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

Optimal. Leaf size=437 \[ \frac{\left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{11/2} d^{9/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{384 b^5 d^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^4}{1024 b^5 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{1536 b^5 d^3}+\frac{(a+b x)^{5/2} (c+d x)^{3/2} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac{(a+b x)^{5/2} (c+d x)^{5/2} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right )}{120 b^3 d^2}-\frac{(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{84 b^2 d^2}+\frac{x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d} \]

[Out]

-((b*c - a*d)^4*(5*b^2*c^2 + 10*a*b*c*d + 9*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x]
)/(1024*b^5*d^4) + ((b*c - a*d)^3*(5*b^2*c^2 + 10*a*b*c*d + 9*a^2*d^2)*(a + b*x)
^(3/2)*Sqrt[c + d*x])/(1536*b^5*d^3) + ((b*c - a*d)^2*(5*b^2*c^2 + 10*a*b*c*d +
9*a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(384*b^5*d^2) + ((b*c - a*d)*(5*b^2*c^
2 + 10*a*b*c*d + 9*a^2*d^2)*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(192*b^4*d^2) + ((5
*b^2*c^2 + 10*a*b*c*d + 9*a^2*d^2)*(a + b*x)^(5/2)*(c + d*x)^(5/2))/(120*b^3*d^2
) - ((7*b*c + 9*a*d)*(a + b*x)^(5/2)*(c + d*x)^(7/2))/(84*b^2*d^2) + (x*(a + b*x
)^(5/2)*(c + d*x)^(7/2))/(7*b*d) + ((b*c - a*d)^5*(5*b^2*c^2 + 10*a*b*c*d + 9*a^
2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(1024*b^(11/2)*
d^(9/2))

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Rubi [A]  time = 0.965973, 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 \[ \frac{\left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{11/2} d^{9/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^2}{384 b^5 d^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^4}{1024 b^5 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^3}{1536 b^5 d^3}+\frac{(a+b x)^{5/2} (c+d x)^{3/2} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)}{192 b^4 d^2}+\frac{(a+b x)^{5/2} (c+d x)^{5/2} \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right )}{120 b^3 d^2}-\frac{(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{84 b^2 d^2}+\frac{x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)^4*(5*b^2*c^2 + 10*a*b*c*d + 9*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x]
)/(1024*b^5*d^4) + ((b*c - a*d)^3*(5*b^2*c^2 + 10*a*b*c*d + 9*a^2*d^2)*(a + b*x)
^(3/2)*Sqrt[c + d*x])/(1536*b^5*d^3) + ((b*c - a*d)^2*(5*b^2*c^2 + 10*a*b*c*d +
9*a^2*d^2)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(384*b^5*d^2) + ((b*c - a*d)*(5*b^2*c^
2 + 10*a*b*c*d + 9*a^2*d^2)*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(192*b^4*d^2) + ((5
*b^2*c^2 + 10*a*b*c*d + 9*a^2*d^2)*(a + b*x)^(5/2)*(c + d*x)^(5/2))/(120*b^3*d^2
) - ((7*b*c + 9*a*d)*(a + b*x)^(5/2)*(c + d*x)^(7/2))/(84*b^2*d^2) + (x*(a + b*x
)^(5/2)*(c + d*x)^(7/2))/(7*b*d) + ((b*c - a*d)^5*(5*b^2*c^2 + 10*a*b*c*d + 9*a^
2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(1024*b^(11/2)*
d^(9/2))

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Rubi in Sympy [A]  time = 96.4778, size = 420, normalized size = 0.96 \[ \frac{x \left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{7}{2}}}{7 b d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{7}{2}} \left (9 a d + 7 b c\right )}{84 b^{2} d^{2}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (9 a^{2} d^{2} + 10 a b c d + 5 b^{2} c^{2}\right )}{120 b^{3} d^{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (9 a^{2} d^{2} + 10 a b c d + 5 b^{2} c^{2}\right )}{192 b^{4} d^{2}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (9 a^{2} d^{2} + 10 a b c d + 5 b^{2} c^{2}\right )}{384 b^{5} d^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (9 a^{2} d^{2} + 10 a b c d + 5 b^{2} c^{2}\right )}{1536 b^{5} d^{3}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4} \left (9 a^{2} d^{2} + 10 a b c d + 5 b^{2} c^{2}\right )}{1024 b^{5} d^{4}} - \frac{\left (a d - b c\right )^{5} \left (9 a^{2} d^{2} + 10 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{1024 b^{\frac{11}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x)**(5/2)*(c + d*x)**(7/2)/(7*b*d) - (a + b*x)**(5/2)*(c + d*x)**(7/2)*
(9*a*d + 7*b*c)/(84*b**2*d**2) + (a + b*x)**(5/2)*(c + d*x)**(5/2)*(9*a**2*d**2
+ 10*a*b*c*d + 5*b**2*c**2)/(120*b**3*d**2) - (a + b*x)**(5/2)*(c + d*x)**(3/2)*
(a*d - b*c)*(9*a**2*d**2 + 10*a*b*c*d + 5*b**2*c**2)/(192*b**4*d**2) + (a + b*x)
**(5/2)*sqrt(c + d*x)*(a*d - b*c)**2*(9*a**2*d**2 + 10*a*b*c*d + 5*b**2*c**2)/(3
84*b**5*d**2) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**3*(9*a**2*d**2 + 10*
a*b*c*d + 5*b**2*c**2)/(1536*b**5*d**3) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c
)**4*(9*a**2*d**2 + 10*a*b*c*d + 5*b**2*c**2)/(1024*b**5*d**4) - (a*d - b*c)**5*
(9*a**2*d**2 + 10*a*b*c*d + 5*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sq
rt(c + d*x)))/(1024*b**(11/2)*d**(9/2))

