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

Optimal. Leaf size=348 \[ \frac{5 (a d+b c) (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{9/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^5}{1024 b^4 d^4}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (a d+b c) (b c-a d)^4}{1536 b^4 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{384 b^4 d^2}-\frac{(a+b x)^{7/2} \sqrt{c+d x} (a d+b c) (b c-a d)^2}{64 b^4 d}-\frac{(a+b x)^{7/2} (c+d x)^{3/2} (a d+b c) (b c-a d)}{24 b^3 d}-\frac{(a+b x)^{7/2} (c+d x)^{5/2} (a d+b c)}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d} \]

[Out]

(-5*(b*c - a*d)^5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1024*b^4*d^4) + (5*(
b*c - a*d)^4*(b*c + a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(1536*b^4*d^3) - ((b*c -
a*d)^3*(b*c + a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(384*b^4*d^2) - ((b*c - a*d)^
2*(b*c + a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(64*b^4*d) - ((b*c - a*d)*(b*c + a*
d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(24*b^3*d) - ((b*c + a*d)*(a + b*x)^(7/2)*(c
+ d*x)^(5/2))/(12*b^2*d) + ((a + b*x)^(7/2)*(c + d*x)^(7/2))/(7*b*d) + (5*(b*c
- a*d)^6*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(
1024*b^(9/2)*d^(9/2))

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Rubi [A]  time = 0.609481, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 (a d+b c) (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{1024 b^{9/2} d^{9/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (a d+b c) (b c-a d)^5}{1024 b^4 d^4}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (a d+b c) (b c-a d)^4}{1536 b^4 d^3}-\frac{(a+b x)^{5/2} \sqrt{c+d x} (a d+b c) (b c-a d)^3}{384 b^4 d^2}-\frac{(a+b x)^{7/2} \sqrt{c+d x} (a d+b c) (b c-a d)^2}{64 b^4 d}-\frac{(a+b x)^{7/2} (c+d x)^{3/2} (a d+b c) (b c-a d)}{24 b^3 d}-\frac{(a+b x)^{7/2} (c+d x)^{5/2} (a d+b c)}{12 b^2 d}+\frac{(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b*c - a*d)^5*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1024*b^4*d^4) + (5*(
b*c - a*d)^4*(b*c + a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(1536*b^4*d^3) - ((b*c -
a*d)^3*(b*c + a*d)*(a + b*x)^(5/2)*Sqrt[c + d*x])/(384*b^4*d^2) - ((b*c - a*d)^
2*(b*c + a*d)*(a + b*x)^(7/2)*Sqrt[c + d*x])/(64*b^4*d) - ((b*c - a*d)*(b*c + a*
d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(24*b^3*d) - ((b*c + a*d)*(a + b*x)^(7/2)*(c
+ d*x)^(5/2))/(12*b^2*d) + ((a + b*x)^(7/2)*(c + d*x)^(7/2))/(7*b*d) + (5*(b*c
- a*d)^6*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(
1024*b^(9/2)*d^(9/2))

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Rubi in Sympy [A]  time = 80.5257, size = 311, normalized size = 0.89 \[ \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{7}{2}}}{7 b d} - \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (a d + b c\right )}{12 b^{2} d} + \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (a d + b c\right )}{24 b^{3} d} - \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (a d + b c\right )}{64 b^{4} d} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (a d + b c\right )}{384 b^{4} d^{2}} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{4} \left (a d + b c\right )}{1536 b^{4} d^{3}} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{5} \left (a d + b c\right )}{1024 b^{4} d^{4}} + \frac{5 \left (a d - b c\right )^{6} \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{1024 b^{\frac{9}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(7/2)*(c + d*x)**(7/2)/(7*b*d) - (a + b*x)**(7/2)*(c + d*x)**(5/2)*(a
*d + b*c)/(12*b**2*d) + (a + b*x)**(7/2)*(c + d*x)**(3/2)*(a*d - b*c)*(a*d + b*c
)/(24*b**3*d) - (a + b*x)**(7/2)*sqrt(c + d*x)*(a*d - b*c)**2*(a*d + b*c)/(64*b*
*4*d) + (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c)**3*(a*d + b*c)/(384*b**4*d**2
) + 5*(a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**4*(a*d + b*c)/(1536*b**4*d**3)
+ 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**5*(a*d + b*c)/(1024*b**4*d**4) + 5
*(a*d - b*c)**6*(a*d + b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))
/(1024*b**(9/2)*d**(9/2))

