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

Optimal. Leaf size=445 \[ -\frac{2 c x^3 \sqrt{a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}-\frac{2 c x^2 \sqrt{a+b x} (a d+b c) \left (7 a^2 d^2-22 a b c d+7 b^2 c^2\right )}{3 b^2 d^2 \sqrt{c+d x} (b c-a d)^4}+\frac{5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{9/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left ((a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )-2 b d x \left (35 a^4 d^4-76 a^3 b c d^3+18 a^2 b^2 c^2 d^2-76 a b^3 c^3 d+35 b^4 c^4\right )\right )}{12 b^4 d^4 (b c-a d)^4}+\frac{2 a x^4 (13 b c-7 a d)}{3 b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

(2*a*x^5)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*(13*b*c - 7*a
*d)*x^4)/(3*b^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b^2*c^2 + 1
4*a*b*c*d - 7*a^2*d^2)*x^3*Sqrt[a + b*x])/(3*b^2*d*(b*c - a*d)^3*(c + d*x)^(3/2)
) - (2*c*(b*c + a*d)*(7*b^2*c^2 - 22*a*b*c*d + 7*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*
b^2*d^2*(b*c - a*d)^4*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*((b*c + a*d)
*(105*b^4*c^4 - 340*a*b^3*c^3*d + 406*a^2*b^2*c^2*d^2 - 340*a^3*b*c*d^3 + 105*a^
4*d^4) - 2*b*d*(35*b^4*c^4 - 76*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 76*a^3*b*c*d^
3 + 35*a^4*d^4)*x))/(12*b^4*d^4*(b*c - a*d)^4) + (5*(7*b^2*c^2 + 10*a*b*c*d + 7*
a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(9/2)*d^
(9/2))

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Rubi [A]  time = 1.4491, antiderivative size = 445, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 c x^3 \sqrt{a+b x} \left (-7 a^2 d^2+14 a b c d+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}-\frac{2 c x^2 \sqrt{a+b x} (a d+b c) \left (7 a^2 d^2-22 a b c d+7 b^2 c^2\right )}{3 b^2 d^2 \sqrt{c+d x} (b c-a d)^4}+\frac{5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{9/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left ((a d+b c) \left (105 a^4 d^4-340 a^3 b c d^3+406 a^2 b^2 c^2 d^2-340 a b^3 c^3 d+105 b^4 c^4\right )-2 b d x \left (35 a^4 d^4-76 a^3 b c d^3+18 a^2 b^2 c^2 d^2-76 a b^3 c^3 d+35 b^4 c^4\right )\right )}{12 b^4 d^4 (b c-a d)^4}+\frac{2 a x^4 (13 b c-7 a d)}{3 b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^5}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*x^5)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*(13*b*c - 7*a
*d)*x^4)/(3*b^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b^2*c^2 + 1
4*a*b*c*d - 7*a^2*d^2)*x^3*Sqrt[a + b*x])/(3*b^2*d*(b*c - a*d)^3*(c + d*x)^(3/2)
) - (2*c*(b*c + a*d)*(7*b^2*c^2 - 22*a*b*c*d + 7*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*
b^2*d^2*(b*c - a*d)^4*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*((b*c + a*d)
*(105*b^4*c^4 - 340*a*b^3*c^3*d + 406*a^2*b^2*c^2*d^2 - 340*a^3*b*c*d^3 + 105*a^
4*d^4) - 2*b*d*(35*b^4*c^4 - 76*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 76*a^3*b*c*d^
3 + 35*a^4*d^4)*x))/(12*b^4*d^4*(b*c - a*d)^4) + (5*(7*b^2*c^2 + 10*a*b*c*d + 7*
a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(9/2)*d^
(9/2))

