3.595 \(\int x^2 (a+b x)^{3/2} (c+d x)^{3/2} \, dx\)
Optimal. Leaf size=349 \[ -\frac{\left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{9/2} d^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^3}{512 b^4 d^4}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^2}{768 b^4 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)}{192 b^4 d^2}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} \left (4 a b c d-7 (a d+b c)^2\right )}{96 b^3 d^2}-\frac{7 (a+b x)^{5/2} (c+d x)^{5/2} (a d+b c)}{60 b^2 d^2}+\frac{x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d} \]
[Out]
((b*c - a*d)^3*(4*a*b*c*d - 7*(b*c + a*d)^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b
^4*d^4) - ((b*c - a*d)^2*(4*a*b*c*d - 7*(b*c + a*d)^2)*(a + b*x)^(3/2)*Sqrt[c +
d*x])/(768*b^4*d^3) - ((b*c - a*d)*(4*a*b*c*d - 7*(b*c + a*d)^2)*(a + b*x)^(5/2)
*Sqrt[c + d*x])/(192*b^4*d^2) - ((4*a*b*c*d - 7*(b*c + a*d)^2)*(a + b*x)^(5/2)*(
c + d*x)^(3/2))/(96*b^3*d^2) - (7*(b*c + a*d)*(a + b*x)^(5/2)*(c + d*x)^(5/2))/(
60*b^2*d^2) + (x*(a + b*x)^(5/2)*(c + d*x)^(5/2))/(6*b*d) - ((b*c - a*d)^4*(4*a*
b*c*d - 7*(b*c + a*d)^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])
])/(512*b^(9/2)*d^(9/2))
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Rubi [A] time = 0.723475, antiderivative size = 349, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227
\[ -\frac{\left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{9/2} d^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^3}{512 b^4 d^4}-\frac{(a+b x)^{3/2} \sqrt{c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)^2}{768 b^4 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (4 a b c d-7 (a d+b c)^2\right ) (b c-a d)}{192 b^4 d^2}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} \left (4 a b c d-7 (a d+b c)^2\right )}{96 b^3 d^2}-\frac{7 (a+b x)^{5/2} (c+d x)^{5/2} (a d+b c)}{60 b^2 d^2}+\frac{x (a+b x)^{5/2} (c+d x)^{5/2}}{6 b d} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2),x]
[Out]
((b*c - a*d)^3*(4*a*b*c*d - 7*(b*c + a*d)^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b
^4*d^4) - ((b*c - a*d)^2*(4*a*b*c*d - 7*(b*c + a*d)^2)*(a + b*x)^(3/2)*Sqrt[c +
d*x])/(768*b^4*d^3) - ((b*c - a*d)*(4*a*b*c*d - 7*(b*c + a*d)^2)*(a + b*x)^(5/2)
*Sqrt[c + d*x])/(192*b^4*d^2) - ((4*a*b*c*d - 7*(b*c + a*d)^2)*(a + b*x)^(5/2)*(
c + d*x)^(3/2))/(96*b^3*d^2) - (7*(b*c + a*d)*(a + b*x)^(5/2)*(c + d*x)^(5/2))/(
60*b^2*d^2) + (x*(a + b*x)^(5/2)*(c + d*x)^(5/2))/(6*b*d) - ((b*c - a*d)^4*(4*a*
b*c*d - 7*(b*c + a*d)^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])
])/(512*b^(9/2)*d^(9/2))
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Rubi in Sympy [A] time = 71.6006, size = 325, normalized size = 0.93 \[ \frac{x \left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}}}{6 b d} - \frac{7 \left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (a d + b c\right )}{60 b^{2} d^{2}} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a b c d - \frac{7 \left (a d + b c\right )^{2}}{4}\right )}{24 b^{3} d^{2}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (4 a b c d - 7 \left (a d + b c\right )^{2}\right )}{192 b^{4} d^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (a b c d - \frac{7 \left (a d + b c\right )^{2}}{4}\right )}{192 b^{4} d^{3}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (a b c d - \frac{7 \left (a d + b c\right )^{2}}{4}\right )}{128 b^{4} d^{4}} - \frac{\left (a d - b c\right )^{4} \left (a b c d - \frac{7 \left (a d + b c\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{128 b^{\frac{9}{2}} d^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)**(3/2)*(d*x+c)**(3/2),x)
