3.26.35 \(\int \frac {(-g+f x^2) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx\)

Optimal. Leaf size=212 \[ -\frac {a (d g+e f) \text {RootSum}\left [\text {$\#$1}^8 d-4 \text {$\#$1}^6 c d-2 \text {$\#$1}^4 b d+6 \text {$\#$1}^4 c^2 d+4 \text {$\#$1}^2 b c d-4 \text {$\#$1}^2 c^3 d+a^2 e+b^2 d-2 b c^2 d+c^4 d\& ,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2 c-b+c^2}\& \right ]}{2 d^2}+\frac {4 f \left (3 a x+3 b-2 c^2\right ) \sqrt {\sqrt {a x+b}+c}}{15 a d}+\frac {4 c f \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}}{15 a d} \]

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Rubi [F]  time = 3.58, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-g + f*x^2)*Sqrt[c + Sqrt[b + a*x]])/(e + d*x^2),x]

[Out]

(-4*c*f*(c + Sqrt[b + a*x])^(3/2))/(3*a*d) + (4*f*(c + Sqrt[b + a*x])^(5/2))/(5*a*d) + (4*a*(e*f + d*g)*Defer[
Subst][Defer[Int][x^4/(-(b^2*d*(1 + (-2*b*c^2*d + c^4*d + a^2*e)/(b^2*d))) - 4*b*c*(1 - c^2/b)*d*x^2 + 2*b*(1
- (3*c^2)/b)*d*x^4 + 4*c*d*x^6 - d*x^8), x], x, Sqrt[c + Sqrt[b + a*x]]])/d + (4*a*c*(e*f + d*g)*Defer[Subst][
Defer[Int][x^2/(b^2*d*(1 + (-2*b*c^2*d + c^4*d + a^2*e)/(b^2*d)) + 4*b*c*(1 - c^2/b)*d*x^2 - 2*b*(1 - (3*c^2)/
b)*d*x^4 - 4*c*d*x^6 + d*x^8), x], x, Sqrt[c + Sqrt[b + a*x]]])/d

Rubi steps

\begin {align*} \int \frac {\left (-g+f x^2\right ) \sqrt {c+\sqrt {b+a x}}}{e+d x^2} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x \sqrt {c+x} \left (-a^2 g+f \left (b-x^2\right )^2\right )}{e+\frac {d \left (b-x^2\right )^2}{a^2}} \, dx,x,\sqrt {b+a x}\right )}{a^3}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x^2 \left (-c+x^2\right ) \left (-a^2 g+f \left (b-\left (c-x^2\right )^2\right )^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (-\frac {a^2 c f x^2}{d}+\frac {a^2 f x^4}{d}+\frac {x^2 \left (a^2 c (e f+d g)-a^2 (e f+d g) x^2\right )}{d \left (e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}\right )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3}\\ &=-\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {4 \operatorname {Subst}\left (\int \frac {x^2 \left (a^2 c (e f+d g)-a^2 (e f+d g) x^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d}\\ &=-\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {4 \operatorname {Subst}\left (\int \frac {a^2 (e f+d g) x^2 \left (c-x^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d}\\ &=-\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {(4 (e f+d g)) \operatorname {Subst}\left (\int \frac {x^2 \left (c-x^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=-\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {(4 (e f+d g)) \operatorname {Subst}\left (\int \left (\frac {a^2 x^4}{-b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )-4 b c \left (1-\frac {c^2}{b}\right ) d x^2+2 b \left (1-\frac {3 c^2}{b}\right ) d x^4+4 c d x^6-d x^8}+\frac {a^2 c x^2}{b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )+4 b c \left (1-\frac {c^2}{b}\right ) d x^2-2 b \left (1-\frac {3 c^2}{b}\right ) d x^4-4 c d x^6+d x^8}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=-\frac {4 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}+\frac {(4 a (e f+d g)) \operatorname {Subst}\left (\int \frac {x^4}{-b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )-4 b c \left (1-\frac {c^2}{b}\right ) d x^2+2 b \left (1-\frac {3 c^2}{b}\right ) d x^4+4 c d x^6-d x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d}+\frac {(4 a c (e f+d g)) \operatorname {Subst}\left (\int \frac {x^2}{b^2 d \left (1+\frac {-2 b c^2 d+c^4 d+a^2 e}{b^2 d}\right )+4 b c \left (1-\frac {c^2}{b}\right ) d x^2-2 b \left (1-\frac {3 c^2}{b}\right ) d x^4-4 c d x^6+d x^8} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d}\\ \end {align*}

