3.762 \(\int \frac{(c+d x)^{5/2}}{x^4 (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=230 \[ \frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} \sqrt{c}}-\frac{5 \sqrt{c+d x} (b c-a d)^2 (7 b c-a d)}{8 a^4 c \sqrt{a+b x}}-\frac{5 (c+d x)^{3/2} (b c-a d) (7 b c-a d)}{24 a^3 c x \sqrt{a+b x}}+\frac{(c+d x)^{5/2} (7 b c-a d)}{12 a^2 c x^2 \sqrt{a+b x}}-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}} \]

[Out]

(-5*(b*c - a*d)^2*(7*b*c - a*d)*Sqrt[c + d*x])/(8*a^4*c*Sqrt[a + b*x]) - (5*(b*c
- a*d)*(7*b*c - a*d)*(c + d*x)^(3/2))/(24*a^3*c*x*Sqrt[a + b*x]) + ((7*b*c - a*
d)*(c + d*x)^(5/2))/(12*a^2*c*x^2*Sqrt[a + b*x]) - (c + d*x)^(7/2)/(3*a*c*x^3*Sq
rt[a + b*x]) + (5*(b*c - a*d)^2*(7*b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(8*a^(9/2)*Sqrt[c])

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Rubi [A]  time = 0.418268, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} \sqrt{c}}-\frac{5 \sqrt{c+d x} (b c-a d)^2 (7 b c-a d)}{8 a^4 c \sqrt{a+b x}}-\frac{5 (c+d x)^{3/2} (b c-a d) (7 b c-a d)}{24 a^3 c x \sqrt{a+b x}}+\frac{(c+d x)^{5/2} (7 b c-a d)}{12 a^2 c x^2 \sqrt{a+b x}}-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^4*(a + b*x)^(3/2)),x]

[Out]

(-5*(b*c - a*d)^2*(7*b*c - a*d)*Sqrt[c + d*x])/(8*a^4*c*Sqrt[a + b*x]) - (5*(b*c
- a*d)*(7*b*c - a*d)*(c + d*x)^(3/2))/(24*a^3*c*x*Sqrt[a + b*x]) + ((7*b*c - a*
d)*(c + d*x)^(5/2))/(12*a^2*c*x^2*Sqrt[a + b*x]) - (c + d*x)^(7/2)/(3*a*c*x^3*Sq
rt[a + b*x]) + (5*(b*c - a*d)^2*(7*b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(8*a^(9/2)*Sqrt[c])

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Rubi in Sympy [A]  time = 18.6437, size = 209, normalized size = 0.91 \[ - \frac{2 b \left (c + d x\right )^{\frac{7}{2}}}{a x^{3} \sqrt{a + b x} \left (a d - b c\right )} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (a d - 7 b c\right )}{3 a^{2} x^{3} \left (a d - b c\right )} - \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - 7 b c\right )}{12 a^{3} x^{2}} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - 7 b c\right ) \left (a d - b c\right )}{8 a^{4} x} - \frac{5 \left (a d - 7 b c\right ) \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{9}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**4/(b*x+a)**(3/2),x)

[Out]

-2*b*(c + d*x)**(7/2)/(a*x**3*sqrt(a + b*x)*(a*d - b*c)) - sqrt(a + b*x)*(c + d*
x)**(5/2)*(a*d - 7*b*c)/(3*a**2*x**3*(a*d - b*c)) - 5*sqrt(a + b*x)*(c + d*x)**(
3/2)*(a*d - 7*b*c)/(12*a**3*x**2) - 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - 7*b*c)*
(a*d - b*c)/(8*a**4*x) - 5*(a*d - 7*b*c)*(a*d - b*c)**2*atanh(sqrt(c)*sqrt(a + b
*x)/(sqrt(a)*sqrt(c + d*x)))/(8*a**(9/2)*sqrt(c))

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Mathematica [A]  time = 0.392987, size = 213, normalized size = 0.93 \[ \frac{-\frac{2 \sqrt{a} \sqrt{c+d x} \left (a^3 \left (8 c^2+26 c d x+33 d^2 x^2\right )+a^2 b x \left (-14 c^2-68 c d x+81 d^2 x^2\right )+5 a b^2 c x^2 (7 c-38 d x)+105 b^3 c^2 x^3\right )}{x^3 \sqrt{a+b x}}+\frac{15 \log (x) (a d-7 b c) (b c-a d)^2}{\sqrt{c}}+\frac{15 (7 b c-a d) (b c-a d)^2 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}}{48 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^4*(a + b*x)^(3/2)),x]

[Out]

