3.661 \(\int \frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=280 \[ \frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{7/2} c^{7/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}{512 a^3 c^3 x}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^4}{768 a^2 c^3 x^2}-\frac{\sqrt{a+b x} (c+d x)^{7/2} (b c-a d)^2}{32 c^3 x^4}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^3}{192 a c^3 x^3}-\frac{(a+b x)^{3/2} (c+d x)^{7/2} (b c-a d)}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6} \]

[Out]

(-5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*a^3*c^3*x) + (5*(b*c - a*d)^
4*Sqrt[a + b*x]*(c + d*x)^(3/2))/(768*a^2*c^3*x^2) - ((b*c - a*d)^3*Sqrt[a + b*x
]*(c + d*x)^(5/2))/(192*a*c^3*x^3) - ((b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(7/2
))/(32*c^3*x^4) - ((b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(12*c^2*x^5) - (
(a + b*x)^(5/2)*(c + d*x)^(7/2))/(6*c*x^6) + (5*(b*c - a*d)^6*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(7/2)*c^(7/2))

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Rubi [A]  time = 0.613213, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{7/2} c^{7/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}{512 a^3 c^3 x}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^4}{768 a^2 c^3 x^2}-\frac{\sqrt{a+b x} (c+d x)^{7/2} (b c-a d)^2}{32 c^3 x^4}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^3}{192 a c^3 x^3}-\frac{(a+b x)^{3/2} (c+d x)^{7/2} (b c-a d)}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x]

[Out]

(-5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*a^3*c^3*x) + (5*(b*c - a*d)^
4*Sqrt[a + b*x]*(c + d*x)^(3/2))/(768*a^2*c^3*x^2) - ((b*c - a*d)^3*Sqrt[a + b*x
]*(c + d*x)^(5/2))/(192*a*c^3*x^3) - ((b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(7/2
))/(32*c^3*x^4) - ((b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(12*c^2*x^5) - (
(a + b*x)^(5/2)*(c + d*x)^(7/2))/(6*c*x^6) + (5*(b*c - a*d)^6*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(7/2)*c^(7/2))

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Rubi in Sympy [A]  time = 65.3364, size = 258, normalized size = 0.92 \[ - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{7}{2}}}{6 c x^{6}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )}{12 a c x^{5}} + \frac{5 \left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{96 a^{2} c x^{4}} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}}{192 a^{2} c^{2} x^{3}} - \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{4}}{256 a^{2} c^{3} x^{2}} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{5}}{512 a^{3} c^{3} x} + \frac{5 \left (a d - b c\right )^{6} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{512 a^{\frac{7}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(5/2)/x**7,x)

[Out]

-(a + b*x)**(5/2)*(c + d*x)**(7/2)/(6*c*x**6) + (a + b*x)**(5/2)*(c + d*x)**(5/2
)*(a*d - b*c)/(12*a*c*x**5) + 5*(a + b*x)**(5/2)*(c + d*x)**(3/2)*(a*d - b*c)**2
/(96*a**2*c*x**4) + 5*(a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d - b*c)**3/(192*a**2
*c**2*x**3) - 5*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**4/(256*a**2*c**3*x**
2) + 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**5/(512*a**3*c**3*x) + 5*(a*d - b
*c)**6*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(512*a**(7/2)*c**(7/
2))

