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

Optimal. Leaf size=346 \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-231 a^2 d^2-2 b d x (5 b c-99 a d)+30 a b c d+5 b^2 c^2\right )}{80 b^4 d^2}+\frac{3 (b c-a d)^2 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{13/2} d^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right )}{128 b^6 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right )}{64 b^5 d^2}+\frac{11 x^2 \sqrt{a+b x} (c+d x)^{5/2}}{5 b^2}-\frac{2 x^3 (c+d x)^{5/2}}{b \sqrt{a+b x}} \]

[Out]

(3*(b*c - a*d)*(b^3*c^3 + 7*a*b^2*c^2*d + 63*a^2*b*c*d^2 - 231*a^3*d^3)*Sqrt[a +
b*x]*Sqrt[c + d*x])/(128*b^6*d^2) + ((b^3*c^3 + 7*a*b^2*c^2*d + 63*a^2*b*c*d^2
- 231*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(64*b^5*d^2) - (2*x^3*(c + d*x)^(5
/2))/(b*Sqrt[a + b*x]) + (11*x^2*Sqrt[a + b*x]*(c + d*x)^(5/2))/(5*b^2) - (Sqrt[
a + b*x]*(c + d*x)^(5/2)*(5*b^2*c^2 + 30*a*b*c*d - 231*a^2*d^2 - 2*b*d*(5*b*c -
99*a*d)*x))/(80*b^4*d^2) + (3*(b*c - a*d)^2*(b^3*c^3 + 7*a*b^2*c^2*d + 63*a^2*b*
c*d^2 - 231*a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(
128*b^(13/2)*d^(5/2))

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Rubi [A]  time = 0.767858, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\sqrt{a+b x} (c+d x)^{5/2} \left (-231 a^2 d^2-2 b d x (5 b c-99 a d)+30 a b c d+5 b^2 c^2\right )}{80 b^4 d^2}+\frac{3 (b c-a d)^2 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{128 b^{13/2} d^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right )}{128 b^6 d^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right )}{64 b^5 d^2}+\frac{11 x^2 \sqrt{a+b x} (c+d x)^{5/2}}{5 b^2}-\frac{2 x^3 (c+d x)^{5/2}}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*c - a*d)*(b^3*c^3 + 7*a*b^2*c^2*d + 63*a^2*b*c*d^2 - 231*a^3*d^3)*Sqrt[a +
b*x]*Sqrt[c + d*x])/(128*b^6*d^2) + ((b^3*c^3 + 7*a*b^2*c^2*d + 63*a^2*b*c*d^2
- 231*a^3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(64*b^5*d^2) - (2*x^3*(c + d*x)^(5
/2))/(b*Sqrt[a + b*x]) + (11*x^2*Sqrt[a + b*x]*(c + d*x)^(5/2))/(5*b^2) - (Sqrt[
a + b*x]*(c + d*x)^(5/2)*(5*b^2*c^2 + 30*a*b*c*d - 231*a^2*d^2 - 2*b*d*(5*b*c -
99*a*d)*x))/(80*b^4*d^2) + (3*(b*c - a*d)^2*(b^3*c^3 + 7*a*b^2*c^2*d + 63*a^2*b*
c*d^2 - 231*a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(
128*b^(13/2)*d^(5/2))

