\(\int \frac {(A+B x) (a+b x+c x^2)^{3/2}}{d-f x^2} \, dx\) [82]

Optimal result

Integrand size = 30, antiderivative size = 448 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=-\frac {\left (10 A b c f+B \left (8 c^2 d+b^2 f+8 a c f\right )\right ) \sqrt {a+b x+c x^2}}{8 c f^2}-\frac {(b B+2 A c) x \sqrt {a+b x+c x^2}}{4 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}+\frac {\left (b^3 B f-6 A b^2 c f-12 b B c (2 c d+a f)-8 A c^2 (2 c d+3 a f)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} f^2}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} f^{5/2}}+\frac {\left (B \sqrt {d}+A \sqrt {f}\right ) \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} f^{5/2}} \] Output:

-1/8*(10*A*b*c*f+B*(8*a*c*f+b^2*f+8*c^2*d))*(c*x^2+b*x+a)^(1/2)/c/f^2-1/4* 
(2*A*c+B*b)*x*(c*x^2+b*x+a)^(1/2)/f-1/3*B*(c*x^2+b*x+a)^(3/2)/f+1/16*(b^3* 
B*f-6*A*b^2*c*f-12*b*B*c*(a*f+2*c*d)-8*A*c^2*(3*a*f+2*c*d))*arctanh(1/2*(2 
*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/f^2-1/2*(B*d^(1/2)-A*f^(1/2)) 
*(c*d-b*d^(1/2)*f^(1/2)+a*f)^(3/2)*arctanh(1/2*(b*d^(1/2)-2*a*f^(1/2)+(2*c 
*d^(1/2)-b*f^(1/2))*x)/(c*d-b*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x+a)^(1/ 
2))/d^(1/2)/f^(5/2)+1/2*(B*d^(1/2)+A*f^(1/2))*(c*d+b*d^(1/2)*f^(1/2)+a*f)^ 
(3/2)*arctanh(1/2*(b*d^(1/2)+2*a*f^(1/2)+(2*c*d^(1/2)+b*f^(1/2))*x)/(c*d+b 
*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x+a)^(1/2))/d^(1/2)/f^(5/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 4.09 (sec) , antiderivative size = 1292, normalized size of antiderivative = 2.88 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx =\text {Too large to display} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]
 

Output:

-1/24*((Sqrt[a + x*(b + c*x)]*(6*A*c*f*(5*b + 2*c*x) + B*(3*b^2*f + 2*c*f* 
(16*a + 7*b*x) + 8*c^2*(3*d + f*x^2))))/c + (3*(-(b^3*B*f) + 6*A*b^2*c*f + 
 12*b*B*c*(2*c*d + a*f) + 8*A*c^2*(2*c*d + 3*a*f))*ArcTanh[(Sqrt[c]*x)/(-S 
qrt[a] + Sqrt[a + x*(b + c*x)])])/c^(3/2) + 12*RootSum[c^2*d - b^2*f + 4*S 
qrt[a]*b*f*#1 - 2*c*d*#1^2 - 4*a*f*#1^2 + d*#1^4 & , (b*B*c^2*d^2*Log[x] + 
 A*c^3*d^2*Log[x] - b^3*B*d*f*Log[x] - A*b^2*c*d*f*Log[x] + 2*a*A*c^2*d*f* 
Log[x] - 2*a*A*b^2*f^2*Log[x] - a^2*b*B*f^2*Log[x] + a^2*A*c*f^2*Log[x] - 
b*B*c^2*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - A*c^3*d^2*Log[- 
Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + b^3*B*d*f*Log[-Sqrt[a] + Sqrt[a 
+ b*x + c*x^2] - x*#1] + A*b^2*c*d*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] 
- x*#1] - 2*a*A*c^2*d*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + 2*a 
*A*b^2*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + a^2*b*B*f^2*Log[ 
-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - a^2*A*c*f^2*Log[-Sqrt[a] + Sqrt 
[a + b*x + c*x^2] - x*#1] + 2*Sqrt[a]*B*c^2*d^2*Log[x]*#1 + 2*Sqrt[a]*b^2* 
B*d*f*Log[x]*#1 + 4*Sqrt[a]*A*b*c*d*f*Log[x]*#1 + 4*a^(3/2)*B*c*d*f*Log[x] 
*#1 + 4*a^(3/2)*A*b*f^2*Log[x]*#1 + 2*a^(5/2)*B*f^2*Log[x]*#1 - 2*Sqrt[a]* 
B*c^2*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 - 2*Sqrt[a]*b^2* 
B*d*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 - 4*Sqrt[a]*A*b*c*d* 
f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 - 4*a^(3/2)*B*c*d*f*Log[ 
-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 - 4*a^(3/2)*A*b*f^2*Log[-Sq...
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {1354, 27, 2140, 27, 2144, 27, 1092, 219, 1366, 25, 27, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx\)

