3.613 \(\int \frac{(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx\)
Optimal. Leaf size=333 \[ -\frac{(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{9/2} c^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^4}{512 a^4 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+7 b c) (b c-a d)^3}{768 a^3 c^3 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+7 b c) (b c-a d)^2}{960 a^2 c^3 x^3}+\frac{\sqrt{a+b x} (c+d x)^{7/2} (5 a d+7 b c) (b c-a d)}{160 a c^3 x^4}+\frac{(a+b x)^{3/2} (c+d x)^{7/2} (5 a d+7 b c)}{60 a c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6} \]
[Out]
((b*c - a*d)^4*(7*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*a^4*c^3*x) - ((
b*c - a*d)^3*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(768*a^3*c^3*x^2) +
((b*c - a*d)^2*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(960*a^2*c^3*x^3)
+ ((b*c - a*d)*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(160*a*c^3*x^4) +
((7*b*c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(60*a*c^2*x^5) - ((a + b*x)^(5
/2)*(c + d*x)^(7/2))/(6*a*c*x^6) - ((b*c - a*d)^5*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[
c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(9/2)*c^(7/2))
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Rubi [A] time = 0.659225, antiderivative size = 333, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182
\[ -\frac{(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{9/2} c^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^4}{512 a^4 c^3 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+7 b c) (b c-a d)^3}{768 a^3 c^3 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+7 b c) (b c-a d)^2}{960 a^2 c^3 x^3}+\frac{\sqrt{a+b x} (c+d x)^{7/2} (5 a d+7 b c) (b c-a d)}{160 a c^3 x^4}+\frac{(a+b x)^{3/2} (c+d x)^{7/2} (5 a d+7 b c)}{60 a c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^7,x]
[Out]
((b*c - a*d)^4*(7*b*c + 5*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*a^4*c^3*x) - ((
b*c - a*d)^3*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(768*a^3*c^3*x^2) +
((b*c - a*d)^2*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(960*a^2*c^3*x^3)
+ ((b*c - a*d)*(7*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(160*a*c^3*x^4) +
((7*b*c + 5*a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(60*a*c^2*x^5) - ((a + b*x)^(5
/2)*(c + d*x)^(7/2))/(6*a*c*x^6) - ((b*c - a*d)^5*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[
c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(512*a^(9/2)*c^(7/2))
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Rubi in Sympy [A] time = 66.5431, size = 304, normalized size = 0.91 \[ - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{7}{2}}}{6 a c x^{6}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (5 a d + 7 b c\right )}{60 a^{2} c x^{5}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (5 a d + 7 b c\right )}{96 a^{3} c x^{4}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (5 a d + 7 b c\right )}{192 a^{4} c x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (5 a d + 7 b c\right )}{768 a^{4} c^{2} x^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{4} \left (5 a d + 7 b c\right )}{512 a^{4} c^{3} x} + \frac{\left (a d - b c\right )^{5} \left (5 a d + 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{512 a^{\frac{9}{2}} c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(d*x+c)**(5/2)/x**7,x)
[Out]
-(a + b*x)**(5/2)*(c + d*x)**(7/2)/(6*a*c*x**6) + (a + b*x)**(5/2)*(c + d*x)**(5
/2)*(5*a*d + 7*b*c)/(60*a**2*c*x**5) + (a + b*x)**(5/2)*(c + d*x)**(3/2)*(a*d -
b*c)*(5*a*d + 7*b*c)/(96*a**3*c*x**4) + (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*
c)**2*(5*a*d + 7*b*c)/(192*a**4*c*x**3) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d -
b*c)**3*(5*a*d + 7*b*c)/(768*a**4*c**2*x**2) - sqrt(a + b*x)*sqrt(c + d*x)*(a*d
- b*c)**4*(5*a*d + 7*b*c)/(512*a**4*c**3*x) + (a*d - b*c)**5*(5*a*d + 7*b*c)*ata
nh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(512*a**(9/2)*c**(7/2))
