3.645 \(\int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx\)
Optimal. Leaf size=315 \[ \frac{(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^4}{512 b^3 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^3}{768 b^3 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^2}{960 b^3 d^2}-\frac{(a+b x)^{7/2} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)}{160 b^3 d}-\frac{(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+7 b c)}{60 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d} \]
[Out]
-((b*c - a*d)^4*(7*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^3*d^4) + ((b
*c - a*d)^3*(7*b*c + 5*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(768*b^3*d^3) - ((b*c
- a*d)^2*(7*b*c + 5*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(960*b^3*d^2) - ((b*c -
a*d)*(7*b*c + 5*a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(160*b^3*d) - ((7*b*c + 5*a
*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(60*b^2*d) + ((a + b*x)^(7/2)*(c + d*x)^(5/
2))/(6*b*d) + ((b*c - a*d)^5*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sq
rt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*d^(9/2))
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Rubi [A] time = 0.512114, 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
\[ \frac{(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^4}{512 b^3 d^4}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^3}{768 b^3 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^2}{960 b^3 d^2}-\frac{(a+b x)^{7/2} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)}{160 b^3 d}-\frac{(a+b x)^{7/2} (c+d x)^{3/2} (5 a d+7 b c)}{60 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]
[Out]
-((b*c - a*d)^4*(7*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^3*d^4) + ((b
*c - a*d)^3*(7*b*c + 5*a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(768*b^3*d^3) - ((b*c
- a*d)^2*(7*b*c + 5*a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(960*b^3*d^2) - ((b*c -
a*d)*(7*b*c + 5*a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(160*b^3*d) - ((7*b*c + 5*a
*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(60*b^2*d) + ((a + b*x)^(7/2)*(c + d*x)^(5/
2))/(6*b*d) + ((b*c - a*d)^5*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sq
rt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*d^(9/2))
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Rubi in Sympy [A] time = 62.8032, size = 287, normalized size = 0.91 \[ \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{5}{2}}}{6 b d} - \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d + 7 b c\right )}{60 b^{2} d} + \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (5 a d + 7 b c\right )}{160 b^{3} d} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (5 a d + 7 b c\right )}{960 b^{3} d^{2}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (5 a d + 7 b c\right )}{768 b^{3} d^{3}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4} \left (5 a d + 7 b c\right )}{512 b^{3} d^{4}} - \frac{\left (a d - b c\right )^{5} \left (5 a d + 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{512 b^{\frac{7}{2}} d^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**(5/2)*(d*x+c)**(3/2),x)
[Out]
(a + b*x)**(7/2)*(c + d*x)**(5/2)/(6*b*d) - (a + b*x)**(7/2)*(c + d*x)**(3/2)*(5
*a*d + 7*b*c)/(60*b**2*d) + (a + b*x)**(7/2)*sqrt(c + d*x)*(a*d - b*c)*(5*a*d +
7*b*c)/(160*b**3*d) - (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c)**2*(5*a*d + 7*b
*c)/(960*b**3*d**2) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**3*(5*a*d + 7*b
*c)/(768*b**3*d**3) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**4*(5*a*d + 7*b*c)
/(512*b**3*d**4) - (a*d - b*c)**5*(5*a*d + 7*b*c)*atanh(sqrt(b)*sqrt(c + d*x)/(s
qrt(d)*sqrt(a + b*x)))/(512*b**(7/2)*d**(9/2))
