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

Optimal. Leaf size=262 \[ -\frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{7/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}{512 b^3 d^3}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^4}{768 b^3 d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^3}{192 b^3 d}+\frac{(a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^2}{32 b^3}+\frac{(a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b} \]

[Out]

(5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^3*d^3) - (5*(b*c - a*d)^4*(
a + b*x)^(3/2)*Sqrt[c + d*x])/(768*b^3*d^2) + ((b*c - a*d)^3*(a + b*x)^(5/2)*Sqr
t[c + d*x])/(192*b^3*d) + ((b*c - a*d)^2*(a + b*x)^(7/2)*Sqrt[c + d*x])/(32*b^3)
+ ((b*c - a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(12*b^2) + ((a + b*x)^(7/2)*(c
+ d*x)^(5/2))/(6*b) - (5*(b*c - a*d)^6*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(512*b^(7/2)*d^(7/2))

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Rubi [A]  time = 0.392524, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{512 b^{7/2} d^{7/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}{512 b^3 d^3}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^4}{768 b^3 d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d)^3}{192 b^3 d}+\frac{(a+b x)^{7/2} \sqrt{c+d x} (b c-a d)^2}{32 b^3}+\frac{(a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)}{12 b^2}+\frac{(a+b x)^{7/2} (c+d x)^{5/2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

(5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*b^3*d^3) - (5*(b*c - a*d)^4*(
a + b*x)^(3/2)*Sqrt[c + d*x])/(768*b^3*d^2) + ((b*c - a*d)^3*(a + b*x)^(5/2)*Sqr
t[c + d*x])/(192*b^3*d) + ((b*c - a*d)^2*(a + b*x)^(7/2)*Sqrt[c + d*x])/(32*b^3)
+ ((b*c - a*d)*(a + b*x)^(7/2)*(c + d*x)^(3/2))/(12*b^2) + ((a + b*x)^(7/2)*(c
+ d*x)^(5/2))/(6*b) - (5*(b*c - a*d)^6*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(512*b^(7/2)*d^(7/2))

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Rubi in Sympy [A]  time = 59.6199, size = 233, normalized size = 0.89 \[ \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{7}{2}}}{6 d} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{7}{2}} \left (a d - b c\right )}{12 d^{2}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{7}{2}} \left (a d - b c\right )^{2}}{32 d^{3}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )^{3}}{192 b d^{3}} - \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{4}}{768 b^{2} d^{3}} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{5}}{512 b^{3} d^{3}} - \frac{5 \left (a d - b c\right )^{6} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{512 b^{\frac{7}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(5/2)*(c + d*x)**(7/2)/(6*d) + (a + b*x)**(3/2)*(c + d*x)**(7/2)*(a*d
- b*c)/(12*d**2) + sqrt(a + b*x)*(c + d*x)**(7/2)*(a*d - b*c)**2/(32*d**3) + sq
rt(a + b*x)*(c + d*x)**(5/2)*(a*d - b*c)**3/(192*b*d**3) - 5*sqrt(a + b*x)*(c +
d*x)**(3/2)*(a*d - b*c)**4/(768*b**2*d**3) + 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d
- b*c)**5/(512*b**3*d**3) - 5*(a*d - b*c)**6*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b
)*sqrt(c + d*x)))/(512*b**(7/2)*d**(7/2))

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Mathematica [A]  time = 0.356383, size = 300, normalized size = 1.15 \[ \frac{2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x} \left (15 a^5 d^5-5 a^4 b d^4 (17 c+2 d x)+2 a^3 b^2 d^3 \left (99 c^2+28 c d x+4 d^2 x^2\right )+6 a^2 b^3 d^2 \left (33 c^3+198 c^2 d x+212 c d^2 x^2+72 d^3 x^3\right )+a b^4 d \left (-85 c^4+56 c^3 d x+1272 c^2 d^2 x^2+1696 c d^3 x^3+640 d^4 x^4\right )+b^5 \left (15 c^5-10 c^4 d x+8 c^3 d^2 x^2+432 c^2 d^3 x^3+640 c d^4 x^4+256 d^5 x^5\right )\right )-15 (b c-a d)^6 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{3072 b^{7/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^5*d^5 - 5*a^4*b*d^4*(17*c +
2*d*x) + 2*a^3*b^2*d^3*(99*c^2 + 28*c*d*x + 4*d^2*x^2) + 6*a^2*b^3*d^2*(33*c^3
+ 198*c^2*d*x + 212*c*d^2*x^2 + 72*d^3*x^3) + a*b^4*d*(-85*c^4 + 56*c^3*d*x + 12
72*c^2*d^2*x^2 + 1696*c*d^3*x^3 + 640*d^4*x^4) + b^5*(15*c^5 - 10*c^4*d*x + 8*c^
3*d^2*x^2 + 432*c^2*d^3*x^3 + 640*c*d^4*x^4 + 256*d^5*x^5)) - 15*(b*c - a*d)^6*L
og[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(3072*b
^(7/2)*d^(7/2))

