{384 d^{2}} + \frac{x \left (c + d x^{2}\right )^{\frac{5}{2}} \left (80 a^{2} d^{2} - 20 a b c d + 3 b^{2} c^{2}\right )}{480 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x*(a + b*x**2)*(c + d*x**2)**(7/2)/(10*d) + 3*b*x*(c + d*x**2)**(7/2)*(4*a*d -
b*c)/(80*d**2) + c**3*(80*a**2*d**2 - 20*a*b*c*d + 3*b**2*c**2)*atanh(sqrt(d)*x
/sqrt(c + d*x**2))/(256*d**(5/2)) + c**2*x*sqrt(c + d*x**2)*(80*a**2*d**2 - 20*a
*b*c*d + 3*b**2*c**2)/(256*d**2) + c*x*(c + d*x**2)**(3/2)*(80*a**2*d**2 - 20*a*
b*c*d + 3*b**2*c**2)/(384*d**2) + x*(c + d*x**2)**(5/2)*(80*a**2*d**2 - 20*a*b*c
*d + 3*b**2*c**2)/(480*d**2)

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Mathematica [A]  time = 0.205084, size = 192, normalized size = 0.8 \[ \frac{15 c^3 \left (80 a^2 d^2-20 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\sqrt{d} x \sqrt{c+d x^2} \left (80 a^2 d^2 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+20 a b d \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )+b^2 \left (-45 c^4+30 c^3 d x^2+744 c^2 d^2 x^4+1008 c d^3 x^6+384 d^4 x^8\right )\right )}{3840 d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^2]*(80*a^2*d^2*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^4) + 20*a*
b*d*(15*c^3 + 118*c^2*d*x^2 + 136*c*d^2*x^4 + 48*d^3*x^6) + b^2*(-45*c^4 + 30*c^
3*d*x^2 + 744*c^2*d^2*x^4 + 1008*c*d^3*x^6 + 384*d^4*x^8)) + 15*c^3*(3*b^2*c^2 -
20*a*b*c*d + 80*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(3840*d^(5/2))

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Maple [A]  time = 0.012, size = 308, normalized size = 1.3 \[{\frac{{a}^{2}x}{6} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}cx}{24} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{c}^{2}x}{16}\sqrt{d{x}^{2}+c}}+{\frac{5\,{a}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}+{\frac{{b}^{2}{x}^{3}}{10\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{b}^{2}cx}{80\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}{c}^{2}x}{160\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{x{b}^{2}{c}^{3}}{128\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{2}{c}^{4}x}{256\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{b}^{2}{c}^{5}}{256}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{abx}{4\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{abcx}{24\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,ab{c}^{2}x}{96\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,ab{c}^{3}x}{64\,d}\sqrt{d{x}^{2}+c}}-{\frac{5\,ab{c}^{4}}{64}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*a^2*x*(d*x^2+c)^(5/2)+5/24*a^2*c*x*(d*x^2+c)^(3/2)+5/16*a^2*c^2*x*(d*x^2+c)^
(1/2)+5/16*a^2*c^3/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/10*b^2*x^3*(d*x^2+c)^
(7/2)/d-3/80*b^2*c/d^2*x*(d*x^2+c)^(7/2)+1/160*b^2*c^2/d^2*x*(d*x^2+c)^(5/2)+1/1
28*b^2*c^3/d^2*x*(d*x^2+c)^(3/2)+3/256*b^2*c^4/d^2*x*(d*x^2+c)^(1/2)+3/256*b^2*c