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Rubi in Sympy [A]  time = 79.4687, size = 352, normalized size = 1.02 \[ - \frac{2 x^{3} \left (c + d x\right )^{\frac{5}{2}}}{b \sqrt{a + b x}} + \frac{11 x^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (\frac{693 a^{2} d^{2}}{8} - \frac{45 a b c d}{4} - \frac{15 b^{2} c^{2}}{8} - \frac{3 b d x \left (99 a d - 5 b c\right )}{4}\right )}{30 b^{4} d^{2}} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (231 a^{3} d^{3} - 63 a^{2} b c d^{2} - 7 a b^{2} c^{2} d - b^{3} c^{3}\right )}{64 b^{5} d^{2}} + \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (231 a^{3} d^{3} - 63 a^{2} b c d^{2} - 7 a b^{2} c^{2} d - b^{3} c^{3}\right )}{128 b^{6} d^{2}} - \frac{3 \left (a d - b c\right )^{2} \left (231 a^{3} d^{3} - 63 a^{2} b c d^{2} - 7 a b^{2} c^{2} d - b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{128 b^{\frac{13}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**3*(c + d*x)**(5/2)/(b*sqrt(a + b*x)) + 11*x**2*sqrt(a + b*x)*(c + d*x)**(5
/2)/(5*b**2) + sqrt(a + b*x)*(c + d*x)**(5/2)*(693*a**2*d**2/8 - 45*a*b*c*d/4 -
15*b**2*c**2/8 - 3*b*d*x*(99*a*d - 5*b*c)/4)/(30*b**4*d**2) - sqrt(a + b*x)*(c +
d*x)**(3/2)*(231*a**3*d**3 - 63*a**2*b*c*d**2 - 7*a*b**2*c**2*d - b**3*c**3)/(6
4*b**5*d**2) + 3*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)*(231*a**3*d**3 - 63*a**
2*b*c*d**2 - 7*a*b**2*c**2*d - b**3*c**3)/(128*b**6*d**2) - 3*(a*d - b*c)**2*(23
1*a**3*d**3 - 63*a**2*b*c*d**2 - 7*a*b**2*c**2*d - b**3*c**3)*atanh(sqrt(d)*sqrt
(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(128*b**(13/2)*d**(5/2))

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Mathematica [A]  time = 0.354276, size = 320, normalized size = 0.92 \[ \frac{3 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{256 b^{13/2} d^{5/2}}+\frac{\sqrt{c+d x} \left (3465 a^5 d^4+105 a^4 b d^3 (11 d x-64 c)-42 a^3 b^2 d^2 \left (-79 c^2+57 c d x+11 d^2 x^2\right )+2 a^2 b^3 d \left (-40 c^3+662 c^2 d x+459 c d^2 x^2+132 d^3 x^3\right )-a b^4 \left (15 c^4+70 c^3 d x+466 c^2 d^2 x^2+512 c d^3 x^3+176 d^4 x^4\right )+b^5 x \left (-15 c^4+10 c^3 d x+248 c^2 d^2 x^2+336 c d^3 x^3+128 d^4 x^4\right )\right )}{640 b^6 d^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(3465*a^5*d^4 + 105*a^4*b*d^3*(-64*c + 11*d*x) - 42*a^3*b^2*d^2*(
-79*c^2 + 57*c*d*x + 11*d^2*x^2) + 2*a^2*b^3*d*(-40*c^3 + 662*c^2*d*x + 459*c*d^
2*x^2 + 132*d^3*x^3) + b^5*x*(-15*c^4 + 10*c^3*d*x + 248*c^2*d^2*x^2 + 336*c*d^3
*x^3 + 128*d^4*x^4) - a*b^4*(15*c^4 + 70*c^3*d*x + 466*c^2*d^2*x^2 + 512*c*d^3*x
^3 + 176*d^4*x^4)))/(640*b^6*d^2*Sqrt[a + b*x]) + (3*(b*c - a*d)^2*(b^3*c^3 + 7*
a*b^2*c^2*d + 63*a^2*b*c*d^2 - 231*a^3*d^3)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*
Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(256*b^(13/2)*d^(5/2))

