3.563 \(\int \frac{\sqrt{a+b x} (c+d x)^{5/2}}{x^7} \, dx\)
Optimal. Leaf size=436 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{160 a^2 x^4}+\frac{\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{11/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{960 a^3 c x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (25 a^4 d^4-20 a^3 b c d^3+262 a^2 b^2 c^2 d^2-308 a b^3 c^3 d+105 b^4 c^4\right )}{3840 a^4 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (75 a^5 d^5-65 a^4 b c d^4-90 a^3 b^2 c^2 d^3+838 a^2 b^3 c^3 d^2-945 a b^4 c^4 d+315 b^5 c^5\right )}{7680 a^5 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{60 a x^5} \]
[Out]
((3*b^2*c^2 - 6*a*b*c*d - 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(160*a^2*x^4)
- ((21*b^3*c^3 - 61*a*b^2*c^2*d + 51*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt
[c + d*x])/(960*a^3*c*x^3) + ((105*b^4*c^4 - 308*a*b^3*c^3*d + 262*a^2*b^2*c^2*d
^2 - 20*a^3*b*c*d^3 + 25*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(3840*a^4*c^2*x^2
) - ((315*b^5*c^5 - 945*a*b^4*c^4*d + 838*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2*d^3 -
65*a^4*b*c*d^4 + 75*a^5*d^5)*Sqrt[a + b*x]*Sqrt[c + d*x])/(7680*a^5*c^3*x) - ((
b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(60*a*x^5) - (Sqrt[a + b*x]*(c + d*x
)^(5/2))/(6*x^6) + ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*ArcTanh[
(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(11/2)*c^(7/2))
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Rubi [A] time = 1.46928, antiderivative size = 436, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273
\[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (-5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{160 a^2 x^4}+\frac{\left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{11/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^3 d^3+51 a^2 b c d^2-61 a b^2 c^2 d+21 b^3 c^3\right )}{960 a^3 c x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (25 a^4 d^4-20 a^3 b c d^3+262 a^2 b^2 c^2 d^2-308 a b^3 c^3 d+105 b^4 c^4\right )}{3840 a^4 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (75 a^5 d^5-65 a^4 b c d^4-90 a^3 b^2 c^2 d^3+838 a^2 b^3 c^3 d^2-945 a b^4 c^4 d+315 b^5 c^5\right )}{7680 a^5 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{6 x^6}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{60 a x^5} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^7,x]
[Out]
((3*b^2*c^2 - 6*a*b*c*d - 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(160*a^2*x^4)
- ((21*b^3*c^3 - 61*a*b^2*c^2*d + 51*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt
[c + d*x])/(960*a^3*c*x^3) + ((105*b^4*c^4 - 308*a*b^3*c^3*d + 262*a^2*b^2*c^2*d
^2 - 20*a^3*b*c*d^3 + 25*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(3840*a^4*c^2*x^2
) - ((315*b^5*c^5 - 945*a*b^4*c^4*d + 838*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2*d^3 -
65*a^4*b*c*d^4 + 75*a^5*d^5)*Sqrt[a + b*x]*Sqrt[c + d*x])/(7680*a^5*c^3*x) - ((
b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(60*a*x^5) - (Sqrt[a + b*x]*(c + d*x
)^(5/2))/(6*x^6) + ((b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*ArcTanh[
(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(11/2)*c^(7/2))
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**7,x)
[Out]
Timed out
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Mathematica [A] time = 0.4558, size = 384, normalized size = 0.88 \[ \frac{-15 x^6 \log (x) (b c-a d)^4 \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right )+15 x^6 (b c-a d)^4 \left (5 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (5 a^5 \left (256 c^5+640 c^4 d x+432 c^3 d^2 x^2+8 c^2 d^3 x^3-10 c d^4 x^4+15 d^5 x^5\right )+a^4 b c x \left (128 c^4+416 c^3 d x+408 c^2 d^2 x^2+40 c d^3 x^3-65 d^4 x^4\right )-2 a^3 b^2 c^2 x^2 \left (72 c^3+244 c^2 d x+262 c d^2 x^2+45 d^3 x^3\right )+2 a^2 b^3 c^3 x^3 \left (84 c^2+308 c d x+419 d^2 x^2\right )-105 a b^4 c^4 x^4 (2 c+9 d x)+315 b^5 c^5 x^5\right )}{15360 a^{11/2} c^{7/2} x^6} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^7,x]
[Out]
(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(315*b^5*c^5*x^5 - 105*a*b^4*c^4
*x^4*(2*c + 9*d*x) + 2*a^2*b^3*c^3*x^3*(84*c^2 + 308*c*d*x + 419*d^2*x^2) - 2*a^
3*b^2*c^2*x^2*(72*c^3 + 244*c^2*d*x + 262*c*d^2*x^2 + 45*d^3*x^3) + a^4*b*c*x*(1
28*c^4 + 416*c^3*d*x + 408*c^2*d^2*x^2 + 40*c*d^3*x^3 - 65*d^4*x^4) + 5*a^5*(256
*c^5 + 640*c^4*d*x + 432*c^3*d^2*x^2 + 8*c^2*d^3*x^3 - 10*c*d^4*x^4 + 15*d^5*x^5
)) - 15*(b*c - a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*x^6*Log[x] + 15*(b*c
- a*d)^4*(21*b^2*c^2 + 14*a*b*c*d + 5*a^2*d^2)*x^6*Log[2*a*c + b*c*x + a*d*x +
2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(15360*a^(11/2)*c^(7/2)*x^6)
