3.1283 \(\int \frac{1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=159 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}+\frac{4}{d^3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}+\frac{4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]

[Out]

4/(5*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)) + 4/((b^2 - 4*a*c)^2*d^3*Sqrt[b*d +
2*c*d*x]) + (2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2
- 4*a*c)^(9/4)*d^(7/2)) - (2*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sq
rt[d])])/((b^2 - 4*a*c)^(9/4)*d^(7/2))

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Rubi [A]  time = 0.368067, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{9/4}}+\frac{4}{d^3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}+\frac{4}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)),x]

[Out]

4/(5*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)) + 4/((b^2 - 4*a*c)^2*d^3*Sqrt[b*d +
2*c*d*x]) + (2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2
- 4*a*c)^(9/4)*d^(7/2)) - (2*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sq
rt[d])])/((b^2 - 4*a*c)^(9/4)*d^(7/2))

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Rubi in Sympy [A]  time = 84.8592, size = 156, normalized size = 0.98 \[ \frac{4}{5 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}}} + \frac{4}{d^{3} \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x}} + \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a),x)

[Out]

4/(5*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(5/2)) + 4/(d**3*(-4*a*c + b**2)**2*sqrt
(b*d + 2*c*d*x)) + 2*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/
(d**(7/2)*(-4*a*c + b**2)**(9/4)) - 2*atanh(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c
+ b**2)**(1/4)))/(d**(7/2)*(-4*a*c + b**2)**(9/4))

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Mathematica [A]  time = 0.307427, size = 142, normalized size = 0.89 \[ \frac{2 \left (4 \sqrt [4]{b^2-4 a c} \left (2 c \left (5 c x^2-a\right )+3 b^2+10 b c x\right )+5 (b+2 c x)^{5/2} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-5 (b+2 c x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{5 d \left (b^2-4 a c\right )^{9/4} (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)),x]

[Out]

(2*(4*(b^2 - 4*a*c)^(1/4)*(3*b^2 + 10*b*c*x + 2*c*(-a + 5*c*x^2)) + 5*(b + 2*c*x
)^(5/2)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] - 5*(b + 2*c*x)^(5/2)*ArcTan
h[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/(5*(b^2 - 4*a*c)^(9/4)*d*(d*(b + 2*c*x)
)^(5/2))

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Maple [B]  time = 0.015, size = 369, normalized size = 2.3 \[ -{\frac{4}{5\,d \left ( 4\,ac-{b}^{2} \right ) } \left ( 2\,cdx+bd \right ) ^{-{\frac{5}{2}}}}+4\,{\frac{1}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}\sqrt{2\,cdx+bd}}}+{\frac{\sqrt{2}}{2\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+{\frac{\sqrt{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-{\frac{\sqrt{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a),x)

[Out]

-4/5/d/(4*a*c-b^2)/(2*c*d*x+b*d)^(5/2)+4/d^3/(4*a*c-b^2)^2/(2*c*d*x+b*d)^(1/2)+1
/2/d^3/(4*a*c-b^2)^2/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*ln((2*c*d*x+b*d-(4*a*c*d^
2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2))/(2*c*d*x
+b*d+(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(
1/2)))+1/d^3/(4*a*c-b^2)^2/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(4*a
*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)-1/d^3/(4*a*c-b^2)^2/(4*a*c*d^2-b^2*
d^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)
+1)

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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(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252522, size = 2037, normalized size = 12.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)),x, algorithm="fricas")

[Out]

1/5*(80*c^2*x^2 + 80*b*c*x + 20*(4*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3*x^2
+ 4*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3*x + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c
^2)*d^3)*sqrt(2*c*d*x + b*d)*(1/((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a
^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589
824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^14))^(1/4)*arctan(-(b^1
4 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504
*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^11*(1/((b^18 - 36*a*b^16*c +
576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5
+ 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)
*d^14))^(3/4)/(sqrt((b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 128
0*a^4*b^2*c^4 - 1024*a^5*c^5)*d^8*sqrt(1/((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2
- 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*
c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^14)) + 2*c*d*x
+ b*d) + sqrt(2*c*d*x + b*d))) - 5*(4*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3*
x^2 + 4*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3*x + (b^6 - 8*a*b^4*c + 16*a^2*b
^2*c^2)*d^3)*sqrt(2*c*d*x + b*d)*(1/((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 53
76*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 -
589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^14))^(1/4)*log((b^1
4 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504
*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^11*(1/((b^18 - 36*a*b^16*c +
576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5
+ 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)
*d^14))^(3/4) + sqrt(2*c*d*x + b*d)) + 5*(4*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)
*d^3*x^2 + 4*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3*x + (b^6 - 8*a*b^4*c + 16*
a^2*b^2*c^2)*d^3)*sqrt(2*c*d*x + b*d)*(1/((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2
- 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*
c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^14))^(1/4)*log
(-(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 -
21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^11*(1/((b^18 - 36*a*b^
16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^
8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^
9*c^9)*d^14))^(3/4) + sqrt(2*c*d*x + b*d)) + 24*b^2 - 16*a*c)/((4*(b^4*c^2 - 8*a
*b^2*c^3 + 16*a^2*c^4)*d^3*x^2 + 4*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3*x +
(b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*d^3)*sqrt(2*c*d*x + 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(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234334, size = 817, normalized size = 5.14 \[ -\frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}} - \frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{6} d^{5} - 12 \, a b^{4} c d^{5} + 48 \, a^{2} b^{2} c^{2} d^{5} - 64 \, a^{3} c^{3} d^{5}} + \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{6} d^{5} - 12 \, \sqrt{2} a b^{4} c d^{5} + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d^{5} - 64 \, \sqrt{2} a^{3} c^{3} d^{5}} - \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{6} d^{5} - 12 \, \sqrt{2} a b^{4} c d^{5} + 48 \, \sqrt{2} a^{2} b^{2} c^{2} d^{5} - 64 \, \sqrt{2} a^{3} c^{3} d^{5}} + \frac{4 \,{\left (b^{2} d^{2} - 4 \, a c d^{2} + 5 \,{\left (2 \, c d x + b d\right )}^{2}\right )}}{5 \,{\left (b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}\right )}{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)),x, algorithm="giac")

[Out]

-sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*
a*c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))/(b^6*d^5 -
12*a*b^4*c*d^5 + 48*a^2*b^2*c^2*d^5 - 64*a^3*c^3*d^5) - sqrt(2)*(-b^2*d^2 + 4*a
*c*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt
(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))/(b^6*d^5 - 12*a*b^4*c*d^5 + 48*a^
2*b^2*c^2*d^5 - 64*a^3*c^3*d^5) + (-b^2*d^2 + 4*a*c*d^2)^(3/4)*ln(2*c*d*x + b*d
+ sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a
*c*d^2))/(sqrt(2)*b^6*d^5 - 12*sqrt(2)*a*b^4*c*d^5 + 48*sqrt(2)*a^2*b^2*c^2*d^5
- 64*sqrt(2)*a^3*c^3*d^5) - (-b^2*d^2 + 4*a*c*d^2)^(3/4)*ln(2*c*d*x + b*d - sqrt
(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2
))/(sqrt(2)*b^6*d^5 - 12*sqrt(2)*a*b^4*c*d^5 + 48*sqrt(2)*a^2*b^2*c^2*d^5 - 64*s
qrt(2)*a^3*c^3*d^5) + 4/5*(b^2*d^2 - 4*a*c*d^2 + 5*(2*c*d*x + b*d)^2)/((b^4*d^3
- 8*a*b^2*c*d^3 + 16*a^2*c^2*d^3)*(2*c*d*x + b*d)^(5/2))