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Maple [B]  time = 0.047, size = 1265, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1280*(d*x+c)^(1/2)*(924*x^2*a^3*b^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3
0*x*b^5*c^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*a*b^4*c^4*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+3465*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*d^5-7875*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*c*d^4+5250*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^2*c^2*d^3-750*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^3*c^3*d^2-75*l
n(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
^4*c^4*d-6930*a^5*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-15*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*b^6*c^5-15*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*b^5*c^5-256*
x^5*b^5*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-20*x^2*b^5*c^3*d*((b*x+a)*(d*x+c
))^(1/2)*(b*d)^(1/2)-2310*x*a^4*b*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+13440*
a^4*b*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6636*a^3*b^2*c^2*d^2*((b*x+a)*(d
*x+c))^(1/2)*(b*d)^(1/2)+160*a^2*b^3*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-7
875*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^2*c*d^4+5250*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^3*c^2*d^3-750*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^4*c^3*d^2-75*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^5*c^4*d+352*x^4*a*b^4*
d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-672*x^4*b^5*c*d^3*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)-528*x^3*a^2*b^3*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-496*x^3*b^
5*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3465*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*d^5+1024*x^3*a*b^4*c*d^3*((b*
x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1836*x^2*a^2*b^3*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)+932*x^2*a*b^4*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+4788*x*a^3*
b^2*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2648*x*a^2*b^3*c^2*d^2*((b*x+a)*(d
*x+c))^(1/2)*(b*d)^(1/2)+140*x*a*b^4*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/
((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/d^2/(b*x+a)^(1/2)/b^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((d*x + c)^(5/2)*x^3/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67178, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2560*(4*(128*b^5*d^4*x^5 - 15*a*b^4*c^4 - 80*a^2*b^3*c^3*d + 3318*a^3*b^2*c^2
*d^2 - 6720*a^4*b*c*d^3 + 3465*a^5*d^4 + 16*(21*b^5*c*d^3 - 11*a*b^4*d^4)*x^4 +
8*(31*b^5*c^2*d^2 - 64*a*b^4*c*d^3 + 33*a^2*b^3*d^4)*x^3 + 2*(5*b^5*c^3*d - 233*
a*b^4*c^2*d^2 + 459*a^2*b^3*c*d^3 - 231*a^3*b^2*d^4)*x^2 - (15*b^5*c^4 + 70*a*b^
4*c^3*d - 1324*a^2*b^3*c^2*d^2 + 2394*a^3*b^2*c*d^3 - 1155*a^4*b*d^4)*x)*sqrt(b*
d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(a*b^5*c^5 + 5*a^2*b^4*c^4*d + 50*a^3*b^3*c^
3*d^2 - 350*a^4*b^2*c^2*d^3 + 525*a^5*b*c*d^4 - 231*a^6*d^5 + (b^6*c^5 + 5*a*b^5
*c^4*d + 50*a^2*b^4*c^3*d^2 - 350*a^3*b^3*c^2*d^3 + 525*a^4*b^2*c*d^4 - 231*a^5*
b*d^5)*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)))/((b^7*d^2*x + a*b^6*d^2)*sqrt(b*d)), 1/1280*(2*(128*b^5*d^4*x^5 - 15*a*b^
4*c^4 - 80*a^2*b^3*c^3*d + 3318*a^3*b^2*c^2*d^2 - 6720*a^4*b*c*d^3 + 3465*a^5*d^
4 + 16*(21*b^5*c*d^3 - 11*a*b^4*d^4)*x^4 + 8*(31*b^5*c^2*d^2 - 64*a*b^4*c*d^3 +
33*a^2*b^3*d^4)*x^3 + 2*(5*b^5*c^3*d - 233*a*b^4*c^2*d^2 + 459*a^2*b^3*c*d^3 - 2
31*a^3*b^2*d^4)*x^2 - (15*b^5*c^4 + 70*a*b^4*c^3*d - 1324*a^2*b^3*c^2*d^2 + 2394
*a^3*b^2*c*d^3 - 1155*a^4*b*d^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*
(a*b^5*c^5 + 5*a^2*b^4*c^4*d + 50*a^3*b^3*c^3*d^2 - 350*a^4*b^2*c^2*d^3 + 525*a^
5*b*c*d^4 - 231*a^6*d^5 + (b^6*c^5 + 5*a*b^5*c^4*d + 50*a^2*b^4*c^3*d^2 - 350*a^
3*b^3*c^2*d^3 + 525*a^4*b^2*c*d^4 - 231*a^5*b*d^5)*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)))/((b^7*d^2*x + a*b^6*d^2)*s
qrt(-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**3*(d*x+c)**(5/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.685256, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x