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Rubi in Sympy [A]  time = 141.776, size = 454, normalized size = 1.02 \[ - \frac{2 a x^{5}}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 a x^{4} \left (7 a d - 13 b c\right )}{3 b^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{2 c x^{3} \sqrt{a + b x} \left (7 a^{2} d^{2} - 14 a b c d - b^{2} c^{2}\right )}{3 b^{2} d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} - \frac{2 c x^{2} \sqrt{a + b x} \left (a d + b c\right ) \left (7 a^{2} d^{2} - 22 a b c d + 7 b^{2} c^{2}\right )}{3 b^{2} d^{2} \sqrt{c + d x} \left (a d - b c\right )^{4}} - \frac{8 \sqrt{a + b x} \sqrt{c + d x} \left (- \frac{3 b d x \left (35 a^{4} d^{4} - 76 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 76 a b^{3} c^{3} d + 35 b^{4} c^{4}\right )}{16} + \left (\frac{3 a d}{32} + \frac{3 b c}{32}\right ) \left (105 a^{4} d^{4} - 340 a^{3} b c d^{3} + 406 a^{2} b^{2} c^{2} d^{2} - 340 a b^{3} c^{3} d + 105 b^{4} c^{4}\right )\right )}{9 b^{4} d^{4} \left (a d - b c\right )^{4}} + \frac{5 \left (7 a^{2} d^{2} + 10 a b c d + 7 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{9}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*x**5/(3*b*(a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d - b*c)) - 2*a*x**4*(7*a*d
- 13*b*c)/(3*b**2*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**2) - 2*c*x**3*sqrt
(a + b*x)*(7*a**2*d**2 - 14*a*b*c*d - b**2*c**2)/(3*b**2*d*(c + d*x)**(3/2)*(a*d
- b*c)**3) - 2*c*x**2*sqrt(a + b*x)*(a*d + b*c)*(7*a**2*d**2 - 22*a*b*c*d + 7*b
**2*c**2)/(3*b**2*d**2*sqrt(c + d*x)*(a*d - b*c)**4) - 8*sqrt(a + b*x)*sqrt(c +
d*x)*(-3*b*d*x*(35*a**4*d**4 - 76*a**3*b*c*d**3 + 18*a**2*b**2*c**2*d**2 - 76*a*
b**3*c**3*d + 35*b**4*c**4)/16 + (3*a*d/32 + 3*b*c/32)*(105*a**4*d**4 - 340*a**3
*b*c*d**3 + 406*a**2*b**2*c**2*d**2 - 340*a*b**3*c**3*d + 105*b**4*c**4))/(9*b**
4*d**4*(a*d - b*c)**4) + 5*(7*a**2*d**2 + 10*a*b*c*d + 7*b**2*c**2)*atanh(sqrt(d
)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(4*b**(9/2)*d**(9/2))

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Mathematica [A]  time = 1.46259, size = 246, normalized size = 0.55 \[ \frac{5 \left (7 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{9/2} d^{9/2}}+\frac{1}{12} \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{8 a^6}{b^4 (a+b x)^2 (b c-a d)^3}-\frac{16 a^5 (5 a d-9 b c)}{b^4 (a+b x) (b c-a d)^4}-\frac{33 (a d+b c)}{b^4 d^4}-\frac{8 c^6}{d^4 (c+d x)^2 (a d-b c)^3}-\frac{16 c^5 (5 b c-9 a d)}{d^4 (c+d x) (b c-a d)^4}+\frac{6 x}{b^3 d^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*((-33*(b*c + a*d))/(b^4*d^4) + (6*x)/(b^3*d^3) - (8
*a^6)/(b^4*(b*c - a*d)^3*(a + b*x)^2) - (16*a^5*(-9*b*c + 5*a*d))/(b^4*(b*c - a*
d)^4*(a + b*x)) - (8*c^6)/(d^4*(-(b*c) + a*d)^3*(c + d*x)^2) - (16*c^5*(5*b*c -
9*a*d))/(d^4*(b*c - a*d)^4*(c + d*x))))/12 + (5*(7*b^2*c^2 + 10*a*b*c*d + 7*a^2*
d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(
8*b^(9/2)*d^(9/2))

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Maple [B]  time = 0.071, size = 3425, normalized size = 7.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(-420*x*a*b^6*c^7*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-270*ln(1/2*(2*b*d*x+2
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4*a^5*b^3*c*d^7+135
*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4
*a^4*b^4*c^2*d^6+60*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*
c)/(b*d)^(1/2))*x^4*a^3*b^5*c^3*d^5+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4*a^2*b^6*c^4*d^4-270*ln(1/2*(2*b*d*x+2*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4*a*b^7*c^5*d^3-330*ln
(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*a^
6*b^2*c*d^7-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(