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Mathematica [A]  time = 0.375011, size = 395, normalized size = 0.9 \[ \frac{\left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) (b c-a d)^5 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2048 b^{11/2} d^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (945 a^6 d^6-210 a^5 b d^5 (16 c+3 d x)+7 a^4 b^2 d^4 \left (527 c^2+314 c d x+72 d^2 x^2\right )-4 a^3 b^3 d^3 \left (150 c^3+583 c^2 d x+436 c d^2 x^2+108 d^3 x^3\right )+3 a^2 b^4 d^2 \left (-175 c^4+100 c^3 d x+608 c^2 d^2 x^2+496 c d^3 x^3+128 d^4 x^4\right )+10 a b^5 d \left (140 c^5-91 c^4 d x+72 c^3 d^2 x^2+3352 c^2 d^3 x^3+4864 c d^4 x^4+1920 d^5 x^5\right )-5 b^6 \left (105 c^6-70 c^5 d x+56 c^4 d^2 x^2-48 c^3 d^3 x^3-4736 c^2 d^4 x^4-7424 c d^5 x^5-3072 d^6 x^6\right )\right )}{107520 b^5 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(945*a^6*d^6 - 210*a^5*b*d^5*(16*c + 3*d*x) + 7*a^4
*b^2*d^4*(527*c^2 + 314*c*d*x + 72*d^2*x^2) - 4*a^3*b^3*d^3*(150*c^3 + 583*c^2*d
*x + 436*c*d^2*x^2 + 108*d^3*x^3) + 3*a^2*b^4*d^2*(-175*c^4 + 100*c^3*d*x + 608*
c^2*d^2*x^2 + 496*c*d^3*x^3 + 128*d^4*x^4) + 10*a*b^5*d*(140*c^5 - 91*c^4*d*x +
72*c^3*d^2*x^2 + 3352*c^2*d^3*x^3 + 4864*c*d^4*x^4 + 1920*d^5*x^5) - 5*b^6*(105*
c^6 - 70*c^5*d*x + 56*c^4*d^2*x^2 - 48*c^3*d^3*x^3 - 4736*c^2*d^4*x^4 - 7424*c*d
^5*x^5 - 3072*d^6*x^6)))/(107520*b^5*d^4) + ((b*c - a*d)^5*(5*b^2*c^2 + 10*a*b*c
*d + 9*a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c
+ d*x]])/(2048*b^(11/2)*d^(9/2))

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Maple [B]  time = 0.037, 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)^(3/2)*(d*x+c)^(5/2),x)

[Out]

-1/215040*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(864*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)-2800*a*b^5*c^5*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1
/2)-38400*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)-74240*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)-768*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)-47360*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)+1260*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^5*d^6*b*(b*d)^(1/
2)-700*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*c^5*b^6*d*(b*d)^(1/2)+6720*(b*d*x^2+a*d
*x+b*c*x+a*c)^(1/2)*a^5*c*d^5*b*(b*d)^(1/2)-7378*(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)+1050*(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)-4
80*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)-1008*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)+560*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)-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))*a^6*c*d^6*b+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))*a^5*c^2*d^5*b^2-15
75*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^4-525*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^3*c^4*b^4*d^3-945*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))*c^5*a^2*b^5*d^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))*
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)-1890*(
b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^6*d^6*(b*d)^(1/2)+1050*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2)*c^6*b^6*(b*d)^(1/2)-4396*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)+4664*(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)-
600*(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)+1820*(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)-2976*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)-67040*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)+3488*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)-3
648*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)-1440*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)+945*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^7-525*b^7*l
n(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^5/d^4/(b*d)^(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 + a)^(3/2)*(d*x + c)^(5/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.321456, 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)^(3/2)*(d*x + c)^(5/2)*x^2,x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.426627, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done