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Maple [B] time = 0.046, size = 1271, normalized size = 2.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^7,x)
[Out]
1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^5/c^3*(-336*c^5*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*b^3*a^2*(a*c)^(1/2)*x^3-6400*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*a^5*(a*c)^(
1/2)*c^4*x-256*c^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*a^4*(a*c)^(1/2)*x+100*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)*d^4*a^5*(a*c)^(1/2)*c*x^4-90*ln((a*d*x+b*c*x+2*(a*c)^
(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^5*b*c*d^5-75*ln((a*d*x+b*c
*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^4*b^2*c^2*d^4-3
00*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a
^3*b^3*c^3*d^3+1125*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)+2*a*c)/x)*x^6*a^2*b^4*c^4*d^2-1050*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a*b^5*c^5*d+420*c^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*b^4*a*(a*c)^(1/2)*x^4-80*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^3*a^5*(a*c)^(1/2
)*c^2*x^3-4320*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^2*a^5*(a*c)^(1/2)*c^3*x^2+288*c
^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^2*a^3*(a*c)^(1/2)*x^2-80*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*d^3*b*a^4*(a*c)^(1/2)*c^2*x^4+976*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*
b^2*d*a^3*(a*c)^(1/2)*c^4*x^3+1048*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^2*b^2*a^3*(
a*c)^(1/2)*c^3*x^4-1232*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*b^3*a^2*(a*c)^(1/2)*c^
4*x^4-816*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*d^2*a^4*(a*c)^(1/2)*c^3*x^3+130*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^4*b*a^4*(a*c)^(1/2)*c*x^5+180*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*d^3*b^2*a^3*(a*c)^(1/2)*c^2*x^5-1676*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*d^2*b^3*a^2*(a*c)^(1/2)*c^3*x^5-832*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*b*a^4*(a*
c)^(1/2)*c^4*x^2+75*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)+2*a*c)/x)*x^6*a^6*d^6+315*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)+2*a*c)/x)*x^6*b^6*c^6-2560*c^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*(a*
c)^(1/2)-150*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^5*a^5*(a*c)^(1/2)*x^5-630*c^5*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*(a*c)^(1/2)*x^5+1890*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*d*b^4*a*(a*c)^(1/2)*c^4*x^5)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/(a*c)^(1/2)/x
^6
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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(sqrt(b*x + a)*(d*x + c)^(5/2)/x^7,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 5.65531, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(d*x + c)^(5/2)/x^7,x, algorithm="fricas")
[Out]
[1/30720*(15*(21*b^6*c^6 - 70*a*b^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*
d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + 5*a^6*d^6)*x^6*log((4*(2*a^2*c^2 + (a*
b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) + (8*a^2*c^2 + (b^2*c^2 + 6*a*b*
c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2) - 4*(1280*a^5*c^5
+ (315*b^5*c^5 - 945*a*b^4*c^4*d + 838*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2*d^3 - 65
*a^4*b*c*d^4 + 75*a^5*d^5)*x^5 - 2*(105*a*b^4*c^5 - 308*a^2*b^3*c^4*d + 262*a^3*
b^2*c^3*d^2 - 20*a^4*b*c^2*d^3 + 25*a^5*c*d^4)*x^4 + 8*(21*a^2*b^3*c^5 - 61*a^3*
b^2*c^4*d + 51*a^4*b*c^3*d^2 + 5*a^5*c^2*d^3)*x^3 - 16*(9*a^3*b^2*c^5 - 26*a^4*b
*c^4*d - 135*a^5*c^3*d^2)*x^2 + 128*(a^4*b*c^5 + 25*a^5*c^4*d)*x)*sqrt(a*c)*sqrt
(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^5*c^3*x^6), 1/15360*(15*(21*b^6*c^6 - 70*a
*b^5*c^5*d + 75*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 - 5*a^4*b^2*c^2*d^4 - 6*a^5
*b*c*d^5 + 5*a^6*d^6)*x^6*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*
x + a)*sqrt(d*x + c)*a*c)) - 2*(1280*a^5*c^5 + (315*b^5*c^5 - 945*a*b^4*c^4*d +
838*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 75*a^5*d^5)*x^5 - 2*
(105*a*b^4*c^5 - 308*a^2*b^3*c^4*d + 262*a^3*b^2*c^3*d^2 - 20*a^4*b*c^2*d^3 + 25
*a^5*c*d^4)*x^4 + 8*(21*a^2*b^3*c^5 - 61*a^3*b^2*c^4*d + 51*a^4*b*c^3*d^2 + 5*a^
5*c^2*d^3)*x^3 - 16*(9*a^3*b^2*c^5 - 26*a^4*b*c^4*d - 135*a^5*c^3*d^2)*x^2 + 128
*(a^4*b*c^5 + 25*a^5*c^4*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*
c)*a^5*c^3*x^6)]
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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((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**7,x)
[Out]
Timed out
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(d*x + c)^(5/2)/x^7,x, algorithm="giac")
[Out]
Exception raised: TypeError