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Mathematica [A]  time = 1.41, size = 180, normalized size = 0.85 \begin {gather*} \frac {8 d f \left (\sqrt {a x+b}+c\right )^{3/2} \left (3 \left (\sqrt {a x+b}+c\right )-5 c\right )-15 a^2 (d g+e f) \text {RootSum}\left [\text {$\#$1}^8 d-4 \text {$\#$1}^6 c d-2 \text {$\#$1}^4 b d+6 \text {$\#$1}^4 c^2 d+4 \text {$\#$1}^2 b c d-4 \text {$\#$1}^2 c^3 d+a^2 e+b^2 d-2 b c^2 d+c^4 d\&,\frac {\text {$\#$1} \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{\text {$\#$1}^4-2 \text {$\#$1}^2 c-b+c^2}\&\right ]}{30 a d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-g + f*x^2)*Sqrt[c + Sqrt[b + a*x]])/(e + d*x^2),x]

[Out]

(8*d*f*(c + Sqrt[b + a*x])^(3/2)*(-5*c + 3*(c + Sqrt[b + a*x])) - 15*a^2*(e*f + d*g)*RootSum[b^2*d - 2*b*c^2*d
+ c^4*d + a^2*e + 4*b*c*d*#1^2 - 4*c^3*d*#1^2 - 2*b*d*#1^4 + 6*c^2*d*#1^4 - 4*c*d*#1^6 + d*#1^8 & , (Log[Sqrt
[c + Sqrt[b + a*x]] - #1]*#1)/(-b + c^2 - 2*c*#1^2 + #1^4) & ])/(30*a*d^2)

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IntegrateAlgebraic [A]  time = 0.34, size = 190, normalized size = 0.90 \begin {gather*} -\frac {4 \sqrt {c+\sqrt {b+a x}} \left (2 c^2 f-c f \sqrt {b+a x}-3 f (b+a x)\right )}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}}{-b+c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-g + f*x^2)*Sqrt[c + Sqrt[b + a*x]])/(e + d*x^2),x]

[Out]

(-4*Sqrt[c + Sqrt[b + a*x]]*(2*c^2*f - c*f*Sqrt[b + a*x] - 3*f*(b + a*x)))/(15*a*d) - (a*(e*f + d*g)*RootSum[b
^2*d - 2*b*c^2*d + c^4*d + a^2*e + 4*b*c*d*#1^2 - 4*c^3*d*#1^2 - 2*b*d*#1^4 + 6*c^2*d*#1^4 - 4*c*d*#1^6 + d*#1
^8 & , (Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1)/(-b + c^2 - 2*c*#1^2 + #1^4) & ])/(2*d^2)

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fricas [B]  time = 1.17, size = 5974, normalized size = 28.18

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x, algorithm="fricas")

[Out]

