3.605 \(\int x (a+b x)^{3/2} (c+d x)^{5/2} \, dx\)
Optimal. Leaf size=315 \[ -\frac{(7 a d+5 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{9/2} d^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (7 a d+5 b c) (b c-a d)^4}{512 b^4 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (7 a d+5 b c) (b c-a d)^3}{768 b^4 d^2}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (7 a d+5 b c) (b c-a d)^2}{192 b^4 d}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} (7 a d+5 b c) (b c-a d)}{96 b^3 d}-\frac{(a+b x)^{5/2} (c+d x)^{5/2} (7 a d+5 b c)}{60 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 b d} \]
[Out]
((b*c - a*d)^4*(5*b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^4*d^3) - ((b*
c - a*d)^3*(5*b*c + 7*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(768*b^4*d^2) - ((b*c
- a*d)^2*(5*b*c + 7*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(192*b^4*d) - ((b*c - a*
d)*(5*b*c + 7*a*d)*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(96*b^3*d) - ((5*b*c + 7*a*d
)*(a + b*x)^(5/2)*(c + d*x)^(5/2))/(60*b^2*d) + ((a + b*x)^(5/2)*(c + d*x)^(7/2)
)/(6*b*d) - ((b*c - a*d)^5*(5*b*c + 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt
[b]*Sqrt[c + d*x])])/(512*b^(9/2)*d^(7/2))
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Rubi [A] time = 0.511241, antiderivative size = 315, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2
\[ -\frac{(7 a d+5 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{9/2} d^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (7 a d+5 b c) (b c-a d)^4}{512 b^4 d^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (7 a d+5 b c) (b c-a d)^3}{768 b^4 d^2}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (7 a d+5 b c) (b c-a d)^2}{192 b^4 d}-\frac{(a+b x)^{5/2} (c+d x)^{3/2} (7 a d+5 b c) (b c-a d)}{96 b^3 d}-\frac{(a+b x)^{5/2} (c+d x)^{5/2} (7 a d+5 b c)}{60 b^2 d}+\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 b d} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x)^(3/2)*(c + d*x)^(5/2),x]
[Out]
((b*c - a*d)^4*(5*b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^4*d^3) - ((b*
c - a*d)^3*(5*b*c + 7*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(768*b^4*d^2) - ((b*c
- a*d)^2*(5*b*c + 7*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(192*b^4*d) - ((b*c - a*
d)*(5*b*c + 7*a*d)*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(96*b^3*d) - ((5*b*c + 7*a*d
)*(a + b*x)^(5/2)*(c + d*x)^(5/2))/(60*b^2*d) + ((a + b*x)^(5/2)*(c + d*x)^(7/2)
)/(6*b*d) - ((b*c - a*d)^5*(5*b*c + 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt
[b]*Sqrt[c + d*x])])/(512*b^(9/2)*d^(7/2))
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Rubi in Sympy [A] time = 63.0108, size = 286, normalized size = 0.91 \[ \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{7}{2}}}{6 b d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (7 a d + 5 b c\right )}{60 b^{2} d} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (7 a d + 5 b c\right )}{96 b^{3} d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (7 a d + 5 b c\right )}{192 b^{4} d} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (7 a d + 5 b c\right )}{768 b^{4} d^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4} \left (7 a d + 5 b c\right )}{512 b^{4} d^{3}} + \frac{\left (a d - b c\right )^{5} \left (7 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{512 b^{\frac{9}{2}} d^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**(3/2)*(d*x+c)**(5/2),x)
[Out]
(a + b*x)**(5/2)*(c + d*x)**(7/2)/(6*b*d) - (a + b*x)**(5/2)*(c + d*x)**(5/2)*(7
*a*d + 5*b*c)/(60*b**2*d) + (a + b*x)**(5/2)*(c + d*x)**(3/2)*(a*d - b*c)*(7*a*d
+ 5*b*c)/(96*b**3*d) - (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c)**2*(7*a*d + 5
*b*c)/(192*b**4*d) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**3*(7*a*d + 5*b*
c)/(768*b**4*d**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**4*(7*a*d + 5*b*c)/
(512*b**4*d**3) + (a*d - b*c)**5*(7*a*d + 5*b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(sq
rt(b)*sqrt(c + d*x)))/(512*b**(9/2)*d**(7/2))