[Out]
x*(a + b*x)**(5/2)*(c + d*x)**(5/2)/(6*b*d) - 7*(a + b*x)**(5/2)*(c + d*x)**(5/2
)*(a*d + b*c)/(60*b**2*d**2) - (a + b*x)**(5/2)*(c + d*x)**(3/2)*(a*b*c*d - 7*(a
*d + b*c)**2/4)/(24*b**3*d**2) + (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c)*(4*a
*b*c*d - 7*(a*d + b*c)**2)/(192*b**4*d**2) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d
- b*c)**2*(a*b*c*d - 7*(a*d + b*c)**2/4)/(192*b**4*d**3) - sqrt(a + b*x)*sqrt(c
+ d*x)*(a*d - b*c)**3*(a*b*c*d - 7*(a*d + b*c)**2/4)/(128*b**4*d**4) - (a*d - b
*c)**4*(a*b*c*d - 7*(a*d + b*c)**2/4)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(
c + d*x)))/(128*b**(9/2)*d**(9/2))
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Mathematica [A] time = 0.293962, size = 319, normalized size = 0.91 \[ \frac{\left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) (b c-a d)^4 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{1024 b^{9/2} d^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^5 d^5+5 a^4 b d^4 (47 c+14 d x)-2 a^3 b^2 d^3 \left (33 c^2+76 c d x+28 d^2 x^2\right )+6 a^2 b^3 d^2 \left (-11 c^3+6 c^2 d x+20 c d^2 x^2+8 d^3 x^3\right )+a b^4 d \left (235 c^4-152 c^3 d x+120 c^2 d^2 x^2+2336 c d^3 x^3+1664 d^4 x^4\right )+b^5 \left (-105 c^5+70 c^4 d x-56 c^3 d^2 x^2+48 c^2 d^3 x^3+1664 c d^4 x^4+1280 d^5 x^5\right )\right )}{7680 b^4 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*x)^(3/2)*(c + d*x)^(3/2),x]
[Out]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^5*d^5 + 5*a^4*b*d^4*(47*c + 14*d*x) - 2*a^3
*b^2*d^3*(33*c^2 + 76*c*d*x + 28*d^2*x^2) + 6*a^2*b^3*d^2*(-11*c^3 + 6*c^2*d*x +
20*c*d^2*x^2 + 8*d^3*x^3) + a*b^4*d*(235*c^4 - 152*c^3*d*x + 120*c^2*d^2*x^2 +
2336*c*d^3*x^3 + 1664*d^4*x^4) + b^5*(-105*c^5 + 70*c^4*d*x - 56*c^3*d^2*x^2 + 4
8*c^2*d^3*x^3 + 1664*c*d^4*x^4 + 1280*d^5*x^5)))/(7680*b^4*d^4) + ((b*c - a*d)^4
*(7*b^2*c^2 + 10*a*b*c*d + 7*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]])/(1024*b^(9/2)*d^(9/2))
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Maple [B] time = 0.027, size = 1240, 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)^(3/2),x)
[Out]
1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-112*x^2*a^3*b^2*d^5*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*(b*d)^(1/2)-112*x^2*b^5*c^3*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^
(1/2)+3328*x^4*a*b^4*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+3328*x^4*b^
5*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+96*x^3*a^2*b^3*d^5*(b*d*x^2+
a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+96*x^3*b^5*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*(b*d)^(1/2)-132*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*b^3*d^2*(b*d)^(1/2