^5/d^(5/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/4*a*b*x*(d*x^2+c)^(7/2)/d-1/24*a*b*c/
d*x*(d*x^2+c)^(5/2)-5/96*a*b*c^2/d*x*(d*x^2+c)^(3/2)-5/64*a*b*c^3/d*x*(d*x^2+c)^
(1/2)-5/64*a*b*c^4/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.531597, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (384 \, b^{2} d^{4} x^{9} + 48 \,{\left (21 \, b^{2} c d^{3} + 20 \, a b d^{4}\right )} x^{7} + 8 \,{\left (93 \, b^{2} c^{2} d^{2} + 340 \, a b c d^{3} + 80 \, a^{2} d^{4}\right )} x^{5} + 10 \,{\left (3 \, b^{2} c^{3} d + 236 \, a b c^{2} d^{2} + 208 \, a^{2} c d^{3}\right )} x^{3} - 15 \,{\left (3 \, b^{2} c^{4} - 20 \, a b c^{3} d - 176 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 15 \,{\left (3 \, b^{2} c^{5} - 20 \, a b c^{4} d + 80 \, a^{2} c^{3} d^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{7680 \, d^{\frac{5}{2}}}, \frac{{\left (384 \, b^{2} d^{4} x^{9} + 48 \,{\left (21 \, b^{2} c d^{3} + 20 \, a b d^{4}\right )} x^{7} + 8 \,{\left (93 \, b^{2} c^{2} d^{2} + 340 \, a b c d^{3} + 80 \, a^{2} d^{4}\right )} x^{5} + 10 \,{\left (3 \, b^{2} c^{3} d + 236 \, a b c^{2} d^{2} + 208 \, a^{2} c d^{3}\right )} x^{3} - 15 \,{\left (3 \, b^{2} c^{4} - 20 \, a b c^{3} d - 176 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 15 \,{\left (3 \, b^{2} c^{5} - 20 \, a b c^{4} d + 80 \, a^{2} c^{3} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{3840 \, \sqrt{-d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(2*(384*b^2*d^4*x^9 + 48*(21*b^2*c*d^3 + 20*a*b*d^4)*x^7 + 8*(93*b^2*c^2
*d^2 + 340*a*b*c*d^3 + 80*a^2*d^4)*x^5 + 10*(3*b^2*c^3*d + 236*a*b*c^2*d^2 + 208
*a^2*c*d^3)*x^3 - 15*(3*b^2*c^4 - 20*a*b*c^3*d - 176*a^2*c^2*d^2)*x)*sqrt(d*x^2
+ c)*sqrt(d) + 15*(3*b^2*c^5 - 20*a*b*c^4*d + 80*a^2*c^3*d^2)*log(-2*sqrt(d*x^2
+ c)*d*x - (2*d*x^2 + c)*sqrt(d)))/d^(5/2), 1/3840*((384*b^2*d^4*x^9 + 48*(21*b^
2*c*d^3 + 20*a*b*d^4)*x^7 + 8*(93*b^2*c^2*d^2 + 340*a*b*c*d^3 + 80*a^2*d^4)*x^5
+ 10*(3*b^2*c^3*d + 236*a*b*c^2*d^2 + 208*a^2*c*d^3)*x^3 - 15*(3*b^2*c^4 - 20*a*
b*c^3*d - 176*a^2*c^2*d^2)*x)*sqrt(d*x^2 + c)*sqrt(-d) + 15*(3*b^2*c^5 - 20*a*b*
c^4*d + 80*a^2*c^3*d^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(sqrt(-d)*d^2)]

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Sympy [A]  time = 170.432, size = 537, normalized size = 2.24 \[ \frac{a^{2} c^{\frac{5}{2}} x \sqrt{1 + \frac{d x^{2}}{c}}}{2} + \frac{3 a^{2} c^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 a^{2} c^{\frac{3}{2}} d x^{3}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 a^{2} \sqrt{c} d^{2} x^{5}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{16 \sqrt{d}} + \frac{a^{2} d^{3} x^{7}}{6 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a b c^{\frac{7}{2}} x}{64 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{133 a b c^{\frac{5}{2}} x^{3}}{192 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{127 a b c^{\frac{3}{2}} d x^{5}}{96 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{23 a b \sqrt{c} d^{2} x^{7}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 a b c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{64 d^{\frac{3}{2}}} + \frac{a b d^{3} x^{9}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{3 b^{2} c^{\frac{9}{2}} x}{256 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{7}{2}} x^{3}}{256 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{129 b^{2} c^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{73 b^{2} c^{\frac{3}{2}} d x^{7}}{160 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{29 b^{2} \sqrt{c} d^{2} x^{9}}{80 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} c^{5} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{256 d^{\frac{5}{2}}} + \frac{b^{2} d^{3} x^{11}}{10 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**(5/2)*x*sqrt(1 + d*x**2/c)/2 + 3*a**2*c**(5/2)*x/(16*sqrt(1 + d*x**2/c))
+ 35*a**2*c**(3/2)*d*x**3/(48*sqrt(1 + d*x**2/c)) + 17*a**2*sqrt(c)*d**2*x**5/(
24*sqrt(1 + d*x**2/c)) + 5*a**2*c**3*asinh(sqrt(d)*x/sqrt(c))/(16*sqrt(d)) + a**
2*d**3*x**7/(6*sqrt(c)*sqrt(1 + d*x**2/c)) + 5*a*b*c**(7/2)*x/(64*d*sqrt(1 + d*x
**2/c)) + 133*a*b*c**(5/2)*x**3/(192*sqrt(1 + d*x**2/c)) + 127*a*b*c**(3/2)*d*x*
*5/(96*sqrt(1 + d*x**2/c)) + 23*a*b*sqrt(c)*d**2*x**7/(24*sqrt(1 + d*x**2/c)) -
5*a*b*c**4*asinh(sqrt(d)*x/sqrt(c))/(64*d**(3/2)) + a*b*d**3*x**9/(4*sqrt(c)*sqr
t(1 + d*x**2/c)) - 3*b**2*c**(9/2)*x/(256*d**2*sqrt(1 + d*x**2/c)) - b**2*c**(7/
2)*x**3/(256*d*sqrt(1 + d*x**2/c)) + 129*b**2*c**(5/2)*x**5/(640*sqrt(1 + d*x**2
/c)) + 73*b**2*c**(3/2)*d*x**7/(160*sqrt(1 + d*x**2/c)) + 29*b**2*sqrt(c)*d**2*x
**9/(80*sqrt(1 + d*x**2/c)) + 3*b**2*c**5*asinh(sqrt(d)*x/sqrt(c))/(256*d**(5/2)
) + b**2*d**3*x**11/(10*sqrt(c)*sqrt(1 + d*x**2/c))

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GIAC/XCAS [A]  time = 0.246865, size = 298, normalized size = 1.24 \[ \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, b^{2} d^{2} x^{2} + \frac{21 \, b^{2} c d^{9} + 20 \, a b d^{10}}{d^{8}}\right )} x^{2} + \frac{93 \, b^{2} c^{2} d^{8} + 340 \, a b c d^{9} + 80 \, a^{2} d^{10}}{d^{8}}\right )} x^{2} + \frac{5 \,{\left (3 \, b^{2} c^{3} d^{7} + 236 \, a b c^{2} d^{8} + 208 \, a^{2} c d^{9}\right )}}{d^{8}}\right )} x^{2} - \frac{15 \,{\left (3 \, b^{2} c^{4} d^{6} - 20 \, a b c^{3} d^{7} - 176 \, a^{2} c^{2} d^{8}\right )}}{d^{8}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (3 \, b^{2} c^{5} - 20 \, a b c^{4} d + 80 \, a^{2} c^{3} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{256 \, d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840*(2*(4*(6*(8*b^2*d^2*x^2 + (21*b^2*c*d^9 + 20*a*b*d^10)/d^8)*x^2 + (93*b^2
*c^2*d^8 + 340*a*b*c*d^9 + 80*a^2*d^10)/d^8)*x^2 + 5*(3*b^2*c^3*d^7 + 236*a*b*c^
2*d^8 + 208*a^2*c*d^9)/d^8)*x^2 - 15*(3*b^2*c^4*d^6 - 20*a*b*c^3*d^7 - 176*a^2*c
^2*d^8)/d^8)*sqrt(d*x^2 + c)*x - 1/256*(3*b^2*c^5 - 20*a*b*c^4*d + 80*a^2*c^3*d^
2)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(5/2)