1/30*(15*a*d*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 + d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^
2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 + d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*
g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^
2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*log(d^8*e^2*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*
d^2*g^2 + d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 + d^6*e^
2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^
4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e
^2)))/(d^3*e))*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a
^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*
e^3)) + (a*e^5*f^5 + 5*a*d*e^4*f^4*g + 10*a*d^2*e^3*f^3*g^2 + 10*a*d^3*e^2*f^2*g^3 + 5*a*d^4*e*f*g^4 + a*d^5*g
^5)*sqrt(c + sqrt(a*x + b))) - 15*a*d*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 + d^3*e*sqrt((b*e^4*f^4 + 4*b
*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 + d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^
7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a
^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*log(-d^8*e^2*sqrt(-(c*
e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 + d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*
f*g^3 + b*d^4*g^4 + d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f
^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^
8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*
a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f
*g^7 + a^2*d^8*g^8)/(d^13*e^3)) + (a*e^5*f^5 + 5*a*d*e^4*f^4*g + 10*a*d^2*e^3*f^3*g^2 + 10*a*d^3*e^2*f^2*g^3 +
5*a*d^4*e*f*g^4 + a*d^5*g^5)*sqrt(c + sqrt(a*x + b))) + 15*a*d*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 - d
^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 + d^6*e^2*sqrt(-(a^
2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*
a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*
e))*log(d^8*e^2*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 - d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2
*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 + d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f
^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8
*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*
a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^
2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)) + (a*e^5*f^5 + 5*a*d*e^4*f^4*g + 10*a*d^2*e^3*f^3*g^2
+ 10*a*d^3*e^2*f^2*g^3 + 5*a*d^4*e*f*g^4 + a*d^5*g^5)*sqrt(c + sqrt(a*x + b))) - 15*a*d*sqrt(-(c*e^2*f^2 + 2*
c*d*e*f*g + c*d^2*g^2 - d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^
4*g^4 + d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*
a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13
*e^3)))/(d^6*e^2)))/(d^3*e))*log(-d^8*e^2*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 - d^3*e*sqrt((b*e^4*f^4 +
4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 + d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^
7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 +
28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*sqrt(-(a^2*e^8*f^8
+ 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e
^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)) + (a*e^5*f^5 + 5*a*d*e^4*f^
4*g + 10*a*d^2*e^3*f^3*g^2 + 10*a*d^3*e^2*f^2*g^3 + 5*a*d^4*e*f*g^4 + a*d^5*g^5)*sqrt(c + sqrt(a*x + b))) - 15
*a*d*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 + d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^
2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 - d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56
*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*
f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*log(d^8*e^2*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2
+ d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 - d^6*e^2*sqrt(-
(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 +
56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d
^3*e))*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e
^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)) +
(a*e^5*f^5 + 5*a*d*e^4*f^4*g + 10*a*d^2*e^3*f^3*g^2 + 10*a*d^3*e^2*f^2*g^3 + 5*a*d^4*e*f*g^4 + a*d^5*g^5)*sqrt
(c + sqrt(a*x + b))) + 15*a*d*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 + d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f
^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 - d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28
*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e
^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*log(-d^8*e^2*sqrt(-(c*e^2*f^2
+ 2*c*d*e*f*g + c*d^2*g^2 + d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 +
b*d^4*g^4 - d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 +
70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(
d^13*e^3)))/(d^6*e^2)))/(d^3*e))*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*
e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a
^2*d^8*g^8)/(d^13*e^3)) + (a*e^5*f^5 + 5*a*d*e^4*f^4*g + 10*a*d^2*e^3*f^3*g^2 + 10*a*d^3*e^2*f^2*g^3 + 5*a*d^4
*e*f*g^4 + a*d^5*g^5)*sqrt(c + sqrt(a*x + b))) - 15*a*d*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 - d^3*e*sqr
t((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 - d^6*e^2*sqrt(-(a^2*e^8*f^
8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*
e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*log(
d^8*e^2*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 - d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2
*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 - d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 +
56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7
*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*
e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^
6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)) + (a*e^5*f^5 + 5*a*d*e^4*f^4*g + 10*a*d^2*e^3*f^3*g^2 + 10*a*
d^3*e^2*f^2*g^3 + 5*a*d^4*e*f*g^4 + a*d^5*g^5)*sqrt(c + sqrt(a*x + b))) + 15*a*d*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*
g + c*d^2*g^2 - d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 -
d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*
e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/
(d^6*e^2)))/(d^3*e))*log(-d^8*e^2*sqrt(-(c*e^2*f^2 + 2*c*d*e*f*g + c*d^2*g^2 - d^3*e*sqrt((b*e^4*f^4 + 4*b*d*e
^3*f^3*g + 6*b*d^2*e^2*f^2*g^2 + 4*b*d^3*e*f*g^3 + b*d^4*g^4 - d^6*e^2*sqrt(-(a^2*e^8*f^8 + 8*a^2*d*e^7*f^7*g
+ 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g^5 + 28*a^2*d
^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)))/(d^6*e^2)))/(d^3*e))*sqrt(-(a^2*e^8*f^8 + 8*a^2
*d*e^7*f^7*g + 28*a^2*d^2*e^6*f^6*g^2 + 56*a^2*d^3*e^5*f^5*g^3 + 70*a^2*d^4*e^4*f^4*g^4 + 56*a^2*d^5*e^3*f^3*g
^5 + 28*a^2*d^6*e^2*f^2*g^6 + 8*a^2*d^7*e*f*g^7 + a^2*d^8*g^8)/(d^13*e^3)) + (a*e^5*f^5 + 5*a*d*e^4*f^4*g + 10
*a*d^2*e^3*f^3*g^2 + 10*a*d^3*e^2*f^2*g^3 + 5*a*d^4*e*f*g^4 + a*d^5*g^5)*sqrt(c + sqrt(a*x + b))) + 8*(3*a*f*x
+ sqrt(a*x + b)*c*f - (2*c^2 - 3*b)*f)*sqrt(c + sqrt(a*x + b)))/(a*d)