b*d)^(1/2))*x^3*a^5*b^3*c^2*d^6+390*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*a^4*b^4*c^3*d^5+390*ln(1/2*(2*b*d*x+2*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*a^3*b^5*c^4*d^4-270*ln(1
/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*a^2*
b^6*c^5*d^3-330*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(
b*d)^(1/2))*x^3*a*b^7*c^6*d^2+150*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^7*b*c*d^7-840*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^6*b^2*c^2*d^6+330*ln(1/2*(2*
b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^5*b^3*c^
3*d^5+510*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(
1/2))*x^2*a^4*b^4*c^4*d^4+330*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^3*b^5*c^5*d^3-840*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^2*b^6*c^6*d^2+150*ln(1/2*(2*
b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a*b^7*c^7*
d-330*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2)
)*x*a^7*b*c^2*d^6-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+
b*c)/(b*d)^(1/2))*x*a^6*b^2*c^3*d^5+390*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^5*b^3*c^4*d^4+390*ln(1/2*(2*b*d*x+2*((b*
x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^4*b^4*c^5*d^3-270*ln(1
/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^3*b^
5*c^6*d^2-330*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*
d)^(1/2))*x*a^2*b^6*c^7*d+470*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^6*b*c^3*d^4-
132*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^5*b^2*c^4*d^3-132*(b*d)^(1/2)*((b*x+a)
*(d*x+c))^(1/2)*a^4*b^3*c^5*d^2+470*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^4*
c^6*d+12*x^5*a^4*b^3*d^7*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*x^5*b^7*c^4*d^3*
(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-280*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*
a^6*b*d^7-280*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*b^7*c^6*d-42*(b*d)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)*x^4*a^5*b^2*d^7-42*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^
4*b^7*c^5*d^2-420*x*a^7*c*d^6*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*x^2*a^7*d^
7*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-414*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^
2*a^3*b^4*c^4*d^3-90*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2*a*b^6*c^6*d+660*(b*
d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^2*b^5*c^6*d+660*(b*d)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)*x*a^6*b*c^2*d^5+372*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^5*b^2*c^3*d
^4-456*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x*a^4*b^3*c^4*d^3+372*(b*d)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)*x*a^3*b^4*c^5*d^2+126*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x
^4*a^4*b^3*c*d^6-84*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^4*a^3*b^4*c^2*d^5-84*(
b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^4*a^2*b^5*c^3*d^4+126*(b*d)^(1/2)*((b*x+a)*
(d*x+c))^(1/2)*x^4*a*b^6*c^4*d^3+552*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^5
*b^2*c*d^6+24*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^4*b^3*c^2*d^5-336*(b*d)^
(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a^3*b^4*c^3*d^4+24*(b*d)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)*x^3*a^2*b^5*c^4*d^3-48*x^5*a^3*b^4*c*d^6*(b*d)^(1/2)*((b*x+a)*(d*x+c))^
