3.644 \(\int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=437 \[ -\frac{\left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{11/2}}+\frac{(a+b x)^{7/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)}{320 b^4 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^4}{1024 b^4 d^5}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^3}{1536 b^4 d^4}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^2}{1920 b^4 d^3}+\frac{(a+b x)^{7/2} (c+d x)^{3/2} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right )}{120 b^3 d^2}-\frac{(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d} \]

[Out]

((b*c - a*d)^4*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])
/(1024*b^4*d^5) - ((b*c - a*d)^3*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^
(3/2)*Sqrt[c + d*x])/(1536*b^4*d^4) + ((b*c - a*d)^2*(9*b^2*c^2 + 10*a*b*c*d + 5
*a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(1920*b^4*d^3) + ((b*c - a*d)*(9*b^2*c^
2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(320*b^4*d^2) + ((9*b
^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(120*b^3*d^2)
- ((9*b*c + 7*a*d)*(a + b*x)^(7/2)*(c + d*x)^(5/2))/(84*b^2*d^2) + (x*(a + b*x)^
(7/2)*(c + d*x)^(5/2))/(7*b*d) - ((b*c - a*d)^5*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*
d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(1024*b^(9/2)*d^(
11/2))

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Rubi [A]  time = 0.982276, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{11/2}}+\frac{(a+b x)^{7/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)}{320 b^4 d^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^4}{1024 b^4 d^5}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^3}{1536 b^4 d^4}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) (b c-a d)^2}{1920 b^4 d^3}+\frac{(a+b x)^{7/2} (c+d x)^{3/2} \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right )}{120 b^3 d^2}-\frac{(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{84 b^2 d^2}+\frac{x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]

[Out]

((b*c - a*d)^4*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])
/(1024*b^4*d^5) - ((b*c - a*d)^3*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^
(3/2)*Sqrt[c + d*x])/(1536*b^4*d^4) + ((b*c - a*d)^2*(9*b^2*c^2 + 10*a*b*c*d + 5
*a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(1920*b^4*d^3) + ((b*c - a*d)*(9*b^2*c^
2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(320*b^4*d^2) + ((9*b
^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(120*b^3*d^2)
- ((9*b*c + 7*a*d)*(a + b*x)^(7/2)*(c + d*x)^(5/2))/(84*b^2*d^2) + (x*(a + b*x)^
(7/2)*(c + d*x)^(5/2))/(7*b*d) - ((b*c - a*d)^5*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*
d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(1024*b^(9/2)*d^(
11/2))

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Rubi in Sympy [A]  time = 95.472, size = 420, normalized size = 0.96 \[ \frac{x \left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{5}{2}}}{7 b d} - \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (7 a d + 9 b c\right )}{84 b^{2} d^{2}} + \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (5 a^{2} d^{2} + 10 a b c d + 9 b^{2} c^{2}\right )}{120 b^{3} d^{2}} - \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (5 a^{2} d^{2} + 10 a b c d + 9 b^{2} c^{2}\right )}{320 b^{4} d^{2}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (5 a^{2} d^{2} + 10 a b c d + 9 b^{2} c^{2}\right )}{1920 b^{4} d^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (5 a^{2} d^{2} + 10 a b c d + 9 b^{2} c^{2}\right )}{1536 b^{4} d^{4}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4} \left (5 a^{2} d^{2} + 10 a b c d + 9 b^{2} c^{2}\right )}{1024 b^{4} d^{5}} + \frac{\left (a d - b c\right )^{5} \left (5 a^{2} d^{2} + 10 a b c d + 9 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{1024 b^{\frac{9}{2}} d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)**(5/2)*(d*x+c)**(3/2),x)

[Out]

x*(a + b*x)**(7/2)*(c + d*x)**(5/2)/(7*b*d) - (a + b*x)**(7/2)*(c + d*x)**(5/2)*
(7*a*d + 9*b*c)/(84*b**2*d**2) + (a + b*x)**(7/2)*(c + d*x)**(3/2)*(5*a**2*d**2
+ 10*a*b*c*d + 9*b**2*c**2)/(120*b**3*d**2) - (a + b*x)**(7/2)*sqrt(c + d*x)*(a*
d - b*c)*(5*a**2*d**2 + 10*a*b*c*d + 9*b**2*c**2)/(320*b**4*d**2) + (a + b*x)**(
5/2)*sqrt(c + d*x)*(a*d - b*c)**2*(5*a**2*d**2 + 10*a*b*c*d + 9*b**2*c**2)/(1920
*b**4*d**3) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**3*(5*a**2*d**2 + 10*a*
b*c*d + 9*b**2*c**2)/(1536*b**4*d**4) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)*
*4*(5*a**2*d**2 + 10*a*b*c*d + 9*b**2*c**2)/(1024*b**4*d**5) + (a*d - b*c)**5*(5
*a**2*d**2 + 10*a*b*c*d + 9*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt
(c + d*x)))/(1024*b**(9/2)*d**(11/2))

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Mathematica [A]  time = 0.404465, size = 393, normalized size = 0.9 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-525 a^6 d^6+350 a^5 b d^5 (4 c+d x)-35 a^4 b^2 d^4 \left (15 c^2+26 c d x+8 d^2 x^2\right )+60 a^3 b^3 d^3 \left (-10 c^3+5 c^2 d x+12 c d^2 x^2+4 d^3 x^3\right )+a^2 b^4 d^2 \left (3689 c^4-2332 c^3 d x+1824 c^2 d^2 x^2+33520 c d^3 x^3+23680 d^4 x^4\right )+2 a b^5 d \left (-1680 c^5+1099 c^4 d x-872 c^3 d^2 x^2+744 c^2 d^3 x^3+24320 c d^4 x^4+18560 d^5 x^5\right )+3 b^6 \left (315 c^6-210 c^5 d x+168 c^4 d^2 x^2-144 c^3 d^3 x^3+128 c^2 d^4 x^4+6400 c d^5 x^5+5120 d^6 x^6\right )\right )}{107520 b^4 d^5}-\frac{(b c-a d)^5 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2048 b^{9/2} d^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-525*a^6*d^6 + 350*a^5*b*d^5*(4*c + d*x) - 35*a^4*
b^2*d^4*(15*c^2 + 26*c*d*x + 8*d^2*x^2) + 60*a^3*b^3*d^3*(-10*c^3 + 5*c^2*d*x +
12*c*d^2*x^2 + 4*d^3*x^3) + a^2*b^4*d^2*(3689*c^4 - 2332*c^3*d*x + 1824*c^2*d^2*
x^2 + 33520*c*d^3*x^3 + 23680*d^4*x^4) + 2*a*b^5*d*(-1680*c^5 + 1099*c^4*d*x - 8
72*c^3*d^2*x^2 + 744*c^2*d^3*x^3 + 24320*c*d^4*x^4 + 18560*d^5*x^5) + 3*b^6*(315
*c^6 - 210*c^5*d*x + 168*c^4*d^2*x^2 - 144*c^3*d^3*x^3 + 128*c^2*d^4*x^4 + 6400*
c*d^5*x^5 + 5120*d^6*x^6)))/(107520*b^4*d^5) - ((b*c - a*d)^5*(9*b^2*c^2 + 10*a*
b*c*d + 5*a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqr
t[c + d*x]])/(2048*b^(9/2)*d^(11/2))

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Maple [B]  time = 0.03, size = 1580, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^(5/2)*(d*x+c)^(3/2),x)

[Out]

1/215040*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(480*x^3*a^3*b^3*d^6*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*(b*d)^(1/2)-6720*a*b^5*c^5*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/
2)+74240*x^5*a*b^5*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+38400*x^5*b^6
*c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+47360*x^4*a^2*b^4*d^6*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+768*x^4*b^6*c^2*d^4*(b*d*x^2+a*d*x+b*c*x+a*
c)^(1/2)*(b*d)^(1/2)+700*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^5*d^6*b*(b*d)^(1/2)
-1260*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*c^5*b^6*d*(b*d)^(1/2)+2800*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*a^5*c*d^5*b*(b*d)^(1/2)-1050*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*
a^4*c^2*b^2*d^4*(b*d)^(1/2)-1200*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*c^3*b^3*d^3
*(b*d)^(1/2)+7378*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*c^4*a^2*b^4*d^2*(b*d)^(1/2)-86
4*x^3*b^6*c^3*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-560*x^2*a^4*b^2*d^
6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+1008*x^2*b^6*c^4*d^2*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-1575*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^6*c*d^6*b+945*ln(1/2*(2*b*d*x+2*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*c^2*d^5*b^2+525*
ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/
2))*a^4*c^3*b^3*d^4+1575*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)
^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*c^4*b^4*d^3-4725*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d
*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c^5*a^2*b^5*d^2+3675*ln(1/
2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c
^6*a*b^6*d+30720*x^6*b^6*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-1050*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^6*d^6*(b*d)^(1/2)+1890*(b*d*x^2+a*d*x+b*c*x+a*c)
^(1/2)*c^6*b^6*(b*d)^(1/2)-1820*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*c*b^2*
(b*d)^(1/2)+600*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*c^2*b^3*d^4*(b*d)^(1/2)-46
64*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*c^3*b^4*d^3*(b*d)^(1/2)+4396*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)*x*c^4*a*b^5*d^2*(b*d)^(1/2)+97280*x^4*a*b^5*c*d^5*(b*d*x^2
+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+67040*x^3*a^2*b^4*c*d^5*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*(b*d)^(1/2)+2976*x^3*a*b^5*c^2*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(
b*d)^(1/2)+1440*x^2*a^3*b^3*c*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+36
48*x^2*a^2*b^4*c^2*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-3488*x^2*a*b^
5*c^3*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+525*d^7*ln(1/2*(2*b*d*x+2*
(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^7-945*b^7*ln
(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2)
)*c^7)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/b^4/(b*d)^(1/2)/d^5

