3.710 \(\int \frac{(c+d x)^{5/2}}{x^6 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=346 \[ \frac{c \sqrt{a+b x} \sqrt{c+d x} (9 b c-13 a d)}{40 a^2 x^4}+\frac{\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{11/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{240 a^3 x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{960 a^4 c x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-45 a^4 d^4-90 a^3 b c d^3+1564 a^2 b^2 c^2 d^2-2310 a b^3 c^3 d+945 b^4 c^4\right )}{1920 a^5 c^2 x}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{5 a x^5} \]

[Out]

(c*(9*b*c - 13*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*a^2*x^4) - ((63*b^2*c^2 - 1
48*a*b*c*d + 93*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(240*a^3*x^3) + ((315*b^3*
c^3 - 749*a*b^2*c^2*d + 481*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x
])/(960*a^4*c*x^2) - ((945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 9
0*a^3*b*c*d^3 - 45*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^5*c^2*x) - (c*S
qrt[a + b*x]*(c + d*x)^(3/2))/(5*a*x^5) + ((b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*
d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^
(11/2)*c^(5/2))

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Rubi [A]  time = 1.18717, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{c \sqrt{a+b x} \sqrt{c+d x} (9 b c-13 a d)}{40 a^2 x^4}+\frac{\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{11/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (93 a^2 d^2-148 a b c d+63 b^2 c^2\right )}{240 a^3 x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-15 a^3 d^3+481 a^2 b c d^2-749 a b^2 c^2 d+315 b^3 c^3\right )}{960 a^4 c x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-45 a^4 d^4-90 a^3 b c d^3+1564 a^2 b^2 c^2 d^2-2310 a b^3 c^3 d+945 b^4 c^4\right )}{1920 a^5 c^2 x}-\frac{c \sqrt{a+b x} (c+d x)^{3/2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^6*Sqrt[a + b*x]),x]

[Out]

(c*(9*b*c - 13*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*a^2*x^4) - ((63*b^2*c^2 - 1
48*a*b*c*d + 93*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(240*a^3*x^3) + ((315*b^3*
c^3 - 749*a*b^2*c^2*d + 481*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x
])/(960*a^4*c*x^2) - ((945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 9
0*a^3*b*c*d^3 - 45*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^5*c^2*x) - (c*S
qrt[a + b*x]*(c + d*x)^(3/2))/(5*a*x^5) + ((b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*
d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^
(11/2)*c^(5/2))

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Rubi in Sympy [A]  time = 143.114, size = 338, normalized size = 0.98 \[ - \frac{c \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{5 a x^{5}} - \frac{c \sqrt{a + b x} \sqrt{c + d x} \left (13 a d - 9 b c\right )}{40 a^{2} x^{4}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (93 a^{2} d^{2} - 148 a b c d + 63 b^{2} c^{2}\right )}{240 a^{3} x^{3}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (15 a^{3} d^{3} - 481 a^{2} b c d^{2} + 749 a b^{2} c^{2} d - 315 b^{3} c^{3}\right )}{960 a^{4} c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (45 a^{4} d^{4} + 90 a^{3} b c d^{3} - 1564 a^{2} b^{2} c^{2} d^{2} + 2310 a b^{3} c^{3} d - 945 b^{4} c^{4}\right )}{1920 a^{5} c^{2} x} - \frac{\left (a d - b c\right )^{3} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{128 a^{\frac{11}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**6/(b*x+a)**(1/2),x)

[Out]