\(\Big \downarrow \) 1354

\(\displaystyle \frac {\int \frac {3 \sqrt {c x^2+b x+a} \left ((b B+2 A c) f x^2+2 (B c d+A b f+a B f) x+b B d+2 a A f\right )}{2 \left (d-f x^2\right )}dx}{3 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a} \left ((b B+2 A c) f x^2+2 (B c d+A b f+a B f) x+b B d+2 a A f\right )}{d-f x^2}dx}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 2140

\(\displaystyle \frac {-\frac {\int -\frac {-f^2 \left (B f b^3-6 A c f b^2-12 B c (2 c d+a f) b-8 A c^2 (2 c d+3 a f)\right ) x^2+16 c f \left (2 A b f (c d+a f)+B \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )\right )\right ) x+f \left (B d f b^3+10 A c d f b^2+4 B c d (2 c d+5 a f) b+8 a A c f (c d+2 a f)\right )}{4 \sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{2 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-f^2 \left (B f b^3-6 A c f b^2-12 B c (2 c d+a f) b-8 A c^2 (2 c d+3 a f)\right ) x^2+16 c f \left (2 A b f (c d+a f)+B \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )\right )\right ) x+f \left (B d f b^3+10 A c d f b^2+4 B c d (2 c d+5 a f) b+8 a A c f (c d+2 a f)\right )}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 2144

\(\displaystyle \frac {\frac {f \left (-8 A c^2 (3 a f+2 c d)-12 b B c (a f+2 c d)-6 A b^2 c f+b^3 B f\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx-\frac {\int -\frac {16 c f^2 \left (A d f b^2+2 B d (c d+a f) b+A (c d+a f)^2+\left (2 A b f (c d+a f)+B \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )\right )\right ) x\right )}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {16 c f \int \frac {A d f b^2+2 B d (c d+a f) b+A (c d+a f)^2+\left (2 A b f (c d+a f)+B \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx+f \left (-8 A c^2 (3 a f+2 c d)-12 b B c (a f+2 c d)-6 A b^2 c f+b^3 B f\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {16 c f \int \frac {A d f b^2+2 B d (c d+a f) b+A (c d+a f)^2+\left (2 A b f (c d+a f)+B \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx+2 f \left (-8 A c^2 (3 a f+2 c d)-12 b B c (a f+2 c d)-6 A b^2 c f+b^3 B f\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {16 c f \int \frac {A d f b^2+2 B d (c d+a f) b+A (c d+a f)^2+\left (2 A b f (c d+a f)+B \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )\right )\right ) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx+\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 A c^2 (3 a f+2 c d)-12 b B c (a f+2 c d)-6 A b^2 c f+b^3 B f\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 1366

\(\displaystyle \frac {\frac {16 c f \left (\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int -\frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}+\frac {\left (A \sqrt {f}+B \sqrt {d}\right ) \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )+\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 A c^2 (3 a f+2 c d)-12 b B c (a f+2 c d)-6 A b^2 c f+b^3 B f\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {16 c f \left (\frac {\left (A \sqrt {f}+B \sqrt {d}\right ) \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )+\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 A c^2 (3 a f+2 c d)-12 b B c (a f+2 c d)-6 A b^2 c f+b^3 B f\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {16 c f \left (\frac {\left (A \sqrt {f}+B \sqrt {d}\right ) \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d} \sqrt {f}}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d} \sqrt {f}}\right )+\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 A c^2 (3 a f+2 c d)-12 b B c (a f+2 c d)-6 A b^2 c f+b^3 B f\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {16 c f \left (\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d} \sqrt {f}}-\frac {\left (A \sqrt {f}+B \sqrt {d}\right ) \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d} \sqrt {f}}\right )+\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 A c^2 (3 a f+2 c d)-12 b B c (a f+2 c d)-6 A b^2 c f+b^3 B f\right )}{\sqrt {c}}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 A c^2 (3 a f+2 c d)-12 b B c (a f+2 c d)-6 A b^2 c f+b^3 B f\right )}{\sqrt {c}}+16 c f \left (\frac {\left (A \sqrt {f}+B \sqrt {d}\right ) \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {d} \sqrt {f}}-\frac {\left (B \sqrt {d}-A \sqrt {f}\right ) \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {d} \sqrt {f}}\right )}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (B \left (8 a c f+b^2 f+8 c^2 d\right )+2 c f x (2 A c+b B)+10 A b c f\right )}{4 c f}}{2 f}-\frac {B \left (a+b x+c x^2\right )^{3/2}}{3 f}\)

