Optimal. Leaf size=406 \[ -\frac{2 \sqrt{b} c \sqrt{\frac{a x^2}{b}+1} \left (a c^2+b d^2\right ) \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{c+d x}}+\frac{2 \sqrt{b} \sqrt{\frac{a x^2}{b}+1} \sqrt{c+d x} \left (a c^2-3 b d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}}}+\frac{2 \left (a x^2+b\right ) (c+d x)^{3/2}}{5 a x \sqrt{a+\frac{b}{x^2}}}+\frac{2 c \left (a x^2+b\right ) \sqrt{c+d x}}{5 a x \sqrt{a+\frac{b}{x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.463681, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1450, 833, 844, 719, 424, 419} \[ -\frac{2 \sqrt{b} c \sqrt{\frac{a x^2}{b}+1} \left (a c^2+b d^2\right ) \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{c+d x}}+\frac{2 \sqrt{b} \sqrt{\frac{a x^2}{b}+1} \sqrt{c+d x} \left (a c^2-3 b d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}}}+\frac{2 \left (a x^2+b\right ) (c+d x)^{3/2}}{5 a x \sqrt{a+\frac{b}{x^2}}}+\frac{2 c \left (a x^2+b\right ) \sqrt{c+d x}}{5 a x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1450
Rule 833
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(c+d x)^{3/2}}{\sqrt{a+\frac{b}{x^2}}} \, dx &=\frac{\sqrt{b+a x^2} \int \frac{x (c+d x)^{3/2}}{\sqrt{b+a x^2}} \, dx}{\sqrt{a+\frac{b}{x^2}} x}\\ &=\frac{2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt{a+\frac{b}{x^2}} x}+\frac{\left (2 \sqrt{b+a x^2}\right ) \int \frac{\left (-\frac{3 b d}{2}+\frac{3 a c x}{2}\right ) \sqrt{c+d x}}{\sqrt{b+a x^2}} \, dx}{5 a \sqrt{a+\frac{b}{x^2}} x}\\ &=\frac{2 c \sqrt{c+d x} \left (b+a x^2\right )}{5 a \sqrt{a+\frac{b}{x^2}} x}+\frac{2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt{a+\frac{b}{x^2}} x}+\frac{\left (4 \sqrt{b+a x^2}\right ) \int \frac{-3 a b c d+\frac{3}{4} a \left (a c^2-3 b d^2\right ) x}{\sqrt{c+d x} \sqrt{b+a x^2}} \, dx}{15 a^2 \sqrt{a+\frac{b}{x^2}} x}\\ &=\frac{2 c \sqrt{c+d x} \left (b+a x^2\right )}{5 a \sqrt{a+\frac{b}{x^2}} x}+\frac{2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt{a+\frac{b}{x^2}} x}+\frac{\left (\left (a c^2-3 b d^2\right ) \sqrt{b+a x^2}\right ) \int \frac{\sqrt{c+d x}}{\sqrt{b+a x^2}} \, dx}{5 a d \sqrt{a+\frac{b}{x^2}} x}-\frac{\left (c \left (a c^2+b d^2\right ) \sqrt{b+a x^2}\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{b+a x^2}} \, dx}{5 a d \sqrt{a+\frac{b}{x^2}} x}\\ &=\frac{2 c \sqrt{c+d x} \left (b+a x^2\right )}{5 a \sqrt{a+\frac{b}{x^2}} x}+\frac{2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt{a+\frac{b}{x^2}} x}+\frac{\left (2 \sqrt{-a} \sqrt{b} \left (a c^2-3 b d^2\right ) \sqrt{c+d x} \sqrt{1+\frac{a x^2}{b}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{-a} \sqrt{b} d x^2}{a c-\sqrt{-a} \sqrt{b} d}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )}{5 a^2 d \sqrt{a+\frac{b}{x^2}} x \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}}}-\frac{\left (2 \sqrt{-a} \sqrt{b} c \left (a c^2+b d^2\right ) \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}} \sqrt{1+\frac{a x^2}{b}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-a} \sqrt{b} d x^2}{a c-\sqrt{-a} \sqrt{b} d}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )}{5 a^2 d \sqrt{a+\frac{b}{x^2}} x \sqrt{c+d x}}\\ &=\frac{2 c \sqrt{c+d x} \left (b+a x^2\right )}{5 a \sqrt{a+\frac{b}{x^2}} x}+\frac{2 (c+d x)^{3/2} \left (b+a x^2\right )}{5 a \sqrt{a+\frac{b}{x^2}} x}+\frac{2 \sqrt{b} \left (a c^2-3 b d^2\right ) \sqrt{c+d x} \sqrt{1+\frac{a x^2}{b}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d \sqrt{a+\frac{b}{x^2}} x \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}}}-\frac{2 \sqrt{b} c \left (a c^2+b d^2\right ) \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}} \sqrt{1+\frac{a x^2}{b}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d \sqrt{a+\frac{b}{x^2}} x \sqrt{c+d x}}\\ \end{align*}
Mathematica [C] time = 3.03285, size = 540, normalized size = 1.33 \[ \frac{\sqrt{c+d x} \left (\frac{2 \left (a x^2+b\right ) (2 c+d x)}{a}+\frac{2 \left (d^2 \sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}} \left (a^2 c^2 x^2+a b \left (c^2-3 d^2 x^2\right )-3 b^2 d^2\right )+\sqrt{a} (c+d x)^{3/2} \left (-i a^{3/2} c^3+a \sqrt{b} c^2 d+3 i \sqrt{a} b c d^2-3 b^{3/2} d^3\right ) \sqrt{\frac{d \left (x+\frac{i \sqrt{b}}{\sqrt{a}}\right )}{c+d x}} \sqrt{-\frac{-d x+\frac{i \sqrt{b} d}{\sqrt{a}}}{c+d x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}}}{\sqrt{c+d x}}\right )|\frac{\sqrt{a} c-i \sqrt{b} d}{\sqrt{a} c+i \sqrt{b} d}\right )-\sqrt{a} \sqrt{b} d (c+d x)^{3/2} \left (4 i \sqrt{a} \sqrt{b} c d+a c^2-3 b d^2\right ) \sqrt{\frac{d \left (x+\frac{i \sqrt{b}}{\sqrt{a}}\right )}{c+d x}} \sqrt{-\frac{-d x+\frac{i \sqrt{b} d}{\sqrt{a}}}{c+d x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}}}{\sqrt{c+d x}}\right )|\frac{\sqrt{a} c-i \sqrt{b} d}{\sqrt{a} c+i \sqrt{b} d}\right )\right )}{a^2 d^2 (c+d x) \sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}}}\right )}{5 x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.108, size = 1145, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{3}{2}}}{\sqrt{a + \frac{b}{x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d x^{3} + c x^{2}\right )} \sqrt{d x + c} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{3}{2}}}{\sqrt{a + \frac{b}{x^{2}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]