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giac [B]  time = 1.14, size = 2516, normalized size = 11.87

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x, algorithm="giac")

[Out]

4/15*(3*a^4*(c + sqrt(a*x + b))^(5/2)*d^4*f - 5*a^4*(c + sqrt(a*x + b))^(3/2)*c*d^4*f)/(a^5*d^5) + 1/2*((a^7*(
c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^5*g + a^7*(c + sqrt((b*
d + sqrt(-d*e)*a)/d))^2*d^4*f*e - a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)
) + sqrt(c + sqrt((b*d + sqrt(-d*e)*a)/d)))/((c + sqrt((b*d + sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d + s
qrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c
+ sqrt((b*d + sqrt(-d*e)*a)/d))) - (a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c + sqrt((b*d + sqr
t(-d*e)*a)/d))*c*d^5*g + a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^4*f*e - a^7*(c + sqrt((b*d + sqrt(-d*e)*a)
/d))*c*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) - sqrt(c + sqrt((b*d + sqrt(-d*e)*a)/d)))/((c + sqrt((b*d + sqrt(-
d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sqrt((b*d + sqrt(-
d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b*d + sqrt(-d*e)*a)/d))) + (a^7*(c - sqrt((b*d + sqrt(-d*e)
*a)/d))^2*d^5*g - a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^5*g + a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^
4*f*e - a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) + sqrt(c - sqrt((b*d + s
qrt(-d*e)*a)/d)))/((c - sqrt((b*d + sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^(5/2)*c*d
+ (3*c^2*d - b*d)*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c - sqrt((b*d + sqrt(-d*e)*
a)/d))) - (a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^5*g + a
^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^4*f*e - a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^4*f*e)*log(sqrt(c
+ sqrt(a*x + b)) - sqrt(c - sqrt((b*d + sqrt(-d*e)*a)/d)))/((c - sqrt((b*d + sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c
- sqrt((b*d + sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^(3/2) - (c^3*d
- b*c*d)*sqrt(c - sqrt((b*d + sqrt(-d*e)*a)/d))) + (a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c +
sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^5*g + a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^4*f*e - a^7*(c + sqrt((b*d
- sqrt(-d*e)*a)/d))*c*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) + sqrt(c + sqrt((b*d - sqrt(-d*e)*a)/d)))/((c + sq
rt((b*d - sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sq
rt((b*d - sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b*d - sqrt(-d*e)*a)/d))) - (a^7*(c + sqrt((
b*d - sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^5*g + a^7*(c + sqrt((b*d - sqrt(-
d*e)*a)/d))^2*d^4*f*e - a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) - sqrt(c
+ sqrt((b*d - sqrt(-d*e)*a)/d)))/((c + sqrt((b*d - sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d - sqrt(-d*e)*
a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b
*d - sqrt(-d*e)*a)/d))) + (a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^5*g - a^7*(c - sqrt((b*d - sqrt(-d*e)*a)
/d))*c*d^5*g + a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^4*f*e - a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^4
*f*e)*log(sqrt(c + sqrt(a*x + b)) + sqrt(c - sqrt((b*d - sqrt(-d*e)*a)/d)))/((c - sqrt((b*d - sqrt(-d*e)*a)/d)
)^(7/2)*d - 3*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c - sqrt((b*d - sqrt(-d*e)*a)/d)
)^(3/2) - (c^3*d - b*c*d)*sqrt(c - sqrt((b*d - sqrt(-d*e)*a)/d))) - (a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*
d^5*g - a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^5*g + a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^4*f*e - a^
7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) - sqrt(c - sqrt((b*d - sqrt(-d*e)*
a)/d)))/((c - sqrt((b*d - sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*
d - b*d)*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c - sqrt((b*d - sqrt(-d*e)*a)/d))))/(
a^6*d^5)