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Maple [B]  time = 0.007, size = 1089, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/1024/d^3*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*
b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*c^6*b^3+1/6/
d*(b*x+a)^(5/2)*(d*x+c)^(7/2)+1/32/d*(b*x+a)^(1/2)*(d*x+c)^(7/2)*a^2+25/512/d^2*
(d*x+c)^(1/2)*(b*x+a)^(1/2)*a*c^4*b+5/192/d^2*(d*x+c)^(3/2)*(b*x+a)^(1/2)*c^3*b*
a+1/64/d^2*(d*x+c)^(5/2)*(b*x+a)^(1/2)*c^2*b*a-1/16/d^2*(b*x+a)^(1/2)*(d*x+c)^(7
/2)*a*b*c-25/512*d/b^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^4*c+1/12/d*(b*x+a)^(3/2)*(d
*x+c)^(7/2)*a+1/192/b*(d*x+c)^(5/2)*(b*x+a)^(1/2)*a^3-5/768*d/b^2*(d*x+c)^(3/2)*
(b*x+a)^(1/2)*a^4-1/192/d^3*(d*x+c)^(5/2)*(b*x+a)^(1/2)*c^3*b^2+25/256*((b*x+a)*
(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2
)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^3*c^3-25/256/d*(d*x+c)^(1/2)*(b
*x+a)^(1/2)*a^2*c^3-5/512/d^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*c^5*b^2-5/768/d^3*(d*x
+c)^(3/2)*(b*x+a)^(1/2)*c^4*b^2+25/256/b*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^3*c^2+5/1
92/b*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^3*c-5/128/d*(d*x+c)^(3/2)*(b*x+a)^(1/2)*a^2*c
^2+1/32/d^3*(b*x+a)^(1/2)*(d*x+c)^(7/2)*b^2*c^2-1/12/d^2*(b*x+a)^(3/2)*(d*x+c)^(
7/2)*b*c+5/512*d^2/b^3*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a^5-1/64/d*(d*x+c)^(5/2)*(b*x
+a)^(1/2)*a^2*c-75/1024/d*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln
((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2
)*a^2*c^4*b+15/512/d^2*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1
/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a
*c^5*b^2+15/512*d^2/b^2*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((
1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*
a^5*c-75/1024*d/b*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*
d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^4*c^
2-5/1024*d^3/b^3*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d
+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*a^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)^(5/2)*(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283222, 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)^(5/2)*(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

[1/6144*(4*(256*b^5*d^5*x^5 + 15*b^5*c^5 - 85*a*b^4*c^4*d + 198*a^2*b^3*c^3*d^2
+ 198*a^3*b^2*c^2*d^3 - 85*a^4*b*c*d^4 + 15*a^5*d^5 + 640*(b^5*c*d^4 + a*b^4*d^5
)*x^4 + 16*(27*b^5*c^2*d^3 + 106*a*b^4*c*d^4 + 27*a^2*b^3*d^5)*x^3 + 8*(b^5*c^3*
d^2 + 159*a*b^4*c^2*d^3 + 159*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 - 2*(5*b^5*c^4*d
- 28*a*b^4*c^3*d^2 - 594*a^2*b^3*c^2*d^3 - 28*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x)*sq
rt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*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 - 6*a^5*b*c*d^5 + a^6*d^6)*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)))/(sqrt(b
*d)*b^3*d^3), 1/3072*(2*(256*b^5*d^5*x^5 + 15*b^5*c^5 - 85*a*b^4*c^4*d + 198*a^2
*b^3*c^3*d^2 + 198*a^3*b^2*c^2*d^3 - 85*a^4*b*c*d^4 + 15*a^5*d^5 + 640*(b^5*c*d^
4 + a*b^4*d^5)*x^4 + 16*(27*b^5*c^2*d^3 + 106*a*b^4*c*d^4 + 27*a^2*b^3*d^5)*x^3
+ 8*(b^5*c^3*d^2 + 159*a*b^4*c^2*d^3 + 159*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 - 2*
(5*b^5*c^4*d - 28*a*b^4*c^3*d^2 - 594*a^2*b^3*c^2*d^3 - 28*a^3*b^2*c*d^4 + 5*a^4
*b*d^5)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(b^6*c^6 - 6*a*b^5*c^5*d
+ 15*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 - 6*a^5*b*c*d^5 +
a^6*d^6)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x +
c)*b*d)))/(sqrt(-b*d)*b^3*d^3)]

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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)**(5/2)*(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.436619, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done