-c*sqrt(a + b*x)*(c + d*x)**(3/2)/(5*a*x**5) - c*sqrt(a + b*x)*sqrt(c + d*x)*(13
*a*d - 9*b*c)/(40*a**2*x**4) - sqrt(a + b*x)*sqrt(c + d*x)*(93*a**2*d**2 - 148*a
*b*c*d + 63*b**2*c**2)/(240*a**3*x**3) - sqrt(a + b*x)*sqrt(c + d*x)*(15*a**3*d*
*3 - 481*a**2*b*c*d**2 + 749*a*b**2*c**2*d - 315*b**3*c**3)/(960*a**4*c*x**2) +
sqrt(a + b*x)*sqrt(c + d*x)*(45*a**4*d**4 + 90*a**3*b*c*d**3 - 1564*a**2*b**2*c*
*2*d**2 + 2310*a*b**3*c**3*d - 945*b**4*c**4)/(1920*a**5*c**2*x) - (a*d - b*c)**
3*(3*a**2*d**2 + 14*a*b*c*d + 63*b**2*c**2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)
*sqrt(c + d*x)))/(128*a**(11/2)*c**(5/2))

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Mathematica [A]  time = 0.37773, size = 319, normalized size = 0.92 \[ \frac{-15 x^5 \log (x) (b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )+15 x^5 (b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (3 a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )-2 a^3 b c x \left (216 c^3+592 c^2 d x+481 c d^2 x^2+45 d^3 x^3\right )+2 a^2 b^2 c^2 x^2 \left (252 c^2+749 c d x+782 d^2 x^2\right )-210 a b^3 c^3 x^3 (3 c+11 d x)+945 b^4 c^4 x^4\right )}{3840 a^{11/2} c^{5/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^6*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(945*b^4*c^4*x^4 - 210*a*b^3*c^3
*x^3*(3*c + 11*d*x) + 2*a^2*b^2*c^2*x^2*(252*c^2 + 749*c*d*x + 782*d^2*x^2) - 2*
a^3*b*c*x*(216*c^3 + 592*c^2*d*x + 481*c*d^2*x^2 + 45*d^3*x^3) + 3*a^4*(128*c^4
+ 336*c^3*d*x + 248*c^2*d^2*x^2 + 10*c*d^3*x^3 - 15*d^4*x^4)) - 15*(b*c - a*d)^3
*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*x^5*Log[x] + 15*(b*c - a*d)^3*(63*b^2*c^2
+ 14*a*b*c*d + 3*a^2*d^2)*x^5*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqr
t[a + b*x]*Sqrt[c + d*x]])/(3840*a^(11/2)*c^(5/2)*x^5)

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Maple [B]  time = 0.045, size = 813, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^6/(b*x+a)^(1/2),x)

[Out]