)+470*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*d*(b*d)^(1/2)+140*d^5*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)*x*a^4*b*(b*d)^(1/2)+140*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*x*b^5*d*(b*d)^(1/2)+470*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c*b*(b*d)^(1/2
)-132*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*b^2*d^3*(b*d)^(1/2)+105*d^6*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+105*c^6*b^6*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))+2560*x^5*b^5*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-2
10*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*(b*d)^(1/2)-210*c^5*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*b^5*(b*d)^(1/2)-270*d^5*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*b+135*c^2*d^4*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*b^2+60*c^
3*a^3*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))*b^3*d^3+135*c^4*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^2*b^4*d^2-270*c^5*a*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))*b^5*d-304*d^4*(b*d*x^2+a
*d*x+b*c*x+a*c)^(1/2)*x*a^3*c*b^2*(b*d)^(1/2)+72*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*x*a^2*b^3*d^3*(b*d)^(1/2)-304*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b^4*d
^2*(b*d)^(1/2)+240*x^2*a^2*b^3*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)
+240*x^2*a*b^4*c^2*d^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+4672*x^3*a*b^
4*c*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)/d^4/b^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)^(3/2)*x^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.288367, 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)^(3/2)*x^2,x, algorithm="fricas")
[Out]
[1/30720*(4*(1280*b^5*d^5*x^5 - 105*b^5*c^5 + 235*a*b^4*c^4*d - 66*a^2*b^3*c^3*d
^2 - 66*a^3*b^2*c^2*d^3 + 235*a^4*b*c*d^4 - 105*a^5*d^5 + 1664*(b^5*c*d^4 + a*b^
4*d^5)*x^4 + 16*(3*b^5*c^2*d^3 + 146*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 - 8*(7*b^5
*c^3*d^2 - 15*a*b^4*c^2*d^3 - 15*a^2*b^3*c*d^4 + 7*a^3*b^2*d^5)*x^2 + 2*(35*b^5*
c^4*d - 76*a*b^4*c^3*d^2 + 18*a^2*b^3*c^2*d^3 - 76*a^3*b^2*c*d^4 + 35*a^4*b*d^5)
*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(7*b^6*c^6 - 18*a*b^5*c^5*d + 9*a
^2*b^4*c^4*d^2 + 4*a^3*b^3*c^3*d^3 + 9*a^4*b^2*c^2*d^4 - 18*a^5*b*c*d^5 + 7*a^6*
d^6)*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^4), 1/15360*(2*(1280*b^5*d^5*x^5 - 105*b^5*c^5 + 235*a*b^4*c^4
*d - 66*a^2*b^3*c^3*d^2 - 66*a^3*b^2*c^2*d^3 + 235*a^4*b*c*d^4 - 105*a^5*d^5 + 1
664*(b^5*c*d^4 + a*b^4*d^5)*x^4 + 16*(3*b^5*c^2*d^3 + 146*a*b^4*c*d^4 + 3*a^2*b^
3*d^5)*x^3 - 8*(7*b^5*c^3*d^2 - 15*a*b^4*c^2*d^3 - 15*a^2*b^3*c*d^4 + 7*a^3*b^2*
d^5)*x^2 + 2*(35*b^5*c^4*d - 76*a*b^4*c^3*d^2 + 18*a^2*b^3*c^2*d^3 - 76*a^3*b^2*
c*d^4 + 35*a^4*b*d^5)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(7*b^6*c^6
- 18*a*b^5*c^5*d + 9*a^2*b^4*c^4*d^2 + 4*a^3*b^3*c^3*d^3 + 9*a^4*b^2*c^2*d^4 - 1
8*a^5*b*c*d^5 + 7*a^6*d^6)*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^4)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x+a)**(3/2)*(d*x+c)**(3/2),x)
[Out]
Integral(x**2*(a + b*x)**(3/2)*(c + d*x)**(3/2), x)
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GIAC/XCAS [A] time = 0.336927, 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)^(3/2)*x^2,x, algorithm="giac")
[Out]
Done