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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 + a)^(5/2)*(d*x + c)^(3/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.325268, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)*x^2,x, algorithm="fricas")

[Out]

[1/430080*(4*(15360*b^6*d^6*x^6 + 945*b^6*c^6 - 3360*a*b^5*c^5*d + 3689*a^2*b^4*
c^4*d^2 - 600*a^3*b^3*c^3*d^3 - 525*a^4*b^2*c^2*d^4 + 1400*a^5*b*c*d^5 - 525*a^6
*d^6 + 1280*(15*b^6*c*d^5 + 29*a*b^5*d^6)*x^5 + 128*(3*b^6*c^2*d^4 + 380*a*b^5*c
*d^5 + 185*a^2*b^4*d^6)*x^4 - 16*(27*b^6*c^3*d^3 - 93*a*b^5*c^2*d^4 - 2095*a^2*b
^4*c*d^5 - 15*a^3*b^3*d^6)*x^3 + 8*(63*b^6*c^4*d^2 - 218*a*b^5*c^3*d^3 + 228*a^2
*b^4*c^2*d^4 + 90*a^3*b^3*c*d^5 - 35*a^4*b^2*d^6)*x^2 - 2*(315*b^6*c^5*d - 1099*
a*b^5*c^4*d^2 + 1166*a^2*b^4*c^3*d^3 - 150*a^3*b^3*c^2*d^4 + 455*a^4*b^2*c*d^5 -
175*a^5*b*d^6)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 105*(9*b^7*c^7 - 35*a
*b^6*c^6*d + 45*a^2*b^5*c^5*d^2 - 15*a^3*b^4*c^4*d^3 - 5*a^4*b^3*c^3*d^4 - 9*a^5
*b^2*c^2*d^5 + 15*a^6*b*c*d^6 - 5*a^7*d^7)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^
2)*sqrt(b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2
+ 8*(b^2*c*d + a*b*d^2)*x)*sqrt(b*d)))/(sqrt(b*d)*b^4*d^5), 1/215040*(2*(15360*b
^6*d^6*x^6 + 945*b^6*c^6 - 3360*a*b^5*c^5*d + 3689*a^2*b^4*c^4*d^2 - 600*a^3*b^3
*c^3*d^3 - 525*a^4*b^2*c^2*d^4 + 1400*a^5*b*c*d^5 - 525*a^6*d^6 + 1280*(15*b^6*c
*d^5 + 29*a*b^5*d^6)*x^5 + 128*(3*b^6*c^2*d^4 + 380*a*b^5*c*d^5 + 185*a^2*b^4*d^
6)*x^4 - 16*(27*b^6*c^3*d^3 - 93*a*b^5*c^2*d^4 - 2095*a^2*b^4*c*d^5 - 15*a^3*b^3
*d^6)*x^3 + 8*(63*b^6*c^4*d^2 - 218*a*b^5*c^3*d^3 + 228*a^2*b^4*c^2*d^4 + 90*a^3
*b^3*c*d^5 - 35*a^4*b^2*d^6)*x^2 - 2*(315*b^6*c^5*d - 1099*a*b^5*c^4*d^2 + 1166*
a^2*b^4*c^3*d^3 - 150*a^3*b^3*c^2*d^4 + 455*a^4*b^2*c*d^5 - 175*a^5*b*d^6)*x)*sq
rt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 105*(9*b^7*c^7 - 35*a*b^6*c^6*d + 45*a^2*
b^5*c^5*d^2 - 15*a^3*b^4*c^4*d^3 - 5*a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 + 15*a^
6*b*c*d^6 - 5*a^7*d^7)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a
)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b^4*d^5)]

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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(x**2*(b*x+a)**(5/2)*(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.41235, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)*x^2,x, algorithm="giac")

[Out]

Done