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Mathematica [A] time = 0.456243, size = 356, normalized size = 1.07 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (5 a^5 \left (256 c^5+640 c^4 d x+432 c^3 d^2 x^2+8 c^2 d^3 x^3-10 c d^4 x^4+15 d^5 x^5\right )+a^4 b c x \left (1664 c^4+4448 c^3 d x+3384 c^2 d^2 x^2+160 c d^3 x^3-245 d^4 x^4\right )+6 a^3 b^2 c^2 x^2 \left (8 c^3+36 c^2 d x+58 c d^2 x^2+25 d^3 x^3\right )-2 a^2 b^3 c^3 x^3 \left (28 c^2+136 c d x+273 d^2 x^2\right )+5 a b^4 c^4 x^4 (14 c+83 d x)-105 b^5 c^5 x^5\right )+15 x^6 \log (x) (b c-a d)^5 (5 a d+7 b c)-15 x^6 (b c-a d)^5 (5 a d+7 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{15360 a^{9/2} c^{7/2} x^6} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^7,x]
[Out]
(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*b^5*c^5*x^5 + 5*a*b^4*c^4*
x^4*(14*c + 83*d*x) - 2*a^2*b^3*c^3*x^3*(28*c^2 + 136*c*d*x + 273*d^2*x^2) + 6*a
^3*b^2*c^2*x^2*(8*c^3 + 36*c^2*d*x + 58*c*d^2*x^2 + 25*d^3*x^3) + a^4*b*c*x*(166
4*c^4 + 4448*c^3*d*x + 3384*c^2*d^2*x^2 + 160*c*d^3*x^3 - 245*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)^5*(7*b*c + 5*a*d)*x^6*Log[x] - 15*(b*c - a*d)^5*(7*b*c +
5*a*d)*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^(9/2)*c^(7/2)*x^6)
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Maple [B] time = 0.036, size = 1271, normalized size = 3.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^7,x)
[Out]
1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^3*(112*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-3328*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^4-270*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
+300*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^3-675*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+450*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-140*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^2-96*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^2-320*(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^4-432*(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^3-696*(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+544*(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^4-6768*(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+490*(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^5-300*(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+1092*(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^5-8896*(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^6-105*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+210*c^5*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^5*(a*c)^(1/2)*x^5-830*(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((b*x + a)^(3/2)*(d*x + c)^(5/2)/x^7,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 5.8155, 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^7,x, algorithm="fricas")
[Out]
[-1/30720*(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)*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 - (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)*x^5 + 2*(35*a*b^4*c^5 - 136*a^2*b^3*c^4*d + 174*a
^3*b^2*c^3*d^2 + 80*a^4*b*c^2*d^3 - 25*a^5*c*d^4)*x^4 - 8*(7*a^2*b^3*c^5 - 27*a^
3*b^2*c^4*d - 423*a^4*b*c^3*d^2 - 5*a^5*c^2*d^3)*x^3 + 16*(3*a^3*b^2*c^5 + 278*a
^4*b*c^4*d + 135*a^5*c^3*d^2)*x^2 + 128*(13*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^4*c^3*x^6), -1/15360*(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)*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 - (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)*x^5 + 2*(35*a*b^4*c^5 - 136*a^2*b^3*c^4*d + 174*a^3*b^2*c^3*d^2 + 80*a^4*b*c^
2*d^3 - 25*a^5*c*d^4)*x^4 - 8*(7*a^2*b^3*c^5 - 27*a^3*b^2*c^4*d - 423*a^4*b*c^3*
d^2 - 5*a^5*c^2*d^3)*x^3 + 16*(3*a^3*b^2*c^5 + 278*a^4*b*c^4*d + 135*a^5*c^3*d^2
)*x^2 + 128*(13*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^4*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)**(3/2)*(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)^(3/2)*(d*x + c)^(5/2)/x^7,x, algorithm="giac")
[Out]
Exception raised: TypeError