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Mathematica [A]  time = 0.460026, size = 337, normalized size = 1.2 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^5 \left (256 c^5+640 c^4 d x+432 c^3 d^2 x^2+8 c^2 d^3 x^3-10 c d^4 x^4+15 d^5 x^5\right )+a^4 b c x \left (640 c^4+1696 c^3 d x+1272 c^2 d^2 x^2+56 c d^3 x^3-85 d^4 x^4\right )+6 a^3 b^2 c^2 x^2 \left (72 c^3+212 c^2 d x+198 c d^2 x^2+33 d^3 x^3\right )+2 a^2 b^3 c^3 x^3 \left (4 c^2+28 c d x+99 d^2 x^2\right )-5 a b^4 c^4 x^4 (2 c+17 d x)+15 b^5 c^5 x^5\right )-15 x^6 \log (x) (b c-a d)^6+15 x^6 (b c-a d)^6 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{3072 a^{7/2} c^{7/2} x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^5*c^5*x^5 - 5*a*b^4*c^4*x^
4*(2*c + 17*d*x) + 2*a^2*b^3*c^3*x^3*(4*c^2 + 28*c*d*x + 99*d^2*x^2) + 6*a^3*b^2
*c^2*x^2*(72*c^3 + 212*c^2*d*x + 198*c*d^2*x^2 + 33*d^3*x^3) + a^4*b*c*x*(640*c^
4 + 1696*c^3*d*x + 1272*c^2*d^2*x^2 + 56*c*d^3*x^3 - 85*d^4*x^4) + 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)^6*x^6*Log[x] + 15*(b*c - a*d)^6*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]])/(3072*a^(7/2)*c^(7/2)*x^6)

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Maple [B]  time = 0.036, size = 1271, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x)

[Out]

1/3072*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(-16*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^3-1280*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*a^5*(a*c)^(1/
2)*c^4*x-1280*c^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*a^4*(a*c)^(1/2)*x+20*(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+225*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^4-30
0*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+225*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^2-90*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*b^5*c^5*d+20*c^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*
b^4*a*(a*c)^(1/2)*x^4-16*(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^3-864*(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^2-864*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^2-112*(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^4-2544*(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^3-2376*(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^4-112*(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
-2544*(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+170*(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^5-396*(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^5-396*(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^5-3392*(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+15*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+15*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^6-512*c^5*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*(a*c)^(1/
2)-30*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^5*a^5*(a*c)^(1/2)*x^5-30*c^5*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*b^5*(a*c)^(1/2)*x^5+170*(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)/x^6/(a*c)^(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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 12.1599, 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^7,x, algorithm="fricas")

[Out]

[1/6144*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*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 - 6*a^5*b*c*d^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*(256*a^5*c^5 + (15*b
^5*c^5 - 85*a*b^4*c^4*d + 198*a^2*b^3*c^3*d^2 + 198*a^3*b^2*c^2*d^3 - 85*a^4*b*c
*d^4 + 15*a^5*d^5)*x^5 - 2*(5*a*b^4*c^5 - 28*a^2*b^3*c^4*d - 594*a^3*b^2*c^3*d^2
- 28*a^4*b*c^2*d^3 + 5*a^5*c*d^4)*x^4 + 8*(a^2*b^3*c^5 + 159*a^3*b^2*c^4*d + 15
9*a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^3 + 16*(27*a^3*b^2*c^5 + 106*a^4*b*c^4*d + 27*a
^5*c^3*d^2)*x^2 + 640*(a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*
x + c))/(sqrt(a*c)*a^3*c^3*x^6), 1/3072*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*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 - 6*a^5*b*c*d^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*(256*a^5*c^5 + (15*b^5*c^5 - 85*a*b^4*c^4*d + 198*a^2*b^3*c^3*d^2 + 198*
a^3*b^2*c^2*d^3 - 85*a^4*b*c*d^4 + 15*a^5*d^5)*x^5 - 2*(5*a*b^4*c^5 - 28*a^2*b^3
*c^4*d - 594*a^3*b^2*c^3*d^2 - 28*a^4*b*c^2*d^3 + 5*a^5*c*d^4)*x^4 + 8*(a^2*b^3*
c^5 + 159*a^3*b^2*c^4*d + 159*a^4*b*c^3*d^2 + a^5*c^2*d^3)*x^3 + 16*(27*a^3*b^2*
c^5 + 106*a^4*b*c^4*d + 27*a^5*c^3*d^2)*x^2 + 640*(a^4*b*c^5 + a^5*c^4*d)*x)*sqr
t(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^3*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((b*x+a)**(5/2)*(d*x+c)**(5/2)/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((b*x + a)^(5/2)*(d*x + c)^(5/2)/x^7,x, algorithm="giac")

[Out]

Exception raised: TypeError