-1/3840*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^5/c^2*(45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((
b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^5*d^5+75*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^4*b*c*d^4+450*ln((a*d*x+b*c*x+2*(a*c)^(1/2)
*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^3*b^2*c^2*d^3-2250*ln((a*d*x+b*c*x+2*(a
*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^2*b^3*c^3*d^2+2625*ln((a*d*x+b
*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a*b^4*c^4*d-945*ln((a*d
*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*b^5*c^5-90*((b*x+a)
*(d*x+c))^(1/2)*d^4*a^4*x^4*(a*c)^(1/2)-180*((b*x+a)*(d*x+c))^(1/2)*d^3*b*c*a^3*
x^4*(a*c)^(1/2)+3128*((b*x+a)*(d*x+c))^(1/2)*d^2*b^2*c^2*a^2*x^4*(a*c)^(1/2)-462
0*((b*x+a)*(d*x+c))^(1/2)*d*b^3*c^3*a*x^4*(a*c)^(1/2)+1890*c^4*((b*x+a)*(d*x+c))
^(1/2)*b^4*x^4*(a*c)^(1/2)+60*((b*x+a)*(d*x+c))^(1/2)*d^3*c*a^4*x^3*(a*c)^(1/2)-
1924*((b*x+a)*(d*x+c))^(1/2)*d^2*b*c^2*a^3*x^3*(a*c)^(1/2)+2996*((b*x+a)*(d*x+c)
)^(1/2)*d*b^2*c^3*a^2*x^3*(a*c)^(1/2)-1260*c^4*((b*x+a)*(d*x+c))^(1/2)*b^3*a*x^3
*(a*c)^(1/2)+1488*((b*x+a)*(d*x+c))^(1/2)*d^2*c^2*a^4*x^2*(a*c)^(1/2)-2368*((b*x
+a)*(d*x+c))^(1/2)*d*b*c^3*a^3*x^2*(a*c)^(1/2)+1008*c^4*((b*x+a)*(d*x+c))^(1/2)*
b^2*a^2*x^2*(a*c)^(1/2)+2016*((b*x+a)*(d*x+c))^(1/2)*d*c^3*a^4*x*(a*c)^(1/2)-864
*c^4*((b*x+a)*(d*x+c))^(1/2)*b*a^3*x*(a*c)^(1/2)+768*c^4*((b*x+a)*(d*x+c))^(1/2)
*a^4*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^5/(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((d*x + c)^(5/2)/(sqrt(b*x + a)*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.86083, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} x^{5} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (384 \, a^{4} c^{4} +{\left (945 \, b^{4} c^{4} - 2310 \, a b^{3} c^{3} d + 1564 \, a^{2} b^{2} c^{2} d^{2} - 90 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4}\right )} x^{4} - 2 \,{\left (315 \, a b^{3} c^{4} - 749 \, a^{2} b^{2} c^{3} d + 481 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (63 \, a^{2} b^{2} c^{4} - 148 \, a^{3} b c^{3} d + 93 \, a^{4} c^{2} d^{2}\right )} x^{2} - 144 \,{\left (3 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, \sqrt{a c} a^{5} c^{2} x^{5}}, \frac{15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (384 \, a^{4} c^{4} +{\left (945 \, b^{4} c^{4} - 2310 \, a b^{3} c^{3} d + 1564 \, a^{2} b^{2} c^{2} d^{2} - 90 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4}\right )} x^{4} - 2 \,{\left (315 \, a b^{3} c^{4} - 749 \, a^{2} b^{2} c^{3} d + 481 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (63 \, a^{2} b^{2} c^{4} - 148 \, a^{3} b c^{3} d + 93 \, a^{4} c^{2} d^{2}\right )} x^{2} - 144 \,{\left (3 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, \sqrt{-a c} a^{5} c^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(sqrt(b*x + a)*x^6),x, algorithm="fricas")

[Out]

[-1/7680*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^
2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*x^5*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*(384*a^4*c^4 + (945*b^4*c^4 - 2
310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 90*a^3*b*c*d^3 - 45*a^4*d^4)*x^4 - 2*(3
15*a*b^3*c^4 - 749*a^2*b^2*c^3*d + 481*a^3*b*c^2*d^2 - 15*a^4*c*d^3)*x^3 + 8*(63
*a^2*b^2*c^4 - 148*a^3*b*c^3*d + 93*a^4*c^2*d^2)*x^2 - 144*(3*a^3*b*c^4 - 7*a^4*
c^3*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^5*c^2*x^5), 1/3840
*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 -
5*a^4*b*c*d^4 - 3*a^5*d^5)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sq
rt(b*x + a)*sqrt(d*x + c)*a*c)) - 2*(384*a^4*c^4 + (945*b^4*c^4 - 2310*a*b^3*c^3
*d + 1564*a^2*b^2*c^2*d^2 - 90*a^3*b*c*d^3 - 45*a^4*d^4)*x^4 - 2*(315*a*b^3*c^4
- 749*a^2*b^2*c^3*d + 481*a^3*b*c^2*d^2 - 15*a^4*c*d^3)*x^3 + 8*(63*a^2*b^2*c^4
- 148*a^3*b*c^3*d + 93*a^4*c^2*d^2)*x^2 - 144*(3*a^3*b*c^4 - 7*a^4*c^3*d)*x)*sqr
t(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^5*c^2*x^5)]

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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)/x**6/(b*x+a)**(1/2),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((d*x + c)^(5/2)/(sqrt(b*x + a)*x^6),x, algorithm="giac")

[Out]

Exception raised: TypeError