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x]
 

Output:

-1/3*(B*(a + b*x + c*x^2)^(3/2))/f + (-1/4*((10*A*b*c*f + B*(8*c^2*d + b^2 
*f + 8*a*c*f) + 2*c*(b*B + 2*A*c)*f*x)*Sqrt[a + b*x + c*x^2])/(c*f) + ((f* 
(b^3*B*f - 6*A*b^2*c*f - 12*b*B*c*(2*c*d + a*f) - 8*A*c^2*(2*c*d + 3*a*f)) 
*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/Sqrt[c] + 16*c*f* 
(-1/2*((B*Sqrt[d] - A*Sqrt[f])*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTa 
nh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b 
*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(Sqrt[d]*Sqrt[f]) + ((B*S 
qrt[d] + A*Sqrt[f])*(c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[ 
d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sq 
rt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]*Sqrt[f])))/(8*c*f^2))/(2* 
f)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1354

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f 
_.)*(x_)^2)^(q_), x_Symbol] :> Simp[h*(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 
 1)/(2*f*(p + q + 1))), x] - Simp[1/(2*f*(p + q + 1))   Int[(a + b*x + c*x^ 
2)^(p - 1)*(d + f*x^2)^q*Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*( 
c*d - a*f) + b*(-2*g*f)*(p + q + 1))*x + (h*p*((-b)*f) + c*(-2*g*f)*(p + q 
+ 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4* 
a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
 

rule 1366

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( 
f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[(h/2 + c*(g/(2*q 
)))   Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[(h/2 - c*(g/( 
2*q)))   Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d 
, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[(-a)*c]
 

rule 2140

Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_ 
), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P 
x, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p) + 2*c*C*f*(p + q + 1)*x) 
*(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 
 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))   Int[(a + b*x + c 
*x^2)^(p - 1)*(d + f*x^2)^q*Simp[p*(b*d)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*( 
2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f) + f*(-2*A*f)*(2 
*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*(2*p + 
2*q + 3)) + (p + q + 1)*((-b)*c*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A* 
f)*(2*p + 2*q + 3))))*x + (p*((-b)*f)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*(2*p 
 + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(-4*d*f)*(2*p + 
q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, 
 b, c, d, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && 
 NeQ[2*p + 2*q + 3, 0] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]
 

rule 2144

Int[(Px_)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), 
x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, 
x, 2]}, Simp[C/c   Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[1/c   Int[(A* 
c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, 
c, d, e, f}, x] && PolyQ[Px, x, 2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(909\) vs. \(2(358)=716\).

Time = 2.59 (sec) , antiderivative size = 910, normalized size of antiderivative = 2.03

method	result	size
risch	\(-\frac {\left (8 B \,c^{2} f \,x^{2}+12 A \,c^{2} f x +14 B b c f x +30 A b c f +32 B a c f +3 B \,b^{2} f +24 B \,c^{2} d \right ) \sqrt {c \,x^{2}+b x +a}}{24 c \,f^{2}}-\frac {16 A \,c^{\frac {5}{2}} d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {b^{3} B f \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+24 A a \,c^{\frac {3}{2}} f \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+6 A \,b^{2} \sqrt {c}\, f \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )+24 B b \,c^{\frac {3}{2}} d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {8 c \left (2 A \sqrt {d f}\, a b \,f^{2}+2 A \sqrt {d f}\, b c d f -A \,a^{2} f^{3}-2 A a c d \,f^{2}-A \,b^{2} d \,f^{2}-A \,c^{2} d^{2} f +B \sqrt {d f}\, a^{2} f^{2}+2 B \sqrt {d f}\, a c d f +B \sqrt {d f}\, b^{2} d f +B \sqrt {d f}\, c^{2} d^{2}-2 B a b d \,f^{2}-2 B b c \,d^{2} f \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}-\frac {8 c \left (2 A \sqrt {d f}\, a b \,f^{2}+2 A \sqrt {d f}\, b c d f +A \,a^{2} f^{3}+2 A a c d \,f^{2}+A \,b^{2} d \,f^{2}+A \,c^{2} d^{2} f +B \sqrt {d f}\, a^{2} f^{2}+2 B \sqrt {d f}\, a c d f +B \sqrt {d f}\, b^{2} d f +B \sqrt {d f}\, c^{2} d^{2}+2 B a b d \,f^{2}+2 B b c \,d^{2} f \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}+12 B a b \sqrt {c}\, f \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c \,f^{2}}\)	\(910\)
default	\(\text {Expression too large to display}\)	\(1506\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