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Mathematica [A] time = 0.297824, size = 306, normalized size = 0.97 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^5 d^5+5 a^4 b d^4 (83 c+14 d x)-2 a^3 b^2 d^3 \left (273 c^2+136 c d x+28 d^2 x^2\right )+6 a^2 b^3 d^2 \left (25 c^3+58 c^2 d x+36 c d^2 x^2+8 d^3 x^3\right )+a b^4 d \left (-245 c^4+160 c^3 d x+3384 c^2 d^2 x^2+4448 c d^3 x^3+1664 d^4 x^4\right )+5 b^5 \left (15 c^5-10 c^4 d x+8 c^3 d^2 x^2+432 c^2 d^3 x^3+640 c d^4 x^4+256 d^5 x^5\right )\right )}{7680 b^4 d^3}-\frac{(b c-a d)^5 (7 a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{1024 b^{9/2} d^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x)^(3/2)*(c + d*x)^(5/2),x]
[Out]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^5*d^5 + 5*a^4*b*d^4*(83*c + 14*d*x) - 2*a^3
*b^2*d^3*(273*c^2 + 136*c*d*x + 28*d^2*x^2) + 6*a^2*b^3*d^2*(25*c^3 + 58*c^2*d*x
+ 36*c*d^2*x^2 + 8*d^3*x^3) + a*b^4*d*(-245*c^4 + 160*c^3*d*x + 3384*c^2*d^2*x^
2 + 4448*c*d^3*x^3 + 1664*d^4*x^4) + 5*b^5*(15*c^5 - 10*c^4*d*x + 8*c^3*d^2*x^2
+ 432*c^2*d^3*x^3 + 640*c*d^4*x^4 + 256*d^5*x^5)))/(7680*b^4*d^3) - ((b*c - a*d)
^5*(5*b*c + 7*a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqr
t[c + d*x]])/(1024*b^(9/2)*d^(7/2))
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Maple [B] time = 0.026, size = 1240, normalized size = 3.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^(3/2)*(d*x+c)^(5/2),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)+80*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)+6400*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)+4320*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)+300*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)-490*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)-100*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/
2)*x*b^5*d*(b*d)^(1/2)+830*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c*b*(b*d)^(1/
2)-1092*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^6-75*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)-
210*d^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*(b*d)^(1/2)+150*c^5*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)*b^5*(b*d)^(1/2)-450*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+675*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^2-300*
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^3-225*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-544*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)+696*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)+320*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)+432*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)+6768*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)+8896*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)/b^4/d^3/(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,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.285773, 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,x, algorithm="fricas")
[Out]
[1/30720*(4*(1280*b^5*d^5*x^5 + 75*b^5*c^5 - 245*a*b^4*c^4*d + 150*a^2*b^3*c^3*d
^2 - 546*a^3*b^2*c^2*d^3 + 415*a^4*b*c*d^4 - 105*a^5*d^5 + 128*(25*b^5*c*d^4 + 1
3*a*b^4*d^5)*x^4 + 16*(135*b^5*c^2*d^3 + 278*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 +
8*(5*b^5*c^3*d^2 + 423*a*b^4*c^2*d^3 + 27*a^2*b^3*c*d^4 - 7*a^3*b^2*d^5)*x^2 - 2
*(25*b^5*c^4*d - 80*a*b^4*c^3*d^2 - 174*a^2*b^3*c^2*d^3 + 136*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*(5*b^6*c^6 - 18*a*b^5*
c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 - 45*a^4*b^2*c^2*d^4 + 30*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^3), 1/15360*(2*(1280*b^5*d^5*x^5 + 75*b^5*c^5 -
245*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 546*a^3*b^2*c^2*d^3 + 415*a^4*b*c*d^4 -
105*a^5*d^5 + 128*(25*b^5*c*d^4 + 13*a*b^4*d^5)*x^4 + 16*(135*b^5*c^2*d^3 + 278*
a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + 8*(5*b^5*c^3*d^2 + 423*a*b^4*c^2*d^3 + 27*a^2
*b^3*c*d^4 - 7*a^3*b^2*d^5)*x^2 - 2*(25*b^5*c^4*d - 80*a*b^4*c^3*d^2 - 174*a^2*b
^3*c^2*d^3 + 136*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*(5*b^6*c^6 - 18*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*
d^3 - 45*a^4*b^2*c^2*d^4 + 30*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^3)]
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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*(b*x+a)**(3/2)*(d*x+c)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.373833, 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,x, algorithm="giac")
[Out]
Done