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Mathematica [A] time = 0.310257, size = 305, normalized size = 0.97 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (75 a^5 d^5-5 a^4 b d^4 (49 c+10 d x)+10 a^3 b^2 d^3 \left (15 c^2+16 c d x+4 d^2 x^2\right )+6 a^2 b^3 d^2 \left (-91 c^3+58 c^2 d x+564 c d^2 x^2+360 d^3 x^3\right )+a b^4 d \left (415 c^4-272 c^3 d x+216 c^2 d^2 x^2+4448 c d^3 x^3+3200 d^4 x^4\right )+b^5 \left (-105 c^5+70 c^4 d x-56 c^3 d^2 x^2+48 c^2 d^3 x^3+1664 c d^4 x^4+1280 d^5 x^5\right )\right )}{7680 b^3 d^4}+\frac{(5 a d+7 b c) (b c-a d)^5 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{1024 b^{7/2} d^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]
[Out]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(75*a^5*d^5 - 5*a^4*b*d^4*(49*c + 10*d*x) + 10*a^3*
b^2*d^3*(15*c^2 + 16*c*d*x + 4*d^2*x^2) + 6*a^2*b^3*d^2*(-91*c^3 + 58*c^2*d*x +
564*c*d^2*x^2 + 360*d^3*x^3) + a*b^4*d*(415*c^4 - 272*c^3*d*x + 216*c^2*d^2*x^2
+ 4448*c*d^3*x^3 + 3200*d^4*x^4) + b^5*(-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)))/(7680*b^3*d^4) + ((b*c - a*d)
^5*(7*b*c + 5*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^(7/2)*d^(9/2))
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Maple [B] time = 0.025, 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)^(5/2)*(d*x+c)^(3/2),x)
[Out]
-1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-80*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)-6400*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)-4320*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)+1092*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)-830*c^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*b^4*d*(b*d)^(1/2)+100*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)+490*d^4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c*b*(b*d)^(
1/2)-300*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)+75*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)
-150*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+225*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^3-675*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+450*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-320*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)+544*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)-6768*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)-432*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)/d^4/b^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)^(5/2)*(d*x + c)^(3/2)*x,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.286927, 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,x, algorithm="fricas")
[Out]
[1/30720*(4*(1280*b^5*d^5*x^5 - 105*b^5*c^5 + 415*a*b^4*c^4*d - 546*a^2*b^3*c^3*
d^2 + 150*a^3*b^2*c^2*d^3 - 245*a^4*b*c*d^4 + 75*a^5*d^5 + 128*(13*b^5*c*d^4 + 2
5*a*b^4*d^5)*x^4 + 16*(3*b^5*c^2*d^3 + 278*a*b^4*c*d^4 + 135*a^2*b^3*d^5)*x^3 -
8*(7*b^5*c^3*d^2 - 27*a*b^4*c^2*d^3 - 423*a^2*b^3*c*d^4 - 5*a^3*b^2*d^5)*x^2 + 2
*(35*b^5*c^4*d - 136*a*b^4*c^3*d^2 + 174*a^2*b^3*c^2*d^3 + 80*a^3*b^2*c*d^4 - 25
*a^4*b*d^5)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(7*b^6*c^6 - 30*a*b^5*
c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*
c*d^5 - 5*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^3*d^4), 1/15360*(2*(1280*b^5*d^5*x^5 - 105*b^5*c^5
+ 415*a*b^4*c^4*d - 546*a^2*b^3*c^3*d^2 + 150*a^3*b^2*c^2*d^3 - 245*a^4*b*c*d^4
+ 75*a^5*d^5 + 128*(13*b^5*c*d^4 + 25*a*b^4*d^5)*x^4 + 16*(3*b^5*c^2*d^3 + 278*a
*b^4*c*d^4 + 135*a^2*b^3*d^5)*x^3 - 8*(7*b^5*c^3*d^2 - 27*a*b^4*c^2*d^3 - 423*a^
2*b^3*c*d^4 - 5*a^3*b^2*d^5)*x^2 + 2*(35*b^5*c^4*d - 136*a*b^4*c^3*d^2 + 174*a^2
*b^3*c^2*d^3 + 80*a^3*b^2*c*d^4 - 25*a^4*b*d^5)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt
(d*x + c) + 15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3
*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*c*d^5 - 5*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^3*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*(b*x+a)**(5/2)*(d*x+c)**(3/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.369698, 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,x, algorithm="giac")
[Out]
Done