(1/2)+72*x^5*a^2*b^5*c^2*d^5*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-48*x^5*a*b^6*c^
3*d^4*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+1098*x^2*a^5*b^2*c^2*d^5*(b*d)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)+1098*x^2*a^2*b^5*c^5*d^2*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)+552*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^3*a*b^6*c^5*d^2-90*(b*d)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)*x^2*a^6*b*c*d^6-414*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*x^2
*a^4*b^3*c^3*d^4+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b
*c)/(b*d)^(1/2))*x^2*a^8*d^8+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)
^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^8*c^8+105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^8*c^2*d^6+105*ln(1/2*(2*b*d*x+2*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^6*c^8-210*x^2*b^7*c^7
*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-210*a^7*c^2*d^5*(b*d)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)-210*a^2*b^5*c^7*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+105*ln(1/2*(2*b*d*x
+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4*a^6*b^2*d^8+105
*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4
*b^8*c^6*d^2+210*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*x^3*a^7*b*d^8+210*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*b^8*c^7*d+210*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^8*c*d^7+210*ln(1/2*(2*b*d*x+2*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a*b^7*c^8-270*ln(1/2*(2*b
*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^7*b*c^3*d^5+1
35*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a
^6*b^2*c^4*d^4+60*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)
/(b*d)^(1/2))*a^5*b^3*c^5*d^3+135*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^4*c^6*d^2-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^5*c^7*d)/((b*x+a)*(d*x+c))^(
1/2)/(a*d-b*c)^4/(b*d)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(3/2)/b^4/d^4

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.62463, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(4*(105*a^2*b^5*c^7 - 235*a^3*b^4*c^6*d + 66*a^4*b^3*c^5*d^2 + 66*a^5*b^2
*c^4*d^3 - 235*a^6*b*c^3*d^4 + 105*a^7*c^2*d^5 - 6*(b^7*c^4*d^3 - 4*a*b^6*c^3*d^
4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^5 + 21*(b^7*c^5*d^2 - 3
*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b
^2*d^7)*x^4 + 4*(35*b^7*c^6*d - 69*a*b^6*c^5*d^2 - 3*a^2*b^5*c^4*d^3 + 42*a^3*b^
4*c^3*d^4 - 3*a^4*b^3*c^2*d^5 - 69*a^5*b^2*c*d^6 + 35*a^6*b*d^7)*x^3 + 3*(35*b^7
*c^7 + 15*a*b^6*c^6*d - 183*a^2*b^5*c^5*d^2 + 69*a^3*b^4*c^4*d^3 + 69*a^4*b^3*c^
3*d^4 - 183*a^5*b^2*c^2*d^5 + 15*a^6*b*c*d^6 + 35*a^7*d^7)*x^2 + 6*(35*a*b^6*c^7
- 55*a^2*b^5*c^6*d - 31*a^3*b^4*c^5*d^2 + 38*a^4*b^3*c^4*d^3 - 31*a^5*b^2*c^3*d
^4 - 55*a^6*b*c^2*d^5 + 35*a^7*c*d^6)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) -
15*(7*a^2*b^6*c^8 - 18*a^3*b^5*c^7*d + 9*a^4*b^4*c^6*d^2 + 4*a^5*b^3*c^5*d^3 +
9*a^6*b^2*c^4*d^4 - 18*a^7*b*c^3*d^5 + 7*a^8*c^2*d^6 + (7*b^8*c^6*d^2 - 18*a*b^7
*c^5*d^3 + 9*a^2*b^6*c^4*d^4 + 4*a^3*b^5*c^3*d^5 + 9*a^4*b^4*c^2*d^6 - 18*a^5*b^
3*c*d^7 + 7*a^6*b^2*d^8)*x^4 + 2*(7*b^8*c^7*d - 11*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5
*d^3 + 13*a^3*b^5*c^4*d^4 + 13*a^4*b^4*c^3*d^5 - 9*a^5*b^3*c^2*d^6 - 11*a^6*b^2*
c*d^7 + 7*a^7*b*d^8)*x^3 + (7*b^8*c^8 + 10*a*b^7*c^7*d - 56*a^2*b^6*c^6*d^2 + 22