-1/24/c*(8*B*c^2*f*x^2+12*A*c^2*f*x+14*B*b*c*f*x+30*A*b*c*f+32*B*a*c*f+3*B 
*b^2*f+24*B*c^2*d)*(c*x^2+b*x+a)^(1/2)/f^2-1/16/c/f^2*(16*A*c^(5/2)*d*ln(( 
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-b^3*B*f*ln((1/2*b+c*x)/c^(1/2)+(c* 
x^2+b*x+a)^(1/2))/c^(1/2)+24*A*a*c^(3/2)*f*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b 
*x+a)^(1/2))+6*A*b^2*c^(1/2)*f*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)) 
+24*B*b*c^(3/2)*d*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-8*c*(2*A*(d* 
f)^(1/2)*a*b*f^2+2*A*(d*f)^(1/2)*b*c*d*f-A*a^2*f^3-2*A*a*c*d*f^2-A*b^2*d*f 
^2-A*c^2*d^2*f+B*(d*f)^(1/2)*a^2*f^2+2*B*(d*f)^(1/2)*a*c*d*f+B*(d*f)^(1/2) 
*b^2*d*f+B*(d*f)^(1/2)*c^2*d^2-2*B*a*b*d*f^2-2*B*b*c*d^2*f)/(d*f)^(1/2)/f/ 
(1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+a*f+c*d)+1/f* 
(-2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d*f)^(1/2)+a*f+c*d))^ 
(1/2)*(c*(x+(d*f)^(1/2)/f)^2+1/f*(-2*c*(d*f)^(1/2)+f*b)*(x+(d*f)^(1/2)/f)+ 
1/f*(-b*(d*f)^(1/2)+a*f+c*d))^(1/2))/(x+(d*f)^(1/2)/f))-8*c*(2*A*(d*f)^(1/ 
2)*a*b*f^2+2*A*(d*f)^(1/2)*b*c*d*f+A*a^2*f^3+2*A*a*c*d*f^2+A*b^2*d*f^2+A*c 
^2*d^2*f+B*(d*f)^(1/2)*a^2*f^2+2*B*(d*f)^(1/2)*a*c*d*f+B*(d*f)^(1/2)*b^2*d 
*f+B*(d*f)^(1/2)*c^2*d^2+2*B*a*b*d*f^2+2*B*b*c*d^2*f)/(d*f)^(1/2)/f/((b*(d 
*f)^(1/2)+a*f+c*d)/f)^(1/2)*ln((2*(b*(d*f)^(1/2)+a*f+c*d)/f+(2*c*(d*f)^(1/ 
2)+f*b)/f*(x-(d*f)^(1/2)/f)+2*((b*(d*f)^(1/2)+a*f+c*d)/f)^(1/2)*(c*(x-(d*f 
)^(1/2)/f)^2+(2*c*(d*f)^(1/2)+f*b)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+a*f+ 
c*d)/f)^(1/2))/(x-(d*f)^(1/2)/f))+12*B*a*b*c^(1/2)*f*ln((1/2*b+c*x)/c^(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=- \int \frac {A a \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {A b x \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {A c x^{2} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {B a x \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {B b x^{2} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {B c x^{3} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**(3/2)/(-f*x**2+d),x)
 

Output:

-Integral(A*a*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(A*b*x*sq 
rt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(A*c*x**2*sqrt(a + b*x + 
c*x**2)/(-d + f*x**2), x) - Integral(B*a*x*sqrt(a + b*x + c*x**2)/(-d + f* 
x**2), x) - Integral(B*b*x**2*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x) - I 
ntegral(B*c*x**3*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(((c*sqrt(4*d*f))/(2*f^2)>0)', se 
e `assume?
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d-f\,x^2} \,d x \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/(d - f*x^2),x)
 

Output:

int(((A + B*x)*(a + b*x + c*x^2)^(3/2))/(d - f*x^2), x)
 

Reduce [F]

\[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\int \frac {\left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{-f \,x^{2}+d}d x \] Input:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)
 

Output:

int((B*x+A)*(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)