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Mathematica [A]  time = 0.460942, size = 376, normalized size = 1.08 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^6 d^6+70 a^5 b d^5 (7 c+d x)-7 a^4 b^2 d^4 \left (113 c^2+46 c d x+8 d^2 x^2\right )+4 a^3 b^3 d^3 \left (75 c^3+127 c^2 d x+64 c d^2 x^2+12 d^3 x^3\right )+a^2 b^4 d^2 \left (-791 c^4+508 c^3 d x+9840 c^2 d^2 x^2+12752 c d^3 x^3+4736 d^4 x^4\right )+2 a b^5 d \left (245 c^5-161 c^4 d x+128 c^3 d^2 x^2+6376 c^2 d^3 x^3+9344 c d^4 x^4+3712 d^5 x^5\right )+b^6 \left (-105 c^6+70 c^5 d x-56 c^4 d^2 x^2+48 c^3 d^3 x^3+4736 c^2 d^4 x^4+7424 c d^5 x^5+3072 d^6 x^6\right )\right )}{21504 b^4 d^4}+\frac{5 (a d+b c) (b c-a d)^6 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2048 b^{9/2} d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^6*d^6 + 70*a^5*b*d^5*(7*c + d*x) - 7*a^4*b^
2*d^4*(113*c^2 + 46*c*d*x + 8*d^2*x^2) + 4*a^3*b^3*d^3*(75*c^3 + 127*c^2*d*x + 6
4*c*d^2*x^2 + 12*d^3*x^3) + a^2*b^4*d^2*(-791*c^4 + 508*c^3*d*x + 9840*c^2*d^2*x
^2 + 12752*c*d^3*x^3 + 4736*d^4*x^4) + 2*a*b^5*d*(245*c^5 - 161*c^4*d*x + 128*c^
3*d^2*x^2 + 6376*c^2*d^3*x^3 + 9344*c*d^4*x^4 + 3712*d^5*x^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)))/(21504*b^4*d^4) + (5*(b*c - a*d)^6*(b*c + a*d)*Log[b*c + a*d
+ 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2048*b^(9/2)*d^(9/2
))

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Maple [B]  time = 0.028, size = 1580, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43008*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(96*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)+980*a*b^5*c^5*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+
14848*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)+14848*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)+9472*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)+9472*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)+140*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^5*d^6*b*(b*d)^(1/2)+14
0*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*c^5*b^6*d*(b*d)^(1/2)+980*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*a^5*c*d^5*b*(b*d)^(1/2)-1582*(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)+600*(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)-1582*(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)+96*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)-112*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)-112*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)-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^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^4-525*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^2-525*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))*c^6*a*b^6*d+6
144*x^6*b^6*d^6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-210*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)*a^6*d^6*(b*d)^(1/2)-210*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*c^6*b^6*
(b*d)^(1/2)-644*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)+1016
*(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)+1016*(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)-644*(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)+37376*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)+25504*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)+25504*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)+512*
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)+19680*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)+512*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)+105*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+105*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)/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)^(5/2)*(d*x + c)^(5/2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.341794, 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)^(5/2)*x,x, algorithm="fricas")

[Out]

[1/86016*(4*(3072*b^6*d^6*x^6 - 105*b^6*c^6 + 490*a*b^5*c^5*d - 791*a^2*b^4*c^4*
d^2 + 300*a^3*b^3*c^3*d^3 - 791*a^4*b^2*c^2*d^4 + 490*a^5*b*c*d^5 - 105*a^6*d^6
+ 7424*(b^6*c*d^5 + a*b^5*d^6)*x^5 + 128*(37*b^6*c^2*d^4 + 146*a*b^5*c*d^5 + 37*
a^2*b^4*d^6)*x^4 + 16*(3*b^6*c^3*d^3 + 797*a*b^5*c^2*d^4 + 797*a^2*b^4*c*d^5 + 3
*a^3*b^3*d^6)*x^3 - 8*(7*b^6*c^4*d^2 - 32*a*b^5*c^3*d^3 - 1230*a^2*b^4*c^2*d^4 -
32*a^3*b^3*c*d^5 + 7*a^4*b^2*d^6)*x^2 + 2*(35*b^6*c^5*d - 161*a*b^5*c^4*d^2 + 2
54*a^2*b^4*c^3*d^3 + 254*a^3*b^3*c^2*d^4 - 161*a^4*b^2*c*d^5 + 35*a^5*b*d^6)*x)*
sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 105*(b^7*c^7 - 5*a*b^6*c^6*d + 9*a^2*b^5
*c^5*d^2 - 5*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 - 5*a^6*b*c
*d^6 + 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^4), 1/43008*(2*(3072*b^6*d^6*x^6 - 105*b^6*c^6 + 49
0*a*b^5*c^5*d - 791*a^2*b^4*c^4*d^2 + 300*a^3*b^3*c^3*d^3 - 791*a^4*b^2*c^2*d^4
+ 490*a^5*b*c*d^5 - 105*a^6*d^6 + 7424*(b^6*c*d^5 + a*b^5*d^6)*x^5 + 128*(37*b^6
*c^2*d^4 + 146*a*b^5*c*d^5 + 37*a^2*b^4*d^6)*x^4 + 16*(3*b^6*c^3*d^3 + 797*a*b^5
*c^2*d^4 + 797*a^2*b^4*c*d^5 + 3*a^3*b^3*d^6)*x^3 - 8*(7*b^6*c^4*d^2 - 32*a*b^5*
c^3*d^3 - 1230*a^2*b^4*c^2*d^4 - 32*a^3*b^3*c*d^5 + 7*a^4*b^2*d^6)*x^2 + 2*(35*b
^6*c^5*d - 161*a*b^5*c^4*d^2 + 254*a^2*b^4*c^3*d^3 + 254*a^3*b^3*c^2*d^4 - 161*a
^4*b^2*c*d^5 + 35*a^5*b*d^6)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 105*(b^
7*c^7 - 5*a*b^6*c^6*d + 9*a^2*b^5*c^5*d^2 - 5*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 - 5*a^6*b*c*d^6 + 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^4)]

_______________________________________________________________________________________

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)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.482107, 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)^(5/2)*x,x, algorithm="giac")

[Out]

Done