*a^3*b^5*c^5*d^3 + 34*a^4*b^4*c^4*d^4 + 22*a^5*b^3*c^3*d^5 - 56*a^6*b^2*c^2*d^6
+ 10*a^7*b*c*d^7 + 7*a^8*d^8)*x^2 + 2*(7*a*b^7*c^8 - 11*a^2*b^6*c^7*d - 9*a^3*b^
5*c^6*d^2 + 13*a^4*b^4*c^5*d^3 + 13*a^5*b^3*c^4*d^4 - 9*a^6*b^2*c^3*d^5 - 11*a^7
*b*c^2*d^6 + 7*a^8*c*d^7)*x)*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)))/((a^2*b^8*c^6*d^4 - 4*a^3*b^7*c^5*d^5 + 6*a^4*b^6*c^4*d
^6 - 4*a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + (b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a
^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^4 + 2*(b^10*c^5*d^5 - 3*a*b^9
*c^4*d^6 + 2*a^2*b^8*c^3*d^7 + 2*a^3*b^7*c^2*d^8 - 3*a^4*b^6*c*d^9 + a^5*b^5*d^1
0)*x^3 + (b^10*c^6*d^4 - 9*a^2*b^8*c^4*d^6 + 16*a^3*b^7*c^3*d^7 - 9*a^4*b^6*c^2*
d^8 + a^6*b^4*d^10)*x^2 + 2*(a*b^9*c^6*d^4 - 3*a^2*b^8*c^5*d^5 + 2*a^3*b^7*c^4*d
^6 + 2*a^4*b^6*c^3*d^7 - 3*a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9)*x)*sqrt(b*d)), -1/24
*(2*(105*a^2*b^5*c^7 - 235*a^3*b^4*c^6*d + 66*a^4*b^3*c^5*d^2 + 66*a^5*b^2*c^4*d
^3 - 235*a^6*b*c^3*d^4 + 105*a^7*c^2*d^5 - 6*(b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*
a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^5 + 21*(b^7*c^5*d^2 - 3*a*b^6
*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7
)*x^4 + 4*(35*b^7*c^6*d - 69*a*b^6*c^5*d^2 - 3*a^2*b^5*c^4*d^3 + 42*a^3*b^4*c^3*
d^4 - 3*a^4*b^3*c^2*d^5 - 69*a^5*b^2*c*d^6 + 35*a^6*b*d^7)*x^3 + 3*(35*b^7*c^7 +
15*a*b^6*c^6*d - 183*a^2*b^5*c^5*d^2 + 69*a^3*b^4*c^4*d^3 + 69*a^4*b^3*c^3*d^4
- 183*a^5*b^2*c^2*d^5 + 15*a^6*b*c*d^6 + 35*a^7*d^7)*x^2 + 6*(35*a*b^6*c^7 - 55*
a^2*b^5*c^6*d - 31*a^3*b^4*c^5*d^2 + 38*a^4*b^3*c^4*d^3 - 31*a^5*b^2*c^3*d^4 - 5
5*a^6*b*c^2*d^5 + 35*a^7*c*d^6)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(
7*a^2*b^6*c^8 - 18*a^3*b^5*c^7*d + 9*a^4*b^4*c^6*d^2 + 4*a^5*b^3*c^5*d^3 + 9*a^6
*b^2*c^4*d^4 - 18*a^7*b*c^3*d^5 + 7*a^8*c^2*d^6 + (7*b^8*c^6*d^2 - 18*a*b^7*c^5*
d^3 + 9*a^2*b^6*c^4*d^4 + 4*a^3*b^5*c^3*d^5 + 9*a^4*b^4*c^2*d^6 - 18*a^5*b^3*c*d
^7 + 7*a^6*b^2*d^8)*x^4 + 2*(7*b^8*c^7*d - 11*a*b^7*c^6*d^2 - 9*a^2*b^6*c^5*d^3
+ 13*a^3*b^5*c^4*d^4 + 13*a^4*b^4*c^3*d^5 - 9*a^5*b^3*c^2*d^6 - 11*a^6*b^2*c*d^7
+ 7*a^7*b*d^8)*x^3 + (7*b^8*c^8 + 10*a*b^7*c^7*d - 56*a^2*b^6*c^6*d^2 + 22*a^3*
b^5*c^5*d^3 + 34*a^4*b^4*c^4*d^4 + 22*a^5*b^3*c^3*d^5 - 56*a^6*b^2*c^2*d^6 + 10*
a^7*b*c*d^7 + 7*a^8*d^8)*x^2 + 2*(7*a*b^7*c^8 - 11*a^2*b^6*c^7*d - 9*a^3*b^5*c^6
*d^2 + 13*a^4*b^4*c^5*d^3 + 13*a^5*b^3*c^4*d^4 - 9*a^6*b^2*c^3*d^5 - 11*a^7*b*c^
2*d^6 + 7*a^8*c*d^7)*x)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x +
a)*sqrt(d*x + c)*b*d)))/((a^2*b^8*c^6*d^4 - 4*a^3*b^7*c^5*d^5 + 6*a^4*b^6*c^4*d^
6 - 4*a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + (b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^
2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^4 + 2*(b^10*c^5*d^5 - 3*a*b^9*
c^4*d^6 + 2*a^2*b^8*c^3*d^7 + 2*a^3*b^7*c^2*d^8 - 3*a^4*b^6*c*d^9 + a^5*b^5*d^10
)*x^3 + (b^10*c^6*d^4 - 9*a^2*b^8*c^4*d^6 + 16*a^3*b^7*c^3*d^7 - 9*a^4*b^6*c^2*d
^8 + a^6*b^4*d^10)*x^2 + 2*(a*b^9*c^6*d^4 - 3*a^2*b^8*c^5*d^5 + 2*a^3*b^7*c^4*d^
6 + 2*a^4*b^6*c^3*d^7 - 3*a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9)*x)*sqrt(-b*d))]

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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(x**6/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.811644, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x