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maple [B]  time = 0.56, size = 198, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {-\frac {4 f \left (-\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}+\frac {c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}\right )}{d}+\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +a^{2} e +b^{2} d \right )}{\sum }\frac {\left (\left (-d g -e f \right ) \textit {\_R}^{4}+c \left (d g +e f \right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\)	\(198\)
default	\(\frac {-\frac {4 f \left (-\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}+\frac {c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}\right )}{d}+\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +a^{2} e +b^{2} d \right )}{\sum }\frac {\left (\left (-d g -e f \right ) \textit {\_R}^{4}+c \left (d g +e f \right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x,method=_RETURNVERBOSE)

[Out]

2/a*(-2*f/d*(-1/5*(c+(a*x+b)^(1/2))^(5/2)+1/3*c*(c+(a*x+b)^(1/2))^(3/2))+1/4*a^2/d^2*sum(((-d*g-e*f)*_R^4+c*(d
*g+e*f)*_R^2)/(_R^7-3*_R^5*c+3*_R^3*c^2-_R^3*b-_R*c^3+_R*b*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(d*_Z^8-
4*c*d*_Z^6+(6*c^2*d-2*b*d)*_Z^4+(-4*c^3*d+4*b*c*d)*_Z^2+c^4*d-2*b*c^2*d+a^2*e+b^2*d)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (f x^{2} - g\right )} \sqrt {c + \sqrt {a x + b}}}{d x^{2} + e}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2-g)*(c+(a*x+b)^(1/2))^(1/2)/(d*x^2+e),x, algorithm="maxima")

[Out]

integrate((f*x^2 - g)*sqrt(c + sqrt(a*x + b))/(d*x^2 + e), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\sqrt {c+\sqrt {b+a\,x}}\,\left (g-f\,x^2\right )}{d\,x^2+e} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c + (b + a*x)^(1/2))^(1/2)*(g - f*x^2))/(e + d*x^2),x)

[Out]

int(-((c + (b + a*x)^(1/2))^(1/2)*(g - f*x^2))/(e + d*x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + \sqrt {a x + b}} \left (f x^{2} - g\right )}{d x^{2} + e}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2-g)*(c+(a*x+b)**(1/2))**(1/2)/(d*x**2+e),x)

[Out]

Integral(sqrt(c + sqrt(a*x + b))*(f*x**2 - g)/(d*x**2 + e), x)

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