((-2*Sqrt[a]*Sqrt[c + d*x]*(105*b^3*c^2*x^3 + 5*a*b^2*c*x^2*(7*c - 38*d*x) + a^3
*(8*c^2 + 26*c*d*x + 33*d^2*x^2) + a^2*b*x*(-14*c^2 - 68*c*d*x + 81*d^2*x^2)))/(
x^3*Sqrt[a + b*x]) + (15*(b*c - a*d)^2*(-7*b*c + a*d)*Log[x])/Sqrt[c] + (15*(b*c
- a*d)^2*(7*b*c - a*d)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b
*x]*Sqrt[c + d*x]])/Sqrt[c])/(48*a^(9/2))

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Maple [B]  time = 0.043, size = 704, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^4/(b*x+a)^(3/2),x)

[Out]

-1/48*(d*x+c)^(1/2)*(15*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^4*a^3*b*d^3-135*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^4*a^2*b^2*c*d^2+225*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^4*a*b^3*c^2*d-105*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^4*b^4*c^3+15*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^3*a^4*d^3-135*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^3*a^3*b*c*d^2+225*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^3*a^2*b^2*c^2*d-105*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^3*a*b^3*c^3+162*x^3*a^2*b*d^2*(a*c)^(1/2)*((b*x+a
)*(d*x+c))^(1/2)-380*x^3*a*b^2*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+210*x^3*b
^3*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+66*x^2*a^3*d^2*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)-136*x^2*a^2*b*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+70*x^2*a*b^2
*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+52*x*a^3*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+
c))^(1/2)-28*x*a^2*b*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+16*a^3*c^2*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2))/a^4/((b*x+a)*(d*x+c))^(1/2)/x^3/(a*c)^(1/2)/(b*x+a)
^(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)/((b*x + a)^(3/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.659387, size = 1, normalized size = 0. \[ \left [-\frac{4 \,{\left (8 \, a^{3} c^{2} +{\left (105 \, b^{3} c^{2} - 190 \, a b^{2} c d + 81 \, a^{2} b d^{2}\right )} x^{3} +{\left (35 \, a b^{2} c^{2} - 68 \, a^{2} b c d + 33 \, a^{3} d^{2}\right )} x^{2} - 2 \,{\left (7 \, a^{2} b c^{2} - 13 \, a^{3} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} + 15 \,{\left ({\left (7 \, b^{4} c^{3} - 15 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} +{\left (7 \, a b^{3} c^{3} - 15 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{96 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )} \sqrt{a c}}, -\frac{2 \,{\left (8 \, a^{3} c^{2} +{\left (105 \, b^{3} c^{2} - 190 \, a b^{2} c d + 81 \, a^{2} b d^{2}\right )} x^{3} +{\left (35 \, a b^{2} c^{2} - 68 \, a^{2} b c d + 33 \, a^{3} d^{2}\right )} x^{2} - 2 \,{\left (7 \, a^{2} b c^{2} - 13 \, a^{3} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} - 15 \,{\left ({\left (7 \, b^{4} c^{3} - 15 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} +{\left (7 \, a b^{3} c^{3} - 15 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{48 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/((b*x + a)^(3/2)*x^4),x, algorithm="fricas")

[Out]

[-1/96*(4*(8*a^3*c^2 + (105*b^3*c^2 - 190*a*b^2*c*d + 81*a^2*b*d^2)*x^3 + (35*a*
b^2*c^2 - 68*a^2*b*c*d + 33*a^3*d^2)*x^2 - 2*(7*a^2*b*c^2 - 13*a^3*c*d)*x)*sqrt(
a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((7*b^4*c^3 - 15*a*b^3*c^2*d + 9*a^2*b^2*c
*d^2 - a^3*b*d^3)*x^4 + (7*a*b^3*c^3 - 15*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 - a^4*d^
3)*x^3)*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)*sq
rt(a*c))/x^2))/((a^4*b*x^4 + a^5*x^3)*sqrt(a*c)), -1/48*(2*(8*a^3*c^2 + (105*b^3
*c^2 - 190*a*b^2*c*d + 81*a^2*b*d^2)*x^3 + (35*a*b^2*c^2 - 68*a^2*b*c*d + 33*a^3
*d^2)*x^2 - 2*(7*a^2*b*c^2 - 13*a^3*c*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x +
c) - 15*((7*b^4*c^3 - 15*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 - a^3*b*d^3)*x^4 + (7*a*b
^3*c^3 - 15*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 - a^4*d^3)*x^3)*arctan(1/2*(2*a*c + (b
*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/((a^4*b*x^4 + a^5*x^
3)*sqrt(-a*c))]

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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**4/(b*x+a)**(3/2),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